# Dynamics of multicritical circle maps

Edson de Faria  Instituto de Matemática e Estatística, Universidade de São
Paulo Rua do Matão 1010, 05508-090, São Paulo SP, Brasil<EMAIL_ADDRESS>and
Pablo Guarino  Instituto de Matemática e Estatística, Universidade Federal
Fluminense Rua Prof. Marcos Waldemar de Freitas Reis, S/N, 24.210-201, Bloco
H, Campus do Gragoatá, Niterói, Rio de Janeiro RJ, Brasil pablo_
<EMAIL_ADDRESS>
###### Abstract.

This paper presents a survey of recent and not so recent results concerning
the study of smooth homeomorphisms of the circle with a finite number of non-
flat critical points, an important topic in the area of One-dimensional
Dynamics. We focus on the analysis of the fine geometric structure of orbits
of such dynamical systems, as well as on certain ergodic-theoretic and
complex-analytic aspects of the subject. Finally, we review some conjectures
and open questions in this field.

###### Key words and phrases:

Rigidity, multicritical circle maps, renormalization

###### 2010 Mathematics Subject Classification:

Primary 37E10; Secondary 37E20, 37C40

The first author has been supported by “Projeto Temático Dinâmica em Baixas
Dimensões” FAPESP Grant 2016/25053-8, while the second author has been
supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -
Brasil (CAPES) grant 23038.009189/2013-05.

##  1\. Introduction

One of the major general goals of the area of Dynamical Systems is to solve
the smooth classification problem: given two smooth dynamical systems which
are topologically equivalent, when are they smoothly equivalent? In somewhat
vague terms, this problem is tantamount to understanding the fine-scale
geometric properties of such systems.

In such general setting, and particularly in higher dimensions, the above
classification problem seems rather daunting (perhaps even hopeless). Hence
one should first attempt to understand low-dimensional systems. At least at an
intuitive level, the problem should be much simpler for one-dimensional
systems; after all, in dimension one the linear order structure and “lack of
ambient space” should impose severe restrictions on the possible geometries of
such systems, thereby facilitating their smooth classification. However, even
here the problem turns out to be rather subtle. A basic distinction that must
be made in the one-dimensional context is between invertible dynamics – to
wit, homeomorphisms of the circle – and non-invertible dynamics, such as the
dynamics of unimodal or multimodal maps of the interval (or the circle).

In the present survey we deal with invertible dynamical systems on the circle.
In the case of smooth diffeomorphisms of the circle, deep results have been
obtained from the mid to late seventies onwards, starting with M. Herman’s
thesis [49] and culminating with the work of J.-C. Yoccoz [106], with
important contributions by Y. Katznelson and D. Ornstein [55], among others.
It was in part due to his deep work on diffeomorphisms that Yoccoz was awarded
his Fields Medal in 1994.

We will not say much about the smooth classification of diffeomorphisms of the
circle in the present paper (but see Section 2.3). We will rather focus on the
case of smooth homeomorphisms with critical points, a topic to which both
authors have made contributions that are deemed significant by the one-
dimensional dynamics community. In this context, the notions of
renormalization, rigidity and universality play a decisive role, and have been
widely studied in the last thirty years. From a strictly mathematical
viewpoint, these studies started with basic topological aspects [48, 105],
then evolved to geometric bounds [51, 95] and further to geometric rigidity
and renormalization aspects [4, 29, 31, 35, 36, 41, 45, 46, 47, 59, 62, 95,
99, 100, 101, 102].

Let us summarize some of those contributions in the following statements: on
the one hand, any two $C^{3}$ circle homeomorphisms with the same irrational
rotation number of _bounded type_ and with a single critical point (of the
same odd power-law type) are conjugate to each other by a $C^{1+\alpha}$
circle diffeomorphism, for some universal $\alpha>0$ [47]. On the other hand,
any two $C^{4}$ circle homeomorphisms with the same irrational rotation number
and with a single critical point (again, of the same odd type), are conjugate
to each other by a $C^{1}$ diffeomorphism [46]. This conjugacy is a
$C^{1+\alpha}$ diffeomorphism for a certain set of rotation numbers that has
full Lebesgue measure, but does not include all irrational rotation numbers
(see Section 11 for its definition and further remarks). These statements can
be regarded as the state of the art concerning renormalization and rigidity of
critical circle maps with _a single_ critical point, and they will be further
discussed in Section 8 of the present survey. In the case of homeomorphisms
with _several_ critical points, the theory remains wide open. However, some
interesting recent results [25, 26, 27, 28, 33, 104] will be described in this
survey (again, see Section 8).

We close this introduction by remarking that the rigidity and renormalization
theories to be (partially) described in this survey have been the subject of
at least six ICM talks, namely [3, 50, 58, 76, 77, 93], and it was in part due
to their seminal contributions to these theories that C. McMullen (1998) and
A. Avila (2014) were awarded their Fields Medals.

##  2\. Circle Diffeomorphisms

The _circle_ is defined as $S^{1}=\mathbb{R}/\mathbb{Z}$ , and it can be
identified with $\partial\mathbb{D}=\big{\\{}z\in\mathbb{C}:|z|=1\big{\\}}$
via the universal covering map $\pi:\mathbb{R}\to\partial\mathbb{D}$ given by
$\pi(x)=e^{2\pi ix}$. We denote by $\operatorname{Hom}_{+}(\mathbb{R})$ the
space of orientation preserving homeomorphisms of the real line, and by
$\operatorname{Hom}_{+}(S^{1})$ the space of orientation preserving circle
homeomorphisms (both endowed with the $C^{0}$ topology). For any given
$f\in\operatorname{Hom}_{+}(S^{1})$ there exist infinitely many
$F\in\operatorname{Hom}_{+}(\mathbb{R})$ satisfying $f\circ\pi=\pi\circ F$.
Such an $F$ is called a _lift_ of $f$, and any two lifts differ by an integer
translation. In fact, $F\in\operatorname{Hom}_{+}(\mathbb{R})$ is the lift of
a circle homeomorphism if, and only if, it commutes with unitary translation:
$F(x+1)=F(x)+1$ for all $x\in\mathbb{R}$.

###  2.1. The rotation number

The first major result in circle dynamics goes back to the end of the
nineteenth century, and is due to Poincaré.

######  Theorem 2.1 (Poincaré).

Let $F\in\operatorname{Hom}_{+}(\mathbb{R})$ be a lift of some
$f\in\operatorname{Hom}_{+}(S^{1})$. Then for all $x\in\mathbb{R}$ the
following limit exists and it is independent of $x$:

$\tau(F)=\lim_{n\to+\infty}\frac{F^{n}(x)-x}{n}\,.$

Since any two lifts $F_{1}$ and $F_{2}$ of $f$ differ by an integer
translation, it is easy to see that $\tau(F_{1})$ and $\tau(F_{2})$ differ by
the same integer. The _rotation number_ of $f$ is then defined as
$\rho(f)=\tau(F)\,\,\textrm{(mod 1)}$, for any given lift $F$ of $f$. The
function $\rho:\operatorname{Hom}_{+}(S^{1})\to[0,1)$ is continuous in the
$C^{0}$ topology. The rotation number is preserved under conjugation with
orientation preserving homeomorphisms, and therefore it is a _topological
invariant_ (in the sense that the equivalence classes of
$\operatorname{Hom}_{+}(S^{1})$ under topological conjugacies are contained in
the level sets of $\rho$).

######  Remark 2.2.

Let us briefly explain a measure-theoretical way to define the rotation number
of a circle homeomorphism. Let $f\in\operatorname{Hom}_{+}(S^{1})$ and let
$F\in\operatorname{Hom}_{+}(\mathbb{R})$ be one of its lifts. Let
$\varphi_{F}:S^{1}\to\mathbb{R}$ be the real function whose lift under $\pi$
is $F-\operatorname{Id}$, that is: $F(x)=x+\varphi_{F}(e^{2\pi ix})$ for all
$x\in\mathbb{R}$ . We claim that for any given $f$-invariant Borel probability
measure $\mu$ we have

$\rho(f)=\int_{S^{1}}\varphi_{F}\,\,d\mu\,\,\,(\operatorname{mod}1)\,.$ (2.1)

Indeed, the point here is that the real function $\widetilde{\varphi_{F}}$
given by

$\widetilde{\varphi_{F}}=\lim_{n\to+\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi_{F}\circ
f^{j}$

is well defined on the whole circle and it is _constant_ , equal to $\tau(F)$
(note that this implies at once the identity (2.1), since
$\int_{S^{1}}\varphi_{F}\,\,d\mu=\int_{S^{1}}\widetilde{\varphi_{F}}\,\,d\mu$
by the $f$-invariance of $\mu$). To see that $\widetilde{\varphi_{F}}$ is well
defined and constant, let $x\in S^{1}$ and let $x_{0}\in\mathbb{R}$ be such
that $\pi(x_{0})=x$. Then

$\displaystyle\sum_{j=0}^{n-1}\varphi_{F}\big{(}f^{j}(x)\big{)}$
$\displaystyle=\sum_{j=0}^{n-1}\varphi_{F}\big{(}\pi(F^{j}(x_{0}))\big{)}=\sum_{j=0}^{n-1}\big{(}F-\operatorname{Id}\big{)}\big{(}F^{j}(x_{0})\big{)}$
$\displaystyle=\sum_{j=0}^{n-1}\big{(}F^{j+1}(x_{0})-F^{j}(x_{0})\big{)}=F^{n}(x_{0})-x_{0}\,.$

By Theorem 2.1 we have $\widetilde{\varphi_{F}}(x)=\tau(F)$, as desired. The
equivalent definition of the rotation number of a circle homeomorphism given
by (2.1) will not be further mentioned in this survey, but it is important in
its own right.

###  2.2. Circle diffeomorphisms with irrational rotation number

The rotation number of $f\in\operatorname{Hom}_{+}(S^{1})$ is rational if, and
only if, $f$ admits at least one periodic orbit. In this case, if
$\rho(f)=p/q$, then any periodic orbit has period equal to $q$ and the
dynamics of $f$ is somehow trivial: any given initial condition is asymptotic
(both for the past and the future) to a periodic orbit.

If $f$ has irrational rotation number $\rho$, its orbit structure is much more
interesting. The fact that $f$ has no periodic orbits implies that its non-
wandering set is _minimal_ , being a Cantor set or the whole circle. Now pick
a non-wandering point $x$ in $S^{1}$ and consider its orbit under $f$, that
is: $\mathscr{O}_{f}(x)=\big{\\{}f^{n}(x)\big{\\}}_{n\in\mathbb{Z}}$. The
function $h_{x}:\mathscr{O}_{f}(x)\to S^{1}$ given by
$h_{x}\big{(}f^{n}(x)\big{)}=\pi(n\rho)=e^{2\pi in\rho}$ identifies $x$ with
the point $1$, and conjugates $f$ with the rigid rotation $R_{\rho}$ along the
orbit of $x$. A crucial fact here is that $f$ and $R_{\rho}$ are
_combinatorially equivalent_ , in the sense that for each $n\in\mathbb{N}$ the
first $n$ elements of the orbit of $x$ under $f$ are ordered in the same way
as the first $n$ elements of the orbit of the point $1$ under the rotation
$R_{\rho}$ (otherwise $f$ must have a periodic orbit, which is impossible in
the irrational case). The combinatorial equivalence between $f$ and $R_{\rho}$
implies that the map $h_{x}$, being monotone, extends continuously to the
closure of $\mathscr{O}_{f}(x)$. This extension is surjective because any
orbit of $R_{\rho}$ is dense in $S^{1}$, and then we can extend it (again) as
a constant function in any connected component of the complement of the
closure of $\mathscr{O}_{f}(x)$. This produces a _semi-conjugacy_ between $f$
and $R_{\rho}$, that is: a continuous surjective map $h_{x}:S^{1}\to S^{1}$
satisfying $R_{\rho}\circ h_{x}=h_{x}\circ f$.

$\begin{CD}S^{1}@>{f}>{}>S^{1}\\\ @V{h_{x}}V{}V@V{}V{h_{x}}V\\\
{S^{1}}@>{}>{R_{\rho}}>{S^{1}}\end{CD}$

From the previous construction it is clear that $h_{x}$ is the _unique_ semi-
conjugacy between $f$ and $R_{\rho}$ which identifies the point $x$ with the
point $1$ (this uniqueness will be useful during the proof of Proposition 2.7
below). Note also that for any given point $z\in S^{1}$ we have
$h_{z}=R_{\beta}\circ h_{x}$, with $\exp(2\pi i\beta)=1/h_{x}(z)$. Therefore,
any semi-conjugacy between $f$ and $R_{\rho}$ is unique up to post-composition
with rotations.

If $f$ is minimal in the whole circle we have
$\overline{\mathscr{O}_{f}(x)}=S^{1}$. In particular, $h_{x}$ is a
homeomorphism (and then $f$ is topologically conjugate to the rotation
$R_{\rho}$). Otherwise, there exists a point $y\in S^{1}$ such that
$I=h_{x}^{-1}\big{(}\\{y\\}\big{)}$ is a non-degenerate closed interval. We
call $I$ a _wandering interval_ since $f^{n}(I)\cap f^{m}(I)=\emptyset$ if
$n\neq m\in\mathbb{Z}$, and since it is not contained in the basin of a
periodic attractor. The second major result in circle dynamics was obtained by
Denjoy in the early thirties [18].

######  Theorem 2.3 (Denjoy).

Let $f$ be a $C^{1}$ circle diffeomorphism with irrational rotation number
$\rho$. If $\log Df$ has bounded variation, then $f$ has no wandering
intervals and therefore it is topologically conjugate to the rigid rotation of
angle $\rho$.

######  Remark 2.4.

In [18], Denjoy also proved that _some_ assumption on the first derivative of
$f$ is necessary: given any irrational number $\rho$ there exists a $C^{1}$
circle diffeomorphism with rotation number $\rho$ and wandering intervals [79,
Section I.2] (actually, examples of this type were given before by Bohl in
[7]). Moreover, counterexamples can be constructed in such a way that the
first derivative is Hölder continuous [49]. See also [52, 55, 56].

######  Remark 2.5.

As explained above, any circle homeomorphism with irrational rotation number
has minimal dynamics when restricted to its non-wandering set, which may be a
Cantor set or the whole circle. For $r\geq 0$, let

$\mathscr{C}_{r}=\big{\\{}K\subset S^{1}:\mbox{$K$ is the minimal Cantor set
for some $C^{r}$ circle diffeomorphism}\big{\\}}.$

Note the obvious fact that $\mathscr{C}_{r}\supseteq\mathscr{C}_{s}$ whenever
$r<s$. Every Cantor set of the unit circle is contained in $\mathscr{C}_{0}$,
since any two Cantor sets in $S^{1}$ are homeomorphic under some circle
homeomorphism. As already mentioned in Remark 2.4, $\mathscr{C}_{1}$ is non-
empty, while Theorem 2.3 implies that $\mathscr{C}_{r}$ is empty for any
$r\geq 2$. The problem of describing $\mathscr{C}_{1}$ seems to go back to
Michael Herman, and it was first addressed by Dusa McDuff in the late
seventies. For instance, she showed in [73] that the standard middle-thirds
Cantor set (more precisely, its isometric copy in
$S^{1}=\mathbb{R}/\mathbb{Z}$) is not in $\mathscr{C}_{1}$. To the best of our
knowledge, a full description of $\mathscr{C}_{1}$ remains open (see [8] and
references therein for partial results and recent developments).

The standard proof of Theorem 2.3 relies on the following simple observation:
any given $f\in\operatorname{Hom}_{+}(S^{1})$ such that some subsequence
$\\{f^{n_{k}}\\}_{k\in\mathbb{N}}$ is _equicontinuous_ , has no wandering
intervals. Indeed, if some interval $I\subset S^{1}$ wanders under $f$, then
the length of $f^{-n}(I)$ goes to zero as $n$ goes to infinity, since the
images of $I$ (past and futire) are pairwise disjoint; but this contradicts
the fact that some sequence of iterates of $f$ is equicontinuous. Therefore,
in order to prove Theorem 2.3 we need first to choose a suitable subsequence
of iterates of $f$. For this purpose, we recall that, being irrational, the
rotation number $\rho\in(0,1)$ has an infinite _continued fraction expansion_
, say

$\rho=[a_{0},a_{1},\cdots]=\cfrac{1}{a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{\ddots}}}\
.$

A classical reference for continued fraction expansion is the monograph [61].
Truncating the expansion at level $n-1$, we obtain a sequence of fractions
$p_{n}/q_{n}$, which are called the _convergents_ of the irrational $\rho$ :

$\frac{p_{n}}{q_{n}}\;=\;[a_{0},a_{1},\cdots,a_{n-1}]\;=\;\dfrac{1}{a_{0}+\dfrac{1}{a_{1}+\dfrac{1}{\ddots\dfrac{1}{a_{n-1}}}}}\,.$

The sequence of denominators $q_{n}$, which we call the _return times_ of
$\rho$, satisfies

$q_{0}=1,\hskip 11.38092ptq_{1}=a_{0},\hskip
11.38092ptq_{n+1}=a_{n}\,q_{n}+q_{n-1}\hskip 8.5359pt\text{for $n\geq 1$}.$

As explained above, Theorem 2.3 follows from the following result.

######  Proposition 2.6.

Let $f$ be a $C^{1}$ circle diffeomorphism with irrational rotation number.
Assume that $\log Df$ has bounded variation $V=\operatorname{var}(\log Df)>0$,
and let $K=e^{V}>1$. Then we have

$\frac{1}{K}\leq Df^{q_{n}}(x)\leq K\quad\mbox{for all $x\in S^{1}$ and all
$n\in\mathbb{N}$,}$

where $\\{q_{n}\\}_{n\in\mathbb{N}}$ is the sequence of return times given by
the rotation number of $f$.

Proposition 2.6 will be a straightforward consequence of two well-known
results (Proposition 2.9 and Lemma 2.11 below) which are interesting in its
own right. Before that, we establish the _unique ergodicity_ of circle
homeomorphisms without periodic orbits.

######  Proposition 2.7 (Unique ergodicity).

If $f\in\operatorname{Hom}_{+}(S^{1})$ has irrational rotation number $\rho$,
it preserves a unique Borel probability measure.

###### Proof of Proposition 2.7.

We claim first that $\nu\big{(}x,f(x)\big{)}=\rho$ for any $x\in S^{1}$ and
any $f$-invariant Borel probability measure $\nu$. Indeed, let
$F\in\operatorname{Hom}_{+}(\mathbb{R})$ be a lift of $f$ and, for any given
$x\in S^{1}$, let $\tilde{x}\in\mathbb{R}$ be one of its lifts. We will prove
that $\tau(F)=\tilde{\nu}\big{(}\tilde{x},F(\tilde{x})\big{)}$, where
$\tilde{\nu}$ is the lift of $\nu$ to the real line (an infinite but
$\sigma$-finite Borel measure). For each $n\in\mathbb{N}$, let us consider
$j_{n}=\lfloor F^{n}(\tilde{x})-\tilde{x}\rfloor$, so that
$F^{n}(\tilde{x})\in(\tilde{x}+j_{n},\tilde{x}+j_{n}+1)$. Then we have:

$F^{n}(\tilde{x})-\tilde{x}-1<j_{n}=\tilde{\nu}(\tilde{x},\tilde{x}+j_{n})<\tilde{\nu}\big{(}\tilde{x},F^{n}(\tilde{x})\big{)}<\tilde{\nu}(\tilde{x},\tilde{x}+j_{n}+1)=j_{n}+1<F^{n}(\tilde{x})-\tilde{x}+1\,,$

and therefore

$\lim_{n\to+\infty}\frac{F^{n}(\tilde{x})-\tilde{x}}{n}=\lim_{n\to+\infty}\frac{\tilde{\nu}\big{(}\tilde{x},F^{n}(\tilde{x})\big{)}}{n}\,.$

On the other hand, since $F$ is monotone and $\tilde{\nu}$ is $F$-invariant,
we see that

$\tilde{\nu}\big{(}\tilde{x},F^{n}(\tilde{x})\big{)}=\sum_{j=0}^{n-1}\tilde{\nu}\big{(}F^{j}(\tilde{x}),F^{j+1}(\tilde{x})\big{)}=n\,\tilde{\nu}\big{(}\tilde{x},F(\tilde{x})\big{)},$

and this tells us that

$\lim_{n\to+\infty}\frac{F^{n}(\tilde{x})-\tilde{x}}{n}=\lim_{n\to+\infty}\frac{\tilde{\nu}\big{(}\tilde{x},F^{n}(\tilde{x})\big{)}}{n}=\tilde{\nu}\big{(}\tilde{x},F(\tilde{x})\big{)}\,.$

Since
$\nu\big{(}x,f(x)\big{)}=\tilde{\nu}\big{(}\tilde{x},F(\tilde{x})\big{)}\,\,\textrm{(mod
1)}$, the claim is proved. Now fix some $x\in S^{1}$ and let $h_{x}:S^{1}\to
S^{1}$ be the semi-conjugacy between $f$ and $R_{\rho}$ constructed above
(before the statement of Theorem 2.3). We define a Borel probability measure
$\mu$ in $S^{1}$ as the _pull-back_ of the Lebesgue measure under $h_{x}$,
that is: $\mu(A)=\lambda\big{(}h_{x}(A)\big{)}$ for any Borel set $A\subset
S^{1}$, where $\lambda$ denotes the normalized Lebesgue measure in the unit
circle. Since $R_{\rho}$ preserves $\lambda$ and since $h_{x}$ is a semi-
conjugacy between $f$ and $R_{\rho}$, we deduce that $f$ preserves $\mu$. We
claim that $\mu$ is the _unique_ Borel probability measure which is invariant
under $f$. Indeed, let $\nu$ be any $f$-invariant probability measure, and
consider $\tilde{h}_{x}:S^{1}\to S^{1}$ defined by

$\tilde{h}_{x}(y)=\exp\left(2\pi i\int_{x}^{y}\\!d\nu\right),$

where, by convention, we measure the arc $(x,y)$ starting from $x$ in the
counter-clockwise sense. Since $\nu$ is $f$-invariant and $f$ has no periodic
orbits, $\nu$ has no points of positive measure, which implies that
$\tilde{h}_{x}$ is continuous, surjective and that $\tilde{h}_{x}(x)=1$.
Moreover, using that $\nu\big{(}x,f(x)\big{)}=\rho$ for any $x\in S^{1}$, we
deduce that $\tilde{h}_{x}\circ f=R_{\rho}\circ\tilde{h}_{x}$, and then it
follows that $\tilde{h}_{x}=h_{x}$. Finally, note that by definition $\nu$ is
the pull-back of the Lebesgue measure under $\tilde{h}_{x}$, that is:
$\lambda\big{(}[\tilde{h}_{x}(x),\tilde{h}_{x}(y)]\big{)}=\lambda\big{(}[1,\tilde{h}_{x}(y)]\big{)}=\nu\big{(}[x,y]\big{)}$,
which implies that it coincides with the measure $\mu$ defined above. ∎

######  Remark 2.8.

It is not difficult to prove that every circle homeomorphism has zero
topological entropy.

######  Proposition 2.9 (Denjoy-Koksma inequality).

Let $f\in\operatorname{Hom}_{+}(S^{1})$ with irrational rotation number
$\rho$, and let $\mu$ be its unique invariant Borel probability measure. Let
$\\{q_{n}\\}_{n\in\mathbb{N}}$ be the sequence of return times given by
$\rho$. For any $\psi:S^{1}\to\mathbb{R}$ (non necessarily continuous) with
finite total variation $\operatorname{var}(\psi)$ we have:

$\left|\sum_{j=0}^{q_{n}-1}\psi\big{(}f^{j}(x)\big{)}-q_{n}\int_{S^{1}}\psi\,d\mu\right|\leq\operatorname{var}(\psi)\quad\mbox{for
all $x\in S^{1}$ and all $n\in\mathbb{N}$.}$

###### Proof of Proposition 2.9.

Fix $x\in S^{1}$ and $n\in\mathbb{N}$. By combinatorics there exist $q_{n}$
pairwise disjoint open intervals $\\{I_{0},I_{1},...,I_{q_{n}-1}\\}$ in the
unit circle such that $R_{\rho}^{j}(1)=e^{2\pi ij\rho}\in\overline{I_{j}}$ and
$\lambda(I_{j})=1/q_{n}$ for all $j\in\\{0,1,...,q_{n}-1\\}$ (just take the
intervals determined by the $q_{n}$-roots of unity, and label them in order to
have $e^{2\pi ij\rho}\in\overline{I_{j}}$ for all
$j\in\\{0,1,...,q_{n}-1\\}$). Let $h=h_{x}$ be the semi-conjugacy between $f$
and $R_{\rho}$ that maps the point $x$ to the point $1$, and for each
$j\in\\{0,1,...,q_{n}-1\\}$ let $J_{j}=h^{-1}(I_{j})$. Note that $f^{j}(x)\in
J_{j}$ and $\mu(J_{j})=1/q_{n}$ for all $j\in\\{0,1,...,q_{n}-1\\}$. Moreover
$\big{\\{}\overline{J_{j}}\big{\\}}_{j=0}^{q_{n}-1}$ is a partition of the
unit circle (modulo boundary points, whose $\mu$-measure is zero since $\mu$
is $f$-invariant and $f$ has no periodic orbits). Therefore:

$\displaystyle\left|\sum_{j=0}^{q_{n}-1}\psi\big{(}f^{j}(x)\big{)}-q_{n}\int_{S^{1}}\psi\,d\mu\right|$
$\displaystyle=\left|\sum_{j=0}^{q_{n}-1}\left(\psi\big{(}f^{j}(x)\big{)}-q_{n}\int_{J_{j}}\psi\,d\mu\right)\right|$
$\displaystyle\leq\sum_{j=0}^{q_{n}-1}\left|\psi\big{(}f^{j}(x)\big{)}-q_{n}\int_{J_{j}}\psi\,d\mu\right|$
$\displaystyle=q_{n}\sum_{j=0}^{q_{n}-1}\left|\int_{J_{j}}\big{(}\psi\big{(}f^{j}(x)\big{)}-\psi\big{)}d\mu\right|$
$\displaystyle\leq
q_{n}\sum_{j=0}^{q_{n}-1}\int_{J_{j}}\big{|}\psi\big{(}f^{j}(x)\big{)}-\psi\big{|}d\mu$
$\displaystyle\leq\sum_{j=0}^{q_{n}-1}\sup_{y\in
J_{j}}\big{|}\psi\big{(}f^{j}(x)\big{)}-\psi(y)\big{|}\leq\operatorname{var}(\psi)\,.$

∎

######  Remark 2.10.

If $f$ is a sufficiently smooth diffeomorphism and the observable $\psi$ is
also smooth, then the Denjoy-Koksma inequality can be considerably improved:
see [5, Section 1.2] and references therein. See also the recent paper [63],
where precise estimates on the (uniform) rate of convergence of the Birkhoff
averages for Hölder observables under Diophantine rotations are given.

######  Lemma 2.11 (Zero Lyapunov exponent).

Let $f\in\operatorname{Diff}_{+}^{1}(S^{1})$ with irrational rotation number,
and let $\mu$ be its unique invariant Borel probability measure. Then:

$\int_{S^{1}}\log Df\,d\mu=0\,.$

###### Proof of Lemma 2.11.

Since $f\in\operatorname{Diff}_{+}^{1}(S^{1})$, the function
$\psi:S^{1}\to\mathbb{R}$ defined by $\psi=\log Df$ is a continuous function
and therefore, by the unique ergodicity of $f$ (Proposition 2.7), the sequence
of functions

$\frac{1}{n}\sum_{j=0}^{n-1}\psi\circ f^{j}$

converges _uniformly_ to the constant $\int_{S^{1}}\log Df\,d\mu$. By the
chain rule:

$\sum_{j=0}^{n-1}\psi\circ f^{j}=\log(Df^{n})\,,$ (2.2)

and therefore the sequence of continuous functions $\log(Df^{n})/n$ converges
to the constant $\int_{S^{1}}\log Df\,d\mu$ uniformly in $S^{1}$. Since
$f^{n}$ is a diffeomorphism for all $n\in\mathbb{N}$, this constant must be
zero. ∎

As we will see in Section 6, Lemma 2.11 still holds true for $C^{3}$ circle
homeomorphisms with critical points (see Theorem 6.4). With Proposition 2.9
and Lemma 2.11 at hand, we can finish Section 2.2. Indeed, to prove
Proposition 2.6 apply the Denjoy-Koksma inequality (Proposition 2.9) with
$\psi=\log Df$, and then use (2.2) combined with Lemma 2.11. As already
explained, Proposition 2.6 immediately implies Denjoy’s Theorem 2.3. In
Section 4 of the present survey we provide another proof of Theorem 2.3, this
time by means of _cross-ratio_ distortion arguments.

###  2.3. Some remarks on smooth rigidity for circle diffeomorphisms

We finish Section 2 by mentioning some results concerning _smooth rigidity_ ,
obtained during the second half of the twentieth century by Arnold, Herman and
Yoccoz. Let $f$ be a $C^{2}$ circle diffeomorphism with irrational rotation
number $\rho$. By Denjoy’s Theorem 2.3, there exists a circle homeomorphism
$h$ which is a topological conjugacy between $f$ and the rigid rotation of
angle $\rho$, that we denote by $R_{\rho}$. The unique invariant Borel
probability measure of $R_{\rho}$ (recall Proposition 2.7) is of course the
normalized Lebesgue measure. Therefore, the unique $f$-invariant measure, say
$\mu$, is given by the _pull-back_ under $h$ of the Lebesgue measure, that is:
$\mu(A)=\lambda\big{(}h(A)\big{)}$ for any Borel set $A$, where $\lambda$
denotes the Lebesgue measure in the unit circle (recall that the conjugacy $h$
is unique up to post-composition with rotations, so the measure $\mu$ is well-
defined).

$\begin{CD}(S^{1},\mu)@>{f}>{}>(S^{1},\mu)\\\ @V{h}V{}V@V{}V{h}V\\\
{(S^{1},\lambda)}@>{}>{R_{\rho}}>{(S^{1},\lambda)}\end{CD}$

In particular, $\mu$ has no atoms and gives positive measure to any open set.
A classical motivation to study the measure $\mu$ comes from Birkhoff’s
Ergodic Theorem: given any point $x\in S^{1}$ and any interval $I\subset
S^{1}$ we have that:

$\lim_{n\to+\infty}\,\frac{1}{n}\,\,\\#\big{\\{}j:0\leq
j<n\quad\mbox{and}\quad f^{j}(x)\in I\big{\\}}=\mu(I)\,.$ (2.3)

Note the following dichotomy: either $\mu$ is _absolutely continuous_ with
respect to Lebesgue, or it is _singular_ (otherwise we would have a
decomposition $\mu=\nu_{1}+\nu_{2}$, where $\nu_{1}$ is absolutely continuous,
$\nu_{2}$ is singular and both are non-zero. Being smooth, $f$ preserves the
class of zero Lebesgue measure sets, and therefore both $\nu_{1}$ and
$\nu_{2}$ are $f$-invariant, which contradicts the unique ergodicity of $f$).

In [2], Arnold gave examples (see also [79, Section I.5]) of real-analytic
circle diffeomorphisms that are minimal, but where the invariant probability
measure $\mu$ is _not_ absolutely continuous with respect to Lebesgue. In
those examples the rotation number is _Liouville_ , and any conjugacy with the
corresponding rotation maps a set of zero Lebesgue measure onto a set of
positive Lebesgue measure. However, in the same work, he proved that any real-
analytic diffeomorphism with _Diophantine_ rotation number, which is a small
perturbation of a rigid rotation, is conjugate to the corresponding rotation
by a real-analytic diffeomorphism. Arnold also conjectured that no restriction
on how close the diffeomorphism is to a rotation should be necessary. This was
later proved by Herman for a large class of Diophantine numbers [49], and
extended by Yoccoz in [106] for all Diophantine numbers, with important
contributions by Y. Katznelson and D. Ornstein [55] (see also [91]). Roughly
speaking, any $C^{2+\varepsilon}$ diffeomorphism with Diophantine rotation
number is conjugate to the corresponding rotation by a $C^{1}$ diffeomorphism
(see [79, Section I.3] for precise statements; see also [60] and references
therein). In these cases, by the Radon-Nikodym theorem, the $\mu$-measure of
any given interval is comparable to its Euclidean length. In particular, for
smooth diffeomorphisms with Diophantine rotation number, the qualitative
notion of minimality can be strengthened to a quantitative one, for (2.3)
implies that the _asymptotic frequency_ with which any given point visits an
open interval is comparable to the Euclidean length of the given interval.

Finally, we remark that the Arnold-Herman-Yoccoz theory described in the
previous paragraph also yields the results that any two $C^{\infty}$ circle
diffeomorphisms with the same Diophantine rotation number are
$C^{\infty}$-conjugate to each other, and that any two real-analytic
diffeomorphisms with the same Diophantine rotation number are conjugate to
each other by a real-analytic diffeomorphism (again, see [79, Section I.3] and
references therein). We refer the reader to the survey [23] for much more on
Yoccoz’s seminal contributions to the theory of circle diffeomorphisms.

##  3\. Multicritical circle maps

Let us now define the maps which are the main object of study in the present
survey. We start with the notion of non-flat critical point.

######  Definition 3.1.

We say that a critical point $c$ of a $C^{r}$ one-dimensional map $f$ is _non-
flat_ of degree $d>1$ if there exists a neighborhood $W$ of the critical point
such that $f(x)=f(c)+\phi(x)\,\big{|}\phi(x)\big{|}^{d-1}$ for all $x\in W$,
where $\phi:W\rightarrow\phi(W)$ is a $C^{r}$ diffeomorphism such that
$\phi(c)=0$. The number $d$ is also called the criticality, the type or the
order of $c$.

Note that every critical point of a real-analytic map is non-flat, and its
criticality must be a positive integer.

######  Definition 3.2.

A _multicritical circle map_ is an orientation preserving $C^{3}$ circle
homeomorphism having $N\geq 1$ critical points, all of which are non-flat.

Being a homeomorphism, a multicritical circle map $f$ has a well defined
rotation number (Theorem 2.1). We will assume that $f$ has _irrational_
rotation number $\rho\in(0,1)$, in which case it follows from Proposition 2.7
that there exists a _unique_ $f$-invariant Borel probability measure $\mu$.

######  Definition 3.3.

We define the _signature_ of $f$ to be the $(2N+2)$-tuple

$(\rho\,;N;\,d_{0},d_{1},\ldots,d_{N-1};\,\delta_{0},\delta_{1},\ldots,\delta_{N-1}),$

where $d_{i}$ is the criticality of the critical point $c_{i}$ for $0\leq
i\leq N-1$, and $\delta_{i}=\mu[c_{i},c_{i+1})$ (with the convention that
$c_{N}=c_{0}$).

In this section we provide some interesting families of real-analytic circle
dynamics.

###  3.1. Blaschke products

Consider the two-parameter family
$f_{a,\omega}:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ of _Blaschke
products_ in the Riemann sphere $\widehat{\mathbb{C}}$ given by:

$f_{a,\omega}(z)=e^{2\pi
i\omega}\,z^{2}\left(\frac{z-a}{1-az}\right)\quad\mbox{for $a\geq 3$ and
$\omega\in[0,1)$.}$ (3.1)

Just as any Blaschke product, every map in this family commutes with the
geometric involution around the unit circle $\Phi(z)=1/\bar{z}$ (note that
$\Phi$ is the identity in the unit circle), and then it leaves invariant the
unit circle (Blaschke products _are_ the rational maps leaving invariant the
unit circle). Moreover, its restriction to $S^{1}$ is a real-analytic
homeomorphism (the fact that $f_{a,\omega}$ has topological degree one, when
restricted to the unit circle, follows from the Argument Principle since it
has two zeros and one pole in the unit disk). When $a>3$, each $f_{a,\omega}$
has four critical points in the Riemann sphere, which are all different and
non-degenerate (quadratic), given by $0$, $\infty$,

$w_{a}=\frac{a^{2}+3}{4a}+\frac{\sqrt{(a+3)(a+1)(a-1)(a-3)}}{4a}\,>1\quad\mbox{
and}$ (3.2)
$1/w_{a}=\frac{a^{2}+3}{4a}-\frac{\sqrt{(a+3)(a+1)(a-1)(a-3)}}{4a}\,\,\,\in(0,1)\,.$
(3.3)

In particular, the restriction of $f_{a,\omega}$ to the unit circle is a real-
analytic diffeomorphism for any $a>3$. When $a\to 3$, both critical points
$w_{a}>1$ and $1/w_{a}\in(0,1)$ collapse to the point $w=1$, as we can see
from (3.2) and (3.3). In other words: when $a\to 3$, the family $f_{a,\omega}$
converges to the _boundary_ of the space of circle diffeomorphisms: for any
$\omega\in[0,1)$, the restriction of $f_{3,\omega}$ to $S^{1}$ is a real-
analytic multicritical circle map with a single critical point at $1$, which
is of cubic type, and with critical value $e^{2\pi i\omega}$.

Now let $p,q\in\mathbb{C}$ with $|p|>1$, $|q|>1$, let $\omega\in[0,1)$ and
consider $g_{p,q,\omega}:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ given by

$g_{p,q,\omega}(z)=e^{2\pi
i\omega}z^{3}\left(\frac{z-p}{1-\overline{p}z}\right)\left(\frac{z-q}{1-\overline{q}z}\right).$
(3.4)

Just as before, every map in this family leaves invariant the unit circle. The
following fact was proved by Zakeri in [108, Section 7].

######  Theorem 3.4.

For any given $\rho\in(0,1)\\!\setminus\\!\mathbb{Q}$ and $\delta\in(0,1)$
there exists a unique $g_{p,q,\omega}$ of the form (3.4) such that
$g_{p,q,\omega}|_{S^{1}}$ is a bi-critical circle map with signature
$(\rho\,;2;3,3;\,\delta,1-\delta)$.

######  Remark 3.5.

It would be interesting to extend Zakeri’s construction in order to obtain
representative families of Blaschke products that restrict to multicritical
circle maps with $N\geq 3$ critical points. Such construction should be useful
to understand rigidity and renormalization problems for multicritical circle
maps with any given number of critical points (see sections 8 to 11).

###  3.2. The Arnold family

Consider the two-parameter family $F_{a,b}:\mathbb{C}\to\mathbb{C}$ of entire
maps in the complex plane given by:

$F_{a,b}(z)=z+a-\left(\frac{b}{2\pi}\right)\sin(2\pi z)\quad\mbox{for
$a\in[0,1)$ and $b\geq 0$.}$

Figure 1. The rotation number, as it varies in the one parameter family
$f_{a,1}:\;x\mapsto x+a-\frac{1}{2\pi}\sin{2\pi x}$, produces a devil
staircase.

Since each $F_{a,b}$ commutes with unitary horizontal translation, it is the
lift of a holomorphic map of the punctured plane
$f_{a,b}:\mathbb{C}\setminus\\{0\\}\to\mathbb{C}\setminus\\{0\\}$ under the
universal cover $z\mapsto e^{2\pi iz}$. Since $F_{a,b}$ preserves the real
axis, $f_{a,b}$ preserves the unit circle. This classical two-parameter family
of real-analytic circle maps was introduced by Arnold in [2], and it is known
as the _Arnold family_.

For $b=0$, the family $f_{a,b}:S^{1}\to S^{1}$ is just the family of rigid
rotations $z\mapsto e^{2\pi ia}z$. As it is easy to check, for $b\in(0,1)$ the
Arnold family is still contained in the space of real-analytic circle
diffeomorphisms. For $b=1$, however, the Arnold family belongs to the
_boundary_ of the space of circle diffeomorphisms: each $F_{a,1}$ projects to
an orientation preserving real-analytic circle homeomorphism $f_{a,1}$, which
presents a critical point (of cubic type) at the point $1$, the projection of
the integers. The rotation number of $f_{a,1}$ varies with the parameter $a$
in a continuous, monotone, non-decreasing way, and the resulting graph is an
example of what one calls a _devil staircase_ ; see Figure 1. Each interval
$\big{\\{}a\in[0,1):\rho(f_{a,1})=\theta\big{\\}}$ degenerates to a point
whenever $\theta$ is irrational and moreover, the set
$\big{\\{}a\in[0,1):\rho(f_{a,1})\in\mathbb{R}\setminus\mathbb{Q}\big{\\}}$
has zero Lebesgue measure [95]. For integers $0\leq p<q$, the set
$\big{\\{}a\in[0,1):\rho(f_{a,1})=p/q\big{\\}}$ is a non-degenerate closed
interval. Its interior is made up of critical circle maps with two periodic
orbits (both of period $q$), one attracting and one repelling, which collapse
to a single parabolic orbit in the boundary of this interval [24]. See Item
(9) in Section 11 for more on this.

Finally, we remark that for $b>1$ the maps $f_{a,b}:S^{1}\to S^{1}$ are no
longer invertible (they present two quadratic critical points). The dynamics
of these maps is much richer than the case of homeomorphisms: the rotation
number becomes a rotation _interval_ , and typical dynamics here have positive
topological entropy, infinitely many periodic orbits (coexisting with dense
orbits) and, under certain conditions on the combinatorics, they preserve an
absolutely continuous probability measure (see [9, 11, 14, 81] and references
therein).

The examples presented in both 3.1 and 3.2 show how multicritical circle maps
arise as bifurcations from circle diffeomorphisms to endomorphisms, and in
particular, from zero to positive topological entropy (compare with infinitely
renormalizable unimodal maps [79, Chapter VI]). This is one of the main
reasons why multicritical circle maps attracted the attention of physicists
and mathematicians interested in the _boundary of chaos_ [19, 39, 53, 64, 65,
68, 69, 82, 85, 86, 87, 88].

##  4\. No wandering intervals

Being a homeomorphism, a multicritical circle map $f$ has a well defined
rotation number. Just as before, we will focus on the case when $f$ has no
periodic orbits. In the early eighties, J.-C. Yoccoz [105] proved that $f$ has
no wandering interval. More precisely, we have the following fundamental
result.

######  Theorem 4.1 (Yoccoz).

Let $f$ be a multicritical circle map with irrational rotation number $\rho$.
Then $f$ is topologically conjugate to the rigid rotation $R_{\rho}$, i.e.,
there exists a homeomorphism $h:S^{1}\rightarrow S^{1}$ such that $h\circ
f=R_{\rho}\circ h.$

It is not possible to remove the non-flatness condition on the critical points
(recall definitions 3.1 and 3.2): in [48], Hall was able to construct
$C^{\infty}$ homeomorphisms of the circle with no periodic points and no dense
orbits (see also [83]).

In the presence of critical points, we should not expect distortion estimates
on the first derivative as the ones developed for diffeomorphisms in Section
2.2 (we certainly can _not_ apply Proposition 2.6 to a multicritical circle
map $f$ since $\log{Df}$ is unbounded, see Figure 2). In this section we
introduce a new notion of distortion, more suitable for one-dimensional
dynamics with non-flat critical points, and we briefly explain the main ideas
in the proof of Yoccoz’s Theorem 4.1 (further applications of this distortion
tool to more general no wandering interval problems, mainly developed in [78,
71], can be found in [79, Chapter IV]).

Figure 2.  The cocycle $\log{Df}$ is unbounded for a multicritical circle map
$f$.

Given two intervals $M\subset T\subset S^{1}$ with $M$ compactly contained in
$T$ (written $M\Subset T$) let us denote by $L$ and $R$ the two connected
components of $T\setminus M$. We define the _cross-ratio_ of the pair $M,T$ as
follows:

$[M,T]=\frac{|L|\,|R|}{|L\cup M|\,|M\cup R|}\in(0,1),$

where $|I|$ denotes the Euclidean length of an interval $I$. Write $M=(b,c)$
and $T=(a,d)$, and let $\phi$ be the Möbius transformation determined by
$\phi(a)=0$, $\phi(c)=1$ and $\phi(d)=\infty$. Then

$[M,T]=\phi(b)=\left(\frac{d-c}{c-a}\right)\left(\frac{b-a}{d-b}\right).$

The cross-ratio distortion of any element $f\in\operatorname{Hom}(S^{1})$ on
the pair of intervals $(M,T)$ is defined to be the ratio

$\operatorname{CrD}(f;M,T)=\frac{\big{[}f(M),f(T)\big{]}}{[M,T]}\,.$

By definition, the cross-ratio is preserved by Möbius transformations.

The main tool available to control cross-ratio distortion around non-flat
critical points is the Schwarzian derivative, which is the differential
operator defined for all $x$ regular point of $f$ by:

$Sf(x)=\dfrac{D^{3}f(x)}{Df(x)}-\dfrac{3}{2}\left(\dfrac{D^{2}f(x)}{Df(x)}\right)^{2}.$

The kernel of the Schwarzian derivative is the group of Möbius
transformations111The fact that the Schwarzian derivative vanishes at Möbius
transformations is a straightforward computation. On the other hand, given an
increasing $C^{3}$ diffeomorphism $f$, consider $g=(Df)^{-1/2}$ and note that
$Sf=-2\,D^{2}g/g$. If $f$ has zero Schwarzian derivative then $g$ is affine,
which implies at once that $f$ is a Möbius transformation.. In particular, the
cross-ratio is preserved by maps with zero Schwarzian derivative. As it turns
out, it is weakly contracted by maps with negative Schwarzian derivative:

######  Lemma 4.2.

If $f$ is a $C^{3}$ diffeomorphism with $Sf<0$, then for any two intervals
$M\subset T$ contained in the domain of $f$ we have
$\operatorname{CrD}(f;M,T)<1$, that is, $\big{[}f(M),f(T)\big{]}<[M,T]$.

A straightforward computation shows that the Schwarzian derivative is always
negative around a non-flat critical point. More precisely, we have the
following fact.

######  Lemma 4.3.

Given $f$ with a non-flat critical point $c$, there exist a neighbourhood $U$
of $c$ and a constant $K=K(f)>0$ such that for all $x\in U\setminus\\{c\\}$ we
have

$Sf(x)<-\,\frac{K}{(x-c)^{2}}\,.$

Proofs of these basic facts can be found, for instance, in the appendix of
[26]. Finally, given a family of intervals $\mathscr{F}$ on $S^{1}$ and a
positive integer $m$, we say that $\mathscr{F}$ has multiplicity of
intersection at most $m$ if each $x\in S^{1}$ belongs to at most $m$ elements
of $\mathscr{F}$.

######  Theorem 4.4.

Given a multicritical critical circle map $f:S^{1}\to S^{1}$, there exists a
constant $C>1$, depending only on $f$, such that the following holds. If
$M_{i}\Subset T_{i}\subset S^{1}$, where $i$ runs through some finite set of
indices $\mathscr{I}$, are intervals on the circle such that the family
$\\{T_{i}:i\in\mathscr{I}\\}$ has multiplicity of intersection at most $m$,
then

$\prod_{i\in\mathscr{I}}\operatorname{CrD}(f;M_{i},T_{i})\leq C^{m}\ .$ (4.1)

Theorem 4.4, usually named _The Cross-Ratio Inequality_ , was first obtained
by Yoccoz in a slightly different form [105, Section 4]. The specific version
stated above can be found in [95, Section 2]. Roughly speaking, its proof goes
as follows: let $\mathscr{U}=\bigcup W_{i}$, where the $W_{i}$’s are as in
Definition 3.1, and let $\mathscr{V}$ be an open set with
$\mathscr{U}\cup\mathscr{V}=S^{1}$ whose closure does not contain any critical
point of $f$. We assume without loss of generality that the maximum length of
the $T_{i}$’s is smaller than the Lebesgue number of the covering
$\\{\mathscr{U},\mathscr{V}\\}$. Write the product on the left-hand side of
(4.1) as $P_{1}\cdot P_{2}$, where

$P_{1}\;=\;\prod_{T_{i}\subseteq\mathscr{V}}\operatorname{CrD}(f;M_{i},T_{i})\
\ \ ,\ \ \
P_{2}\;=\;\prod_{T_{i}\subseteq\mathscr{U}}\operatorname{CrD}(f;M_{i},T_{i})\
.$

Then on the one hand we claim that $P_{1}\leq e^{2mV}$, where
$V=\operatorname{var}(\log{Df}|_{\mathscr{V}})$. Indeed:

$\displaystyle\log P_{1}$
$\displaystyle=\sum_{T_{i}\subseteq\mathscr{V}}\log\operatorname{CrD}(f;M_{i},T_{i})$
(4.2) $\displaystyle=\sum_{T_{i}\subseteq\mathscr{V}}\log Df(w_{i})-\log
Df(x_{i})+\log Df(y_{i})-\log Df(z_{i})\leq 2mV\,,$

where the points $w_{i}$, $x_{i}$, $y_{i}$ and $z_{i}$ belong to $T_{i}$ and
are given by the Mean Value Theorem. On the other hand, the factors making up
$P_{2}$ are of two types: those such that $f|_{T_{i}}$ is a diffeomorphism
onto its image, and those such that $T_{i}$ contains some critical point of
$f$. By Lemma 4.3, all factors of the first type are diffeomorphisms with
negative Schwarzian and therefore, by Lemma 4.2, satisfy
$\operatorname{CrD}(f;M_{i},T_{i})<1$. Factors of the second type are easily
controlled by the non-flatness condition:
$\operatorname{CrD}(f;M_{i},T_{i})\leq 9d^{2}$, where $d>1$ is the order of
the critical point that belongs to $T_{i}$ (see for instance [26, Lemma
2.2(4)]). Since there are at most $mN$ such factors (where $N$ is the number
of critical points of $f$), the result follows. For more details, see [95,
Section 2].

Theorem 4.4 is a great tool to estimate cross-ratio distortion of large
iterates of multicritical circle maps. Indeed, note first that given two
intervals $M\Subset T$ and given $n\in\mathbb{N}$ we have the following _chain
rule_ :

$\operatorname{CrD}(f^{n};M,T)=\prod_{i=0}^{n-1}\operatorname{CrD}\big{(}f;f^{i}(M),f^{i}(T)\big{)}.$

With this at hand, Theorem 4.4 can be used to prove that _dynamically
symmetric_ intervals are _comparable_. More precisely, we have the following
fact, which is [25, Lemma 3.3].

######  Lemma 4.5.

Given a multicritical circle map $f$ there exists a constant $C>1$ depending
only on $f$ such that, for all $n\geq 0$ and all $x\in S^{1}$, we have

$C^{-1}\,\big{|}x-f^{-q_{n}}(x)\big{|}\leq\big{|}f^{q_{n}}(x)-x\big{|}\leq
C\,\big{|}x-f^{-q_{n}}(x)\big{|}\,.$

A proof of Lemma 4.5 can be found in [25, Section 3.1]. Now we claim that
Theorem 4.1 is a straightforward consequence of Lemma 4.5. Indeed, assume, by
contradiction, that there exists a wandering interval $J=(a,b)$. For each
$n\in\mathbb{N}$, consider the dynamically symmetric intervals
$L_{n}=\big{(}f^{-q_{n}}(a),a\big{)}$ and
$R_{n}=\big{(}a,f^{q_{n}}(a)\big{)}$. Since we are assuming that $J$ is a
wandering interval, $R_{n}$ contains $J$ for all $n\in\mathbb{N}$, and then we
deduce from Lemma 4.5 that the sequence
$\big{\\{}|L_{n}|\big{\\}}_{n\in\mathbb{N}}$ is bounded away from zero.
However, if we assume that $J$ is a _maximal_ wandering interval, the point
$a$ is recurrent (both for the past and the future) and then
$\big{\\{}|L_{n}|\big{\\}}_{n\in\mathbb{N}}\to 0$ as $n$ goes to infinity.
This contradiction shows that no such wandering interval $J$ exists, finishing
the proof of Theorem 4.1.

######  Remark 4.6.

The argument above gives a new proof of Denjoy’s Theorem 2.3, since the Cross-
Ratio Inequality (Theorem 4.4) certainly holds true whenever $f$ is a $C^{1}$
diffeomorphism and $\log Df$ has bounded variation (note that the Schwarzian
derivative is not needed in this case: estimate (4.2) holds on the whole
circle).

##  5\. Real bounds

Figure 3.  Dynamical partition $\mathscr{P}_{1}(x)$ of a circle homeomorphism
with rotation number $\rho(f)=\sqrt{2}-1=[2,2,2,\ldots]$.

Let $f$ be a $C^{3}$ multicritical circle map with irrational rotation number
$\rho\in(0,1)$, and let us fix some base point $x\in S^{1}$. For each non-
negative integer $n$, let $I_{n}$ be the interval with endpoints $x$ and
$f^{q_{n}}(x)$ containing $f^{q_{n+2}}(x)$, namely,
$I_{n}=\big{[}x,f^{q_{n}}(x)\big{]}$ and
$I_{n+1}=\big{[}f^{q_{n+1}}(x),x\big{]}$ (recall that the sequence of return
times $\\{q_{n}\\}$ has been constructed in Section 2). As it is not difficult
to prove, for each $n\geq 0$, the collection of intervals

$\mathscr{P}_{n}(x)\;=\;\left\\{f^{i}(I_{n}):\;0\leq i\leq
q_{n+1}-1\right\\}\cup\left\\{f^{j}(I_{n+1}):\;0\leq j\leq q_{n}-1\right\\}$

is a partition (modulo endpoints) of the unit circle (see for instance the
appendix in [26]), called the $n$-th dynamical partition associated to $x$.
The intervals of the form $f^{i}(I_{n})$ are called _long_ , whereas those of
the form $f^{j}(I_{n+1})$ are called _short_. The initial partition
$\mathscr{P}_{0}(x)$ is given by

$\mathscr{P}_{0}(x)=\left\\{\big{[}f^{i}(x),f^{i+1}(x)\big{]}:\,i\in\\{0,...,a_{0}-1\\}\right\\}\cup\big{\\{}\big{[}f^{a_{0}}(x),x\big{]}\big{\\}},$

where $a_{0}$ is the integer part of $1/\rho$ (see Section 2).

######  Example 5.1.

Figure 3 shows the dynamical partition $\mathscr{P}_{1}(x)$ associated to a
circle homeomorphism with rotation number $\rho(f)=\sqrt{2}-1=[2,2,2,\ldots]$,
for which $q_{1}=2$ and $q_{2}=5$. Thus, explicitly,
$\mathscr{P}_{1}(x)=\left\\{I_{1},f(I_{1}),f^{2}(I_{1}),f^{3}(I_{1}),f^{4}(I_{1})\right\\}\cup\left\\{I_{2},f(I_{2})\right\\}$.

The following fundamental result was obtained by Herman and Świątek in the
eighties [51, 95].

######  Theorem 5.2 (Real Bounds).

Given $N\geq 1$ in $\mathbb{N}$ and $d>1$ there exists a universal constant
$C=C(N,d)>1$ with the following property: for any given multicritical circle
map $f$ with irrational rotation number, and with at most $N$ critical points
whose criticalities are bounded by $d$, there exists
$n_{0}=n_{0}(f)\in\mathbb{N}$ such that for each critical point $c$ of $f$,
for all $n\geq n_{0}$, and for every pair $I,J$ of adjacent atoms of
$\mathscr{P}_{n}(c)$ we have:

$C^{-1}\,|I|\leq|J|\leq C\,|I|\,,$

where $|I|$ denotes the Euclidean length of an interval $I$.

Note that for a rigid rotation we have $|I_{n}|=a_{n+1}\,|I_{n+1}|+|I_{n+2}|$.
If $a_{n+1}$ is big, then $I_{n}$ is much larger than $I_{n+1}$. Thus, even
for rigid rotations, real bounds do not hold in general.

The proof of Theorem 5.2 follows by combining the Cross-Ratio Inequality
(Theorem 4.4) and Lemma 4.5 with a careful combinatorial analysis that the
reader can find in [25, Sections 3.2-3.3]. To obtain bounds independent of $f$
in the statement of Theorem 5.2, we must replace Theorem 4.4 by the following
result, obtained in [26, Theorem B].

######  Theorem 5.3.

Given $N\geq 1$ in $\mathbb{N}$ and $d>1$ there exists a constant $B=B(N,d)>1$
with the following property: given a multicritical circle map $f$, with at
most $N$ critical points whose criticalities are bounded by $d$, there exists
$n_{0}=n_{0}(f)$ such that for all $n\geq n_{0}$,
$\Delta\in\mathscr{P}_{n}(c)$ and $k\in\mathbb{N}$ such that $f^{i}(\Delta)$
is contained in an element of $\mathscr{P}_{n}(c)$ for all $1\leq i\leq k$, we
have that

$\operatorname{CrD}(f^{k};\Delta,\Delta^{*})\leq B\,,$

where $\Delta^{*}$ denotes the union of $\Delta$ with its left and right
neighbours in $\mathscr{P}_{n}(c)$.

See [25, 26] for more details. For more on the relation between the Schwarzian
derivative and cross-ratio distortion we refer the reader to [79, Chapter IV]
and to [37, Chapter 6], where a proof of Theorem 5.2 is also sketched.

We finish Section 5 by pointing out that it is essential for Theorem 5.2 to be
true to consider dynamical partitions _around critical points_. Indeed, let us
agree to say that $f$ has _bounded geometry_ at $x\in S^{1}$ if there exists
$K>1$ such that for all $n\in\mathbb{N}$ and for every pair $I,J$ of adjacent
atoms of $\mathscr{P}_{n}(x)$ we have

$K^{-1}\,|I|\leq|J|\leq K\,|I|\,.$

Following [33, Section 1.4], we consider the set

$\mathscr{A}=\mathscr{A}(f)=\\{x\in S^{1}:\,\mbox{$f$ has bounded geometry at
$x$}\\}\,.$

As explained in [33, Section 1.4], the set $\mathscr{A}$ is $f$-invariant.
Moreover, as stated in Theorem 5.2, all critical points of $f$ belong to
$\mathscr{A}$. Being $f$-invariant and non-empty, the set $\mathscr{A}$ is
dense in the unit circle. However, even in the case of maps with a single
critical point, $\mathscr{A}$ can be rather small. Indeed, the following is
[33, Theorem D].

######  Theorem 5.4.

There exists a full Lebesgue measure set $\bm{R}\subset(0,1)$ of irrational
numbers with the following property: let $f$ be a $C^{3}$ critical circle map
with a single (non-flat) critical point and rotation number $\rho\in\bm{R}$.
Then the set $\mathscr{A}(f)$ is meagre (in the sense of Baire) and it has
zero $\mu$-measure (where $\mu$ denotes the unique $f$-invariant probability
measure).

As it is not difficult to prove, the full Lebesgue measure set
$\bm{R}\subset(0,1)$ given by Theorem 5.4 contains no bounded type numbers
(see [33] for much more).

##  6\. Some ergodic aspects

In this section we examine multicritical circle maps from the point of view of
measurable dynamics.

###  6.1. No $\sigma$-finite measures

As it was proved in Proposition 2.7, a $C^{3}$ multicritical circle map $f$
with irrational rotation number is uniquely ergodic. Moreover, as we saw in
Section 4, a theorem due to J.-C. Yoccoz (Theorem 4.1) asserts that $f$ is
minimal and therefore topologically conjugate to the corresponding rigid
rotation. Yoccoz’s theorem implies that the support of the unique invariant
Borel probability measure of a multicritical circle map with irrational
rotation number is the whole circle. As it turns out, this measure is _purely
singular_ with respect to Lebesgue measure. This result was proved by Khanin
in the late eighties, by means of a certain thermodynamic formalism [57,
Theorem 4] (see also [43, Proposition 1]). In fact, in [34] the authors have
proved a more general result (namely, Theorem 6.3 below), which we proceed to
state. We need a definition and an auxiliary result. By a _nested sequence of
partitions_ we understand a sequence $\\{\mathscr{Q}_{n}\\}_{n\geq 0}$ of
finite interval partitions of $S^{1}$ (modulo endpoints) which is nested in
the sense that each atom of $\mathscr{Q}_{n}$ is a union of atoms of
$\mathscr{Q}_{n+1}$, for all $n\geq 0$, and is such that
$\mathrm{mesh}(\mathscr{Q}_{n})\to 0$ as $n\to\infty$222Here, as usual, the
_mesh_ of a partition is the maximum length of its atoms..

######  Definition 6.1.

A $C^{1}$ circle homeomorphism $f$ has the _Katznelson property_ if there
exist a nested sequence of partitions $\\{\mathscr{Q}_{n}\\}_{n\geq 0}$ and
constants $\alpha,\beta>1$ for which the following is true for all $n\geq 0$.
For every atom $\Delta\in\mathscr{Q}_{n}$, there exist two disjoint closed
subintervals $J_{1},J_{2}\subset\Delta$ and a positive integer $k$ such that
(i) $|J_{1}|\geq\alpha|J_{2}|$; (ii) we have $f^{k}(J_{1})=J_{2}$, and (iii)
$f^{k}:J_{1}\to J_{2}$ is diffeomorphism whose distortion is bounded by
$\beta$.

######  Theorem 6.2.

Let $f:S^{1}\to S^{1}$ be a $C^{1}$ minimal homeomorphism, and suppose that
$f$ has the Katznelson property. Then $f$ does not admit a $\sigma$-finite
invariant measure which is absolutely continuous with respect to Lebesgue
measure.

For a proof of this result, see [34, Th. 3.1]. In that paper, we refer to this
theorem as the _Katznelson criterion_ , because it is a slight generalization
of a result proved by Katznelson in [54].

######  Theorem 6.3.

Let $f:S^{1}\to S^{1}$ be a $C^{3}$ multicritical circle map with irrational
rotation number. Then $f$ does not admit a $\sigma$-finite invariant measure
which is absolutely continuous with respect to Lebesgue measure.

The strategy to prove this theorem is, not surprisingly, to show that $f$
possesses the Katznelson property, and then invoke Theorem 6.2. In veryfying
Katznelson’s property, one can restrict attention to the nested sequence of
partitions given by the dynamical partitions $\mathscr{P}_{n}(c)$ of some
critical point of $f$ (recall Section 5). The required geometric control comes
from the real bounds (Theorem 5.2), plus the negative Schwarzian property of
first return maps. When the rotation number of $f$ is of unbounded type, it is
also necessary to use Yoccoz’s inequality to control almost parabolic returns
(see Lemma 7.10 below). A detailed proof of Theorem 6.3 can be found in [34, §
4].

###  6.2. Lyapunov exponents

As mentioned in Section 2.2, the analogous statement to Lemma 2.11 also holds
for multicritical circle maps. More precisely, we have the following result.

######  Theorem 6.4 (Zero Lyapunov exponent).

Let $f:S^{1}\to S^{1}$ be a $C^{3}$ multicritical circle map with irrational
rotation number, and let $\mu$ be its unique invariant Borel probability
measure. Then $\log Df$ belongs to $L^{1}(\mu)$ and:

$\int_{S^{1}}\\!\log Df\,d\mu=0\,.$

The proof of Theorem 6.4 is considerably more difficult than the one of Lemma
2.11, since in this case $\log Df$ is not a continuous function: it is defined
only in the complement of the critical set of $f$, and it is unbounded (see
Figure 2). We refer the reader to [32], where Theorem 6.4 is obtained as a
consequence of the real bounds (Theorem 5.2).

###  6.3. Hausdorff dimension

As we have seen above, the unique invariant probability measure $\mu$ under a
multicritical circle map $f$ is purely singular with respect to Lebesgue
measure. The _Hausdorff dimension_ of $\mu$, denoted $\dim_{H}(\mu)$, is by
definition the smallest of the Hausdorff dimensions of all measurable sets
having full $\mu$-measure . More precisely,

$\dim_{H}(\mu)=\inf\,\\{\dim_{H}(E)\,:\;E\subset S^{1}\ \text{is measurable
and}\ \mu(E)=1\\}\ .$

A natural question to ask is: how does the Hausdorff dimension of $\mu$ vary
with $f$? Obviously, a priori it should only depend on the bi-Lipschitz
conjugacy class of $f$. In a recent paper [96], Trujillo establishes lower and
upper bounds for $\dim_{H}(\mu)$ that depend only on the Diophantine nature of
the rotation number of $f$. In order to state his result, we first recall that
an irrational number $\alpha$ is said to be _Diophantine of type $\omega\geq
0$_ if there exists a positive constant $C=C(\alpha)>0$ such that

$\left|\alpha-\frac{p}{q}\right|\;\geq\;\frac{C}{q^{2+\omega}}\ ,\ \ \text{for
all}\ p,q\in\mathbb{Z},\ q\neq 0\ .$

We denote by $\mathscr{D}_{\omega}$ the set of all Diophantine numbers of type
$\omega$. It is a well-known fact that $\mathscr{D}_{0}$ has zero Lebesgue
measure333The set $\mathscr{D}_{0}$ is precisely the set of numbers of
_bounded type_ , as previously defined., whereas for each $\omega>0$ the set
$\mathscr{D}_{\omega}$ has full Lebesgue measure.

######  Theorem 6.5.

If $f$ is a $C^{3}$ multicritical circle map with irrational rotation number
$\rho=\rho(f)$ and $\mu$ is its unique invariant probability measure, then the
following holds.

  1. (i)

If $\rho\in\mathscr{D}_{\omega}$ for some $\omega\geq 0$, then there exists
$\nu>0$ such that

$\dim_{H}(\mu)\;\geq\;\frac{1}{2\omega+\nu}\ .$

  2. (ii)

If $\rho\notin\mathscr{D}_{\omega}$ for some $\omega>0$, then

$\dim_{H}(\mu)\;\leq\;\frac{1}{\omega+1}\ .$

Note that the above theorem does not provide an upper bound for
$\dim_{H}(\mu)$ in the case when the rotation number $\rho(f)$ is of bounded
type (i.e., lies in $\mathscr{D}_{0}$). Nevertheless, it had already been
known since the work of Graczyk and Swiatek [43] that in the bounded type case
the Hausdorff dimension of $\mu$ lies strictly between $0$ and $1$.

##  7\. Quasi-symmetric rigidity

An important first step towards _smooth rigidity_ of multicritical circle maps
(to be examined in Section 8) is to answer the question: when are two
topologically conjugate multicritical circle maps _quasi-symmetrically_
conjugate?

###  7.1. Quasi-symmetry

Recall that an orientation-preserving homeomorphism of
$S^{1}=\mathbb{R}/\mathbb{Z}$, say $h:S^{1}\to S^{1}$, is said to be _quasi-
symmetric_ if there exists a constant $K\geq 1$ such that

$\frac{1}{K}\;\leq\;\frac{h(x+t)-h(x)}{h(x)-h(x-t)}\;\leq\;K\ ,\ \ \text{for
all}\ x\in S^{1}\,\text{and all}\ t>0\ .$ (7.1)

If $K$ is such that $h$ satisfies (7.1) for this $K$, then we say that $h$ is
_$K$ -quasi-symmetric_. The smallest $K$ with this property is called the
_quasi-symmetric distortion_ of $h$.

Quasi-symmetric homeomorphisms arise as boundary values of _quasi-conformal_
homeomorphisms of the unit disk (see [1, ch. 4]).

Let us describe a criterion for quasi-symmetry that is particularly useful in
the study of circle maps. In order to formulate it, we first need a
definition, reproduced almost verbatim from [25, Def. 5.1].

######  Definition 7.1.

A _fine grid_ is a nested sequence $\\{\mathscr{Q}_{n}\\}_{n\geq 0}$ of finite
interval partitions of the circle (modulo endpoints) having the following
properties.

  1. (a)

Each $\mathscr{Q}_{n+1}$ is a strict refinement of $\mathscr{Q}_{n}$.

  2. (b)

There exists an integer $a\geq 2$ such that each atom
$\Delta\in\mathscr{Q}_{n}$ is the disjoint union of at most $a$ atoms of
$\mathscr{Q}_{n+1}$.

  3. (c)

There exists $\sigma>1$ such that
$\sigma^{-1}|\Delta|\leq|\Delta^{\prime}|\leq\sigma|\Delta|$ for each pair of
adjacent atoms $\Delta,\Delta^{\prime}\in\mathscr{Q}_{n}$.

The numbers $a,\sigma$ are called _fine grid constants_.

This notion of fine grid was first introduced in [35, §4]. Its usefulness lies
in the fact that one can sometimes tell how regular a homeomorphism is by
looking at the effect it has on a suitable fine grid. As an example, here is
the criterion for quasi-symmetry that we promised above.

######  Proposition 7.2.

Let $\\{\mathscr{Q}_{n}\\}_{n\geq 0}$ be a fine grid in $S^{1}$ whose fine
grid constants are $a,\sigma$, and let $h:S^{1}\to S^{1}$ be an orientation-
preserving homeomorphism such that

$\left|\frac{|h(\Delta^{\prime})|}{|h(\Delta^{\prime\prime})|}-\frac{|\Delta^{\prime}|}{|\Delta^{\prime\prime}|}\right|\;\leq\;\lambda\
,$

for each pair of adjacent atoms
$\Delta^{\prime},\Delta^{\prime\prime}\in\mathscr{Q}_{n}$, for all $n\geq 0$,
where $\lambda$ is a positive constant. Then there exists
$K=K(a,\sigma,\lambda)>1$ such that $h$ is $K$-quasisymmetric.

A proof of this result can be found in [25, pp. 5611–5612].

Now, let us agree to say that an orientation-preserving homeomorphism
$h:S^{1}\to S^{1}$ is a _fine grid isomorphism_ if it maps fine grids to fine
grids. Then the criterion for quasi-symmetry given by Proposition 7.2 has the
following consequence.

######  Corollary 7.3.

Let $h:S^{1}\to S^{1}$ be an orientation-preserving homeomorphism. Then the
following are equivalent.

  1. (i)

$h$ is quasi-symmetric;

  2. (ii)

$h$ maps _some_ fine grid onto another fine grid;

  3. (iii)

$h$ is a fine grid isomorphism.

As we shall see in the sequence, the characterization of quasi-symmetry
provided by Corollary 7.3 is extremely helpful in the study of critical circle
maps.

###  7.2. Quasi-symmetric conjugacies

The facts presented above allow us to give a short proof of the following
theorem, originally due to Herman [51]. In this section, since we will
consider dynamical partitions associated to different maps, we shall use the
notation $\mathscr{P}_{n}(x,f)$, $I_{n}(x,f)$, instead of
$\mathscr{P}_{n}(x)$, $I_{n}(x)$, etc. to emphasize the dependency on $f$.

######  Theorem 7.4.

A multicritical circle map without periodic points is quasi-symmetrically
conjugate to a rigid rotation if and only if its rotation number is of bounded
type.

###### Proof.

Let us first assume that $f:S^{1}\to S^{1}$ is a multicritical circle map
whose rotation number $\rho=[a_{0},a_{1},a_{2},\ldots]$ is an irrational of
bounded type, say $a_{n}\leq A$ for all $n$. Let $c\in S^{1}$ be a critical
point of $f$. We claim that the dynamical partitions $\mathscr{P}_{2n}(c,f)$,
$n\geq 0$ constitute a fine grid. Indeed, every atom of
$\mathscr{P}_{2n}(c,f)$ is partitioned into at least $2$ and at most
$(a_{2n+1}+1)(a_{2n+2}+1)\leq(A+1)^{2}$ atoms of $\mathscr{P}_{2n+2}(c,f)$,
and these are all comparable by Theorem 5.2. Hence conditions (a), (b) and (c)
of Definition 7.1 are met, and the claim is proved. Let $h:S^{1}\to S^{1}$ be
a topological conjugacy between $f$ and the rigid rotation $R_{\rho}$ (say
$h\circ f=R_{\rho}\circ h$), which exists by Yoccoz’s theorem. Then one can
easily check that the dynamical partitions $\mathscr{P}_{2n}(h(c),R_{\rho})$,
$n\geq 0$, also constitute a fine grid (for $R_{\rho}$). But then $h$
satisfies property (ii) of Corollary 7.3, and therefore it must be quasi-
symmetric.

For the converse, suppose $h:S^{1}\to S^{1}$ is a homeomorphism satisfying
$h\circ f=R_{\rho}\circ h$, and suppose the rotation number of $f$ is not of
bounded type. Then there exists a subsequence $(n_{i})$ with
$a_{n_{i}+1}\to\infty$ as $i\to\infty$. Again we take $c$ to be a critical
point of $f$, and let $x=h(c)$. By the real bounds, the scaling ratios
$|I_{n_{i}+1}(c,f)|/|I_{n_{i}}(c,f)|$ for $f$ remain bounded, whereas for the
rigid rotation we have

$\frac{|h(I_{n_{i}+1}(c,f))|}{|h(I_{n_{i}}(c,f))|}\;=\;\frac{|I_{n_{i}+1}(x,R_{\rho})|}{|I_{n_{i}}(x,R_{\rho})|}\;>\;a_{n_{i}+1}\;\to\;\infty\
\ \text{as}\ i\to\infty\ ,$

and therefore $h$ cannot be quasi-symmetric. ∎

######  Remark 7.5.

In the above proof, the only reason we did not use the full collection of
dynamical partitions as our fine grid is that $\mathscr{P}_{n+1}(c,f)$ is
_not_ a strict refinement of $\mathscr{P}_{n}(c,f)$ (the _short_ atoms of
$\mathscr{P}_{n}(c,f)$ are not decomposed at all in the next step; they become
_long_ atoms of $\mathscr{P}_{n+1}(c,f)$). This is why we skipped every other
level.

######  Remark 7.6.

An interesting application of Theorem 7.4 to holomorphic dynamics goes as
follows. In the complex quadratic family $P_{\theta}:\,z\mapsto e^{2\pi
i\theta}z+z^{2}$, one knows that for each Diophantine $\theta$ the fixed point
at the origin is linearizable, so it belongs to the Fatou set of $P_{\theta}$.
The component of the Fatou set containing $0$ is a _Siegel disk_ ; call it
$\Omega_{\theta}$. In [20], Douady proved that if $\theta$ is a number of
bounded type, then $\partial\Omega_{\theta}$ is a _quasi-circle_ that contains
the critical point of $P_{\theta}$. The rough idea is to start with a Blaschke
product $B$ from the family introduced in section 3.1 (see eq. (3.1)) whose
restriction to $S^{1}$ is a critical circle map $f$ with rotation number
$\theta$. Then, using Theorem 7.4, one applies quasi-conformal surgery to $B$,
cutting out the unit disk and glueing it back in using as sewing map the
quasi-symmetric conjugacy $h$ between $f$ and the rigid rotation with the same
rotation number. Redefining the map in the interior of the unit disk to be
that same rotation, and applying the measurable Riemann mapping theorem, the
unit circle is mapped onto a quasi-circle, and the post-surgery map becomes
$P_{\theta}$, thereby proving Douady’s theorem. This result was later
generalized by Petersen and Zakeri in [84]. Their theorem allows the rotation
number $\theta$ to belong to a certain class of unbounded type numbers, and
the proof is accomplished through the use of _trans-quasiconformal surgery_.

More important for our purposes in the present survey is the following
immediate consequence of Theorem 7.4.

######  Corollary 7.7.

Any two multicritical circle maps $f$ and $g$ with the same irrational
rotation number _of bounded type_ are quasisymmetrically conjugate, and in
fact _every_ topological conjugacy between $f$ and $g$ is a quasi-symmetric
homeomorphism.

Note that in Corollary 7.7 the number of critical points of $f$ and the number
of critical points of $g$ need not be the same! But the bounded type
hypothesis on the rotation number is essential. In full generality, the above
statement is most definitely _false_ for unbounded combinatorics; see Section
7.3.

What can be said, then, for arbitrary irrational rotation numbers? If $f$ and
$g$ have the same number of critical points and there is a conjugacy between
$f$ and $g$ that maps each critical point of $f$ to a critical point of $g$,
the first part of the statement of Corollary 7.7 continues to hold. More
precisely, we have the following result, which is the main theorem in [25].

######  Theorem 7.8.

Let $f,g:S^{1}\to S^{1}$ be two $C^{3}$ multicritical circle maps with the
same irrational rotation number and the same number of (non-flat) critical
points, and let $h:S^{1}\to S^{1}$ be a homeomorphism conjugating f to g,
i.e., such that $h\circ f=g\circ h$. If $h$ maps each critical point of $f$ to
a corresponding critical point of $g$, then $h$ is quasisymmetric.

In the special case of maps with a single critical point, this theorem was
first proved by Yoccoz (unpublished, but see [35, Corollary 4.6]). Here, the
presence of _at least one_ critical point is absolutely crucial: the statement
is _false_ for diffeomorphisms. Indeed, there exist diffeomorphisms of the
circle, even analytic ones, that are topologically conjugate to an irrational
rotation and yet no conjugacy between them is quasi-symmetric; see [79, p.
75].

Let us give a brief description of the main steps in the proof of Theorem 7.8.

#### The basic idea

The basic idea is to build for each multicritical circle map $f$ an associated
fine grid in a _canonical_ way, and then apply Corollary 7.3. By canonical
here we mean that the partitions making up this fine grid must be defined in a
dynamically invariant way, i.e., in purely combinatorial terms. An obvious
first attempt is to use the dynamical partitions $\mathscr{P}_{n}(c,f)$, where
$c$ is a critical point of $f$, all of whose vertices lie in the forward orbit
of $c$. But even if we skip levels (to circumvent the fact that
$\mathscr{P}_{n+1}(c,f)$ is not a strict refinement of $\mathscr{P}_{n}(c,f)$)
and look at a subsequence of this sequence of partitions, we are in trouble
because, whenever a partial quotient $a_{n+1}$ is very large, there are atoms
of $\mathscr{P}_{n+1}(c,f)$ which are much smaller than the atoms of
$\mathscr{P}_{n}(c,f)$ in which they are contained. We need to group some of
these small atoms together, but to do that we first need to understand their
geometry.

#### Bridges and critical spots

Let $\Delta_{0}=f^{q_{n}}(I_{n+1}(c))\subset I_{n}(c)$, and consider its
images under $f^{q_{n+1}}$, namely $\Delta_{i}=f^{iq_{n+1}}(\Delta_{0})$, for
$0\leq i\leq a_{n+1}-1$. These intervals constitute an exact partition of
$I_{n}(c)\setminus I_{n+2}(c)$, and they are _fundamental domains_ of the
first return map $f^{q_{n+1}}:I_{n}(c)\to f^{q_{n+1}}(I_{n}(c))$. Some of
these fundamental domains may contain a number of critical points of the
return map (at most equal to the total number of critical points of $f$).
These fundamental domains are called _critical spots_. We also think of the
interval $I_{n+2}(c)\subset I_{n}(c)$ as a critical spot. The intervals
between two consecutive critical spots are called _bridges_. A fundamental
lemma proved in [25, Lemmas 4.2 and 4.3] as a consequence of the real bounds
states that the lengths of critical spots and (non-empty) bridges are always
comparable with the length of $I_{n}(c)$. The cross-ratio inequality implies
that the same comparability is true for the images of these intervals under
$f^{j}$ with respect to the corresponding long atoms $f^{j}(I_{n}(c))$, $0\leq
j\leq q_{n+1}-1$, (we also refer to these images as critical spots and
bridges). Hence for each $n$ we can define a partition
$\mathscr{P}_{n}^{*}(c,f)$ whose atoms are the short atoms of
$\mathscr{P}_{n}(c,f)$ together with all bridges and critical spots that make
up the long atoms of $\mathscr{P}_{n}(c,f)$. This new partition still has the
property that any two of its atoms that are adjacent are comparable. Moreover,
one easily sees that $\mathscr{P}_{n}^{*}(c,f)$ refines $\mathscr{P}_{n}(c,f)$
and is refined by $\mathscr{P}_{n+1}(c,f)$. This fact tells us that neither
the sequence $\\{\mathscr{P}_{n}^{*}(c,f)\\}_{n\geq 0}$ nor any of its
subsequences can be a fine grid. Hence we are not done yet. We need to
understand the geometry of bridges, and here is where things get more
interesting.

#### Almost parabolic maps

Each bridge is made up of a union of a number $\ell$ of fundamental domains
for $f^{q_{n+1}}$. If $\ell$ is, say, less than $1,000$, we say that the
bridge is _short_ ; otherwise we say that the bridge is _long_. Whenever
$a_{n+1}$ is large, some bridges will certainly be long, and the restriction
of $f^{q_{n+1}}$ to each one of these bridges will be very nearly a parabolic
map at the center of a saddle-node bifurcation (see Figure 4). Such _almost
parabolic maps_ can be described abstractly as follows (see [35, p. 354] or
[25, Def. 4.1]).

Figure 4.  A bridge $B$ between two critical spots, $\Delta^{\prime}$ and
$\Delta^{\prime\prime}$.

######  Definition 7.9.

An _almost parabolic map_ is a $C^{3}$ diffeomorphism

$\phi:\,J_{1}\cup J_{2}\cup\cdots\cup J_{\ell}\;\to\;J_{2}\cup
J_{3}\cup\cdots\cup J_{\ell+1}\ ,$

where $J_{1},J_{2},\ldots,J_{\ell+1}$ are consecutive intervals on the circle
(or on the line), with the following properties.

  1. (i)

One has $\phi(J_{k})=J_{k+1}$ for all $1\leq k\leq\ell$;

  2. (ii)

The Schwarzian derivative of $\phi$ is everywhere negative.

The positive integer $\ell$ is called the _length_ of $\phi$, and the positive
real number

$\sigma=\min\left\\{\frac{|J_{1}|}{|\cup_{k=1}^{\ell}J_{k}|}\,,\,\frac{|J_{\ell}|}{|\cup_{k=1}^{\ell}J_{k}|}\right\\}$

is called the _width_ of $\phi$.

The basic geometric control of an almost parabolic map is provided by the
following fundamental inequality due to Yoccoz, a complete proof of which can
be found in [35, App. B, pp. 386-389].

######  Lemma 7.10 (Yoccoz).

Let $\phi:\bigcup_{j=1}^{\ell}J_{k}\to\bigcup_{k=2}^{\ell+1}J_{k}$ be an
almost parabolic map with length $\ell$ and width $\sigma$. There exists a
constant $C_{\sigma}>1$ (depending on $\sigma$ but not on $\ell$) such that,
for all $k=1,2,\ldots,\ell$, we have

$\frac{C_{\sigma}^{-1}|I|}{[\min\\{k,\ell+1-k\\}]^{2}}\;\leq\;|J_{k}|\;\leq\;\frac{C_{\sigma}|I|}{[\min\\{k,\ell+1-k\\}]^{2}}\
,$ (7.2)

where $I=\bigcup_{k=1}^{\ell}J_{k}$ is the domain of $\phi$.

This lemma can be applied to $f^{q_{n+1}}|_{G}$ for every long bridge $G$,
provided such restrictions have negative Schwarzian derivative. It turns out
that the negative Schwarzian condition is always satisfied provided $n$ is
sufficiently large – this fact is proved in [25, Lemma 4.6], see also [26,
Lemma 4.1] and [34, Proposition A.7].

#### Balanced decomposition of bridges

Given an almost parabolic map $\phi:I\to\phi(I)$ as in Definition 7.9, we
perform a suitable decomposition of its domain $I=\bigcup_{k=1}^{\ell}J_{k}$
as follows. Let $d\in\mathbb{N}$ be largest with the property that
$2^{d+1}\leq\ell/2$, and consider the descending chain of (closed) intervals

$I=M_{0}\supset M_{1}\supset\cdots\supset M_{d+1}$

inductively defined as follows. Having defined $M_{i}$, the interval
$M_{i+1}\subset M_{i}$ is chosen so that, if $L_{i},R_{i}$ denote the left and
right connected components of $M_{i}\setminus M_{i+1}$, then both $L_{i}$ and
$R_{i}$ are the unions of exactly $2^{i}$ consecutive fundamental domains of
$\phi$. Then Yoccoz’s Lemma 7.10 implies that
$|L_{i}|\asymp|M_{i+1}|\asymp|R_{i}|$ for each $0\leq i\leq d$, with
comparability constants depending only on the width $\sigma$ of $\phi$. Note
also that

$I\;=\;\bigcup_{i=0}^{d}L_{i}\;\cup\;M_{d+1}\;\cup\;\bigcup_{i=0}^{d}R_{i}\ .$
(7.3)

The intervals $M_{i}$ ($0\leq i\leq d+1$) are called _central intervals_ ,
whereas the intervals $L_{i},R_{i}$ ($0\leq i\leq d$) are called _lateral
intervals_ of the almost parabolic map $\phi$. Note that this construction is
canonical in the sense that, whenever two almost parabolic maps are
topologically conjugate (by a homeomorphism mapping fundamental domains to
fundamental domains), the conjugacy will send the central (resp. lateral)
intervals of one map to the central (resp. lateral) intervals of the other.
The decomposition of the domain of $\phi$ that we have just decribed is called
the _balanced decomposition_ of (the domain of) our almost parabolic map.

Now, for each _long_ bridge $G\in\mathscr{P}_{n}^{*}(c,f)$, consider the above
balanced decomposition for $\phi=f^{q_{n+1}}|_{G}$. Here we are assuming that
$n$ is large enough that $S\phi<0$. We define, for each such $n$, a new
collection of intervals $\mathscr{P}_{n}^{**}(c,f)$ whose elements are: all
short atoms of $\mathscr{P}_{n}(c,f)$, all _short bridges_ and critical spots
of $\mathscr{P}_{n}^{*}(c,f)$, together with all the central intervals and all
lateral intervals of every _long bridge_ of $\mathscr{P}_{n}^{*}(c,f)$. Note
that this new collection $\mathscr{P}_{n}^{**}(c,f)$ is _not_ a partition of
the circle, since, for example, the central intervals of each long bridge are
_nested_.

#### A suitable fine grid

The desired fine grid $\\{\mathscr{Q}_{n}(c,f)\\}$ can now be constructed by a
recursive procedure, which we state below. Informally, each partition
$\mathscr{Q}_{n}(c,f)$ of the grid will be made up of those atoms in
$\mathscr{P}_{m}^{*}(c,f)$ that are not long bridges together with elements of
the balanced partitions of long bridges, which belong to
$\mathscr{P}_{m}^{**}(c,f)$ for various levels $m\leq n$. The precise
statement (taken from [25, Prop. 5.2]) is the following.

######  Proposition 7.11.

There exists a fine grid $\\{\mathscr{Q}_{n}\\}$ in $S^{1}$ with the following
properties.

  1. ($a$)

Every atom of $\mathscr{Q}_{n}$ is the union of at most $a=4N+3$ atoms of
$\mathscr{Q}_{n+1}$, where $N$ is the number of critical points of $f$.

  2. ($b$)

Every atom $\Delta\in\mathscr{Q}_{n}$ is a union of atoms of
$\mathscr{P}_{m}^{*}(c,f)$ for some $m\leq n$, and there are three
possibilities:

    1. ($b_{1}$)

$\Delta$ is a single atom of $\mathscr{P}_{m}^{*}(c,f)$;

    2. ($b_{2}$)

$\Delta$ is a central interval of $\mathscr{P}_{m}^{**}(c,f)$;

    3. ($b_{3}$)

$\Delta$ is the union of at least two atoms of $\mathscr{P}_{m+1}^{*}(c,f)$
contained in a single atom of $\mathscr{P}_{m}^{**}(c,f)$.

The proof of Proposition 7.11 is given in detail in [25, pp. 5613-5614] (see
also a similar construction in [27, Section 4]). The original precursor of
this result for _unicritical_ circle maps appears in [35, Prop. 4.5]. The
situation there is a bit simpler, because there are no critical spots to be
considered, hence no bridges in between.

#### Finishing the proof

The proof of Theorem 7.8 should now be clear. By hypothesis, the conjugacy $h$
sets a bijective correspondence between the critical points of $f$ and the
critical points of $g$. Let $c$ be a critical point of $f$, and let $h(c)$ be
the corresponding critical point of $g$. Then $h$ maps each partition
$\mathscr{P}_{n}(c,f)$ onto the corresponding partition
$\mathscr{P}_{n}(h(c),g)$, sending critical spots to critical spots and
bridges to bridges. Therefore if $\mathscr{G}_{f}=\\{\mathscr{Q}_{n}(c,f)\\}$
and $\mathscr{G}_{g}=\\{\mathscr{Q}_{n}(h(c),g)\\}$ are the fine grids for $f$
and $g$, respectively, given by Proposition 7.11, it follows that $h$ maps
$\mathscr{G}_{f}$ bijectively onto $\mathscr{G}_{g}$. But then, by Corollary
7.3, $h$ is quasisymmetric. This finishes the proof.

###  7.3. Orbit rigidity and flexibility

As we have just pointed out, the rigidity phenomenon embodied by Corollary 7.7
is quite specific to the bounded combinatorics scenario. Indeed, the following
result was recently obtained in [33, Theorem C].

######  Theorem 7.12.

There exists a set $\bm{R}\subset[0,1]$ of irrational numbers, which contains
a residual set (in the Baire sense) and has full Lebesgue measure, for which
the following holds. For each $\rho\in\bm{R}$ there exist two $C^{\infty}$
multicritical circle maps $f,g:S^{1}\to S^{1}$ with the following properties:

  1. (1)

Both maps have the same rotation number $\rho$;

  2. (2)

The map $f$ has exactly one critical point $c_{0}$, whereas the map $g$ has
exactly two critical points $c_{1}$ and $c_{2}$;

  3. (3)

The topological conjugacy between $f$ and $g$ that takes $c_{0}$ to $c_{1}$ is
not quasisymmetric.

Of course, by the above mentioned consequence of the real bounds, the set
$\bm{R}$ only contains irrationals numbers of _unbounded_ type. Therefore, the
geometric orbit structure is much richer in the unbounded case than in the
bounded one. Another way to see this phenomenon is the following: let us say
that two given orbits of a minimal circle homeomorphism $f$ are said to be
_geometrically equivalent_ if there exists a quasisymmetric circle
homeomorphism identifying both orbits and commuting with $f$. By the well-
known theorem due to Herman and Yoccoz discussed in Section 2.3, if $f$ is a
smooth diffeomorphism with Diophantine rotation number, then any two orbits
are geometrically equivalent. As already mentioned in this section (Corollary
7.7), the same holds if $f$ is a critical circle map with rotation number of
bounded type. By contrast, it was recently proved in [33] that if $f$ is a
critical circle map whose rotation number belongs to a certain full Lebesgue
measure set in $(0,1)$, then the number of equivalence classes is
_uncountable_.

##  8\. Smooth rigidity

The notion of _smooth rigidity_ first appeared in hyperbolic geometry in the
sixties, through the seminal work of Mostow, who showed that the fundamental
group (the _topology_) of a complete, finite-volume hyperbolic manifold of
dimension greater than two, completely determines its _geometry_. In dynamical
systems, smooth rigidity means that a finite number of dynamical invariants
determines the fine scale structure of orbits. More precisely, maps that are
topologically conjugate and share these invariants are in fact smoothly
conjugate. Numerical observations [39, 82, 88] suggested in the early eighties
that this was the case for $C^{3}$ critical circle maps with a single critical
point and with irrational rotation number of bounded type. This was posed as a
conjecture in several works by Lanford [64, 65], Rand [82, 85, 86, 87] and
Shenker [39, 88] among others. We proceed to state the most recent results in
this area, namely Theorem 8.1 and Theorem 8.2 below.

######  Theorem 8.1.

Let $f$ and $g$ be two $C^{4}$ circle homeomorphisms with the same irrational
rotation number and with a unique critical point of the same odd type. Let $h$
be the unique topological conjugacy between $f$ and $g$ that maps the critical
point of $f$ to the critical point of $g$. Then:

  1. (1)

$h$ is a $C^{1}$ diffeomorphism.

  2. (2)

$h$ is $C^{1+\alpha}$ at the critical point of $f$ for a universal $\alpha>0$.

  3. (3)

There exists a full Lebesgue measure set of rotation numbers (containing those
of bounded type) for which the conjugacy $h$ is a global $C^{1+\alpha}$
diffeomorphism.

By Theorem 4.1, the rotation number is the unique invariant of the $C^{0}$
conjugacy classes of critical circle maps with no periodic orbits. Theorem 8.1
is saying that, inside each topological class, the order of the critical point
is the unique invariant of the $C^{1}$ conjugacy classes. This is what we call
_rigidity_.

A delicate problem is to precisely determine “how smooth" is the conjugacy
$h$. By comparing with Section 2.3 we see that, on one hand, the presence of
the critical point gives us more rigidity than in the case of diffeomorphisms:
smooth conjugacy is obtained for all irrational rotation numbers, with no
Diophantine conditions. On the other hand, examples have been constructed [4,
35] showing that $h$ may not be globally $C^{1+\alpha}$ in general, even for
real-analytic dynamics. It might be possible, but probably quite difficult, to
obtain a sharp arithmetical condition on the rotation number that would allow
us to decide whether the conjugacy is “better than $C^{1}$" (see Section 11
for further remarks).

######  Theorem 8.2.

Any two $C^{3}$ critical circle maps with a single critical point, with the
same irrational rotation number of bounded type and with the same odd
criticality are conjugate to each other by a $C^{1+\alpha}$ circle
diffeomorphism, for some universal $\alpha>0$.

We remark that the statement of Theorem 8.2 was the precise statement of the
_rigidity conjecture_ mentioned above. Together, Theorems 8.1 and 8.2 can be
regarded as the state of the art concerning rigidity of critical circle maps
with a single critical point. Theorem 8.1 was proved in [46] while Theorem 8.2
was proved in [47]. Both papers build on earlier work of Herman, Świątek, the
first named author, de Melo, Yampolsky, Khanin and Teplinsky among others [49,
95, 29, 31, 35, 36, 99, 100, 101, 102, 62, 59]. While the proofs of both
results are far beyond the scope of this survey, some ideas will be provided
in sections 9 and 10 below.

What about dynamics with more critical points? Let $f$ be a $C^{3}$
multicritical circle map with irrational rotation number $\rho\in(0,1)$,
unique invariant Borel probability measure $\mu$ and $N\geq 1$ critical points
$c_{i}$, for $0\leq i\leq N-1$. As before, all critical points are assumed to
be non-flat: in $C^{3}$ local coordinates around $c_{i}$, the map $f$ can be
written as $t\mapsto t\,|t|^{d_{i}-1}$ for some $d_{i}>1$ (Definition 3.1).
Moreover, just as in Section 3 (recall Definition 3.3), we define the
_signature_ of $f$ to be the $(2N+2)$-tuple

$(\rho\,;N;\,d_{0},d_{1},\ldots,d_{N-1};\,\delta_{0},\delta_{1},\ldots,\delta_{N-1}),$

where $d_{i}$ is the criticality of the critical point $c_{i}$, and
$\delta_{i}=\mu[c_{i},c_{i+1})$ (with the convention that $c_{N}=c_{0}$).

Now consider two multicritical circle maps, say $f$ and $g$, with the same
irrational rotation number. By Theorem 4.1, they are topologically conjugate
to each other. By elementary reasons, if $f$ and $g$ have the same signature
there exists a circle homeomorphism $h$, which is a topological conjugacy
between $f$ and $g$, identifying each critical point of $f$ with one of $g$
having the same criticality. As explained in Section 7, such conjugacy $h$ is
a quasisymmetric homeomorphism (Theorem 7.8).

######  Question 8.3.

Is this conjugacy a smooth diffeomorphism?

Of course $h$ is the _unique_ conjugacy between $f$ and $g$ that can be smooth
(in fact, as mentioned in Section 7.3, for Lebesgue almost every rotation
number most conjugacies between $f$ and $g$ fail to be even quasisymmetric,
see [33] for precise statements). The following result follows by combining
the recent papers [27, 28, 104].

######  Theorem 8.4.

Let $f$ and $g$ be real-analytic bi-critical circle maps with the same
irrational rotation number, both critical points of cubic type and with the
same signature. If their common rotation number is of bounded type, then the
topological conjugacy $h$ is a $C^{1+\alpha}$ diffeomorphism.

To the best of our knowledge, Theorem 8.4 is the first rigidity statement
available for maps with more than one critical point. In other words, Question
8.3 remains widely open.

##  9\. Renormalization Theory

The main tool available to study rigidity problems in low dimensional dynamics
is renormalization theory. Renormalization of a dynamical system with a marked
point (usually a critical point) means a (suitably rescaled) first return map
to a neighbourhood of such point. Thus, renormalization can be thought as a
_supra_ dynamical system, acting on an infinite dimensional phase space (see
Section 9.1 for precise definitions in the context of multicritical circle
maps). In the context of one dimensional dynamics, the renormalization program
was initiated by Dennis Sullivan in the eighties [93, 94], and then carried
out by mathematicians such as Yoccoz, Douady, Hubbard, Shishikura, McMullen,
Lyubich, Martens, the first named author, de Melo, Yampolsky, van Strien and
Avila among others.

A fundamental principle in this theory states that exponential convergence of
renormalization orbits implies rigidity: topological conjugacies are actually
smooth (when restricted to the attractors of the original systems). We refer
the reader to [79, Section VI.9] for the seminal case of unimodal maps with
bounded combinatorics (more precisely, see Theorem 9.4). For critical circle
maps with a single critical point, this principle has been established by the
first named author and de Melo for Lebesgue almost every irrational rotation
number [35, First Main Theorem], and extended later by Khanin and Teplinsky to
cover all irrational numbers [59, Theorem 2]. Adapting these previous
approaches, this fundamental principle has been recently established for
multicritical circle maps [27, Theorem A].

So the main goal in order to obtain rigidity, as in Section 8 above, is to
establish geometric contraction of renormalization along critical circle maps
with the same irrational rotation number and the same criticality. The
dynamics of renormalization, however, is usually difficult to understand. To
begin with, its phase space is neither bounded nor locally compact. Therefore,
no recurrence is given a priori. This makes some basic dynamical questions,
such as existence of attractors and periodic orbits, quite difficult to
solve444As a first step, the real bounds (Theorem 5.2) can be used to
establish $C^{r}$ bounds for return maps, see for instance [35, Appendix A]. A
standard Arzelà-Ascoli argument gives then pre-compactness of renormalization
orbits. (see Section 11 for further comments). In particular, to prove
exponential contraction is a challenging problem. In the case of a single
critical point and real-analytic dynamics, exponential contraction was
obtained in [36] for rotation numbers of bounded type, and extended in [62] to
cover all irrational rotation numbers (see Theorem 9.3 below). Both papers
lean on complex dynamics techniques (that will be discussed in Section 10
below), and therefore an additional hypothesis is required: the criticality at
both critical points has to be an odd integer. These results have been
recently extended in at least two directions: in [41] exponential contraction
is obtained allowing non-integer criticalities which are close enough to an
odd integer, while in [46] and [47] exponential contraction is established for
finitely smooth critical circle maps (still with odd integer criticalities,
see Theorems 9.6 and 9.7 below). Finally, in the case of two critical points,
it was recently proved in [104] both existence of periodic orbits and
hyperbolicity (under renormalization) of those periodic orbits, for real-
analytic bi-critical circle maps (with both critical points of cubic type).
These results were later extended to bounded combinatorics in [28].

In the remainder of this section and the next one we sketch, with no details,
the proofs of both theorems 8.1 and 8.2.

###  9.1. Renormalization of commuting pairs

As it was already clear in the early eighties [39, 82] it is convenient to
construct a renormalization operator (see Definition 9.2) acting not on the
space of critical circle maps but on a suitable space of _critical commuting
pairs_ , whose precise definition is the following.

######  Definition 9.1.

A $C^{r}$ _critical commuting pair_ $\zeta=(\eta,\xi)$ consists of two $C^{r}$
orientation preserving homeomorphisms $\eta:I_{\eta}\to\eta(I_{\eta})$ and
$\xi:I_{\xi}\to\xi(I_{\xi})$ where:

  1. (1)

$I_{\eta}=[0,\xi(0)]$ and $I_{\xi}=[\eta(0),0]$ are compact intervals in the
real line;

  2. (2)

$\big{(}\eta\circ\xi\big{)}(0)=\big{(}\xi\circ\eta\big{)}(0)\neq 0$;

  3. (3)

$D\eta(x)>0$ for all $x\in I_{\eta}\\!\setminus\\!\\{0\\}$ and $D\xi(x)>0$ for
all $x\in I_{\xi}\\!\setminus\\!\\{0\\}$;

  4. (4)

The origin is a non-flat critical point for both $\eta$ and $\xi$, with the
same criticality.

  5. (5)

Both $\eta$ and $\xi$ extend to $C^{r}$ homeomorphisms, defined on interval
neighbourhoods of their respective domains, which commute around the origin.

Figure 5.  A critical commuting pair and its underlying interval exchange.

Any critical circle map $f$ with irrational rotation number $\rho$ induces a
sequence of critical commuting pairs in a natural way: let $F$ be the lift of
$f$ to the real line (for the canonical covering $t\mapsto e^{2\pi it}$)
satisfying $DF(0)=0$ and $0<F(0)<1$. For each $n\geq 1$ let $\widehat{I}_{n}$
be the closed interval in the real line, adjacent to the origin, that projects
under $t\mapsto e^{2\pi it}$ to $I_{n}$. Let $T:\mathbb{R}\to\mathbb{R}$ be
the translation $x\mapsto x+1$, and define $\eta:\widehat{I}_{n}\to\mathbb{R}$
and $\xi:\widehat{I}_{n+1}\to\mathbb{R}$ as:

$\eta=T^{-p_{n+1}}\circ F^{q_{n+1}}\quad\mbox{and}\quad\xi=T^{-p_{n}}\circ
F^{q_{n}}\,,$

where $\\{p_{n}/q_{n}\\}$ is the sequence of convergents defined in Section 2.
It is not difficult to check that
$(\eta|_{\widehat{I}_{n}},\xi|_{\widehat{I}_{n+1}})$ is a critical commuting
pair, usually denoted by $(f^{q_{n+1}}|_{I_{n}},f^{q_{n}}|_{I_{n+1}})$.

For a commuting pair $\zeta=(\eta,\xi)$ we denote by $\widetilde{\zeta}$ the
pair
$(\widetilde{\eta}|_{\widetilde{I_{\eta}}},\widetilde{\xi}|_{\widetilde{I_{\xi}}})$,
where tilde means _rescaling_ by the linear factor $1/|I_{\xi}|$. In other
words, $|\widetilde{I_{\xi}}|=1$ and the length of $\widetilde{I_{\eta}}$
equals the ratio between those of $I_{\eta}$ and $I_{\xi}$.

Let $\zeta=(\eta,\xi)$ be a critical commuting pair according to Definition
9.1, and recall that
$\big{(}\eta\circ\xi\big{)}(0)=\big{(}\xi\circ\eta\big{)}(0)\neq 0$. Let us
suppose that $\big{(}\xi\circ\eta\big{)}(0)\in I_{\eta}$ (see Figure 5) and
define the _height_ $\chi(\zeta)$ of $\zeta$ as $a\in\mathbb{N}$ if

$\eta^{a+1}\big{(}\xi(0)\big{)}<0\leq\eta^{a}\big{(}\xi(0)\big{)}\,,$

and $\chi(\zeta)=\infty$ if no such $a$ exists. The height of the pair
$(f^{q_{n+1}}|_{I_{n}},f^{q_{n}}|_{I_{n+1}})$ induced by a critical circle map
$f$ is exactly $a_{n+1}$, where $\rho(f)=[a_{0},a_{1},...]$. For
$\zeta=(\eta,\xi)$ with $\big{(}\xi\circ\eta\big{)}(0)\in I_{\eta}$ and
$\chi(\zeta)=a<\infty$, the pair

$\big{(}\eta|_{[0,\eta^{a}(\xi(0))]}\,,\,\eta^{a}\circ\xi|_{I_{\xi}}\big{)}$

is again a commuting pair, and if $\zeta$ is induced by a critical circle map,
_i.e._ ,

$\zeta=\big{(}f^{q_{n+1}}|_{I_{n}}\,,\,f^{q_{n}}|_{I_{n+1}}\big{)}\,,$

then we have

$\big{(}\eta|_{[0,\eta^{a}(\xi(0))]}\,,\,\eta^{a}\circ\xi|_{I_{\xi}}\big{)}=\big{(}f^{q_{n+1}}|_{I_{n+2}}\,,\,f^{q_{n+2}}|_{I_{n+1}}\big{)}\,.$

This motivates the following definition.

######  Definition 9.2.

Let $\zeta=(\eta,\xi)$ be a critical commuting pair with
$\big{(}\xi\circ\eta\big{)}(0)\in I_{\eta}$. We say that $\zeta$ is
_renormalizable_ if $\chi(\zeta)=a<\infty$. In this case we define the
_renormalization_ of $\zeta$ as

$\mathscr{R}(\zeta)=\left(\widetilde{\eta}|_{\widetilde{[0,\eta^{a}(\xi(0))]}}\,,\,\widetilde{\eta^{a}\circ\xi}|_{\widetilde{I_{\xi}}}\right).$

A critical commuting pair is a special case of a _generalized interval
exchange map_ of two intervals, and the renormalization operator defined above
is just the restriction of the _Zorich accelerated version_ of the _Rauzy-
Veech renormalization_ for interval exchange maps (see for instance [107]).
However we will keep in this survey the classical terminology for critical
commuting pairs.

If $\chi(\mathscr{R}^{j}(\zeta))<\infty$ for $j\in\\{0,1,...,n-1\\}$ we say
that $\zeta$ is _n-times renormalizable_ , and if
$\chi(\mathscr{R}^{j}(\zeta))<\infty$ for all $j\in\mathbb{N}$ we say that
$\zeta$ is _infinitely renormalizable_. The space of all infinitely
renormalizable commuting pairs is the natural phase-space for renormalization.
For such a pair, the irrational number whose continued fraction expansion
equals

$\big{[}\chi\big{(}\zeta\big{)},\chi\big{(}\mathscr{R}(\zeta)\big{)},...,\chi\big{(}\mathscr{R}^{n}(\zeta)\big{)},\chi\big{(}\mathscr{R}^{n+1}(\zeta)\big{)},...\big{]}$

is, by definition, the _rotation number_ of the critical commuting pair
$\zeta$ (note that if $\zeta$ is induced by a critical circle map with
irrational rotation number, then it is infinitely renormalizable and both
definitions of rotation number coincide).

For any real number $x$ denote by $\lfloor x\rfloor$ the _integer part_ of
$x$, that is, the greatest integer less than or equal to $x$. Also, denote by
$\\{x\\}$ the _fractional part_ of $x$, that is, $\\{x\\}=x-\lfloor
x\rfloor\in[0,1)$. Recall that the _Gauss map_ $G:[0,1]\to[0,1]$ is given by

$G(\rho)=\left\\{\frac{1}{\rho}\right\\}\mbox{ for $\rho\neq 0$\,, and
$G(0)=0$.}$

Both $\mathbb{Q}\cap[0,1]$ and $[0,1]\setminus\mathbb{Q}$ are $G$-invariant.
Under the action of $G$, all rational numbers in $[0,1]$ eventually land on
the fixed point at the origin, while the irrationals remain in the union
$\bigcup_{k\geq 1}\left(\frac{1}{k+1},\frac{1}{k}\right)$. Moreover, for any
$\rho\in(0,1)\setminus\mathbb{Q}$ and any $j\in\mathbb{N}$ we have that
$G^{j}(\rho)\in\left(\frac{1}{k+1},\frac{1}{k}\right)$ if, and only if,
$a_{j}=k$, where as before $a_{j}$ denotes the partial quotients of $\rho$.
Indeed, if $\rho=[a_{0},a_{1},a_{2},\cdots]$ belongs to
$\big{(}1/(k+1),1/k\big{)}$, then $1/\rho=a_{0}+[a_{1},a_{2},\cdots]$ and then
$a_{0}=\left\lfloor\frac{1}{\rho}\right\rfloor=k$ and
$G(\rho)=[a_{1},a_{2},\cdots]$. In particular, the Gauss map acts as a _left
shift_ on the continued fraction expansion of $\rho$, and therefore the action
of the renormalization operator on the rotation number is given by

$\rho\big{(}\mathscr{R}(\zeta)\big{)}=G\big{(}\rho(\zeta)\big{)}=\sigma\big{(}[a_{0},a_{1},\dots]\big{)}=[a_{1},\dots]\,.$
(9.1)

In particular, the renormalization operator $\mathscr{R}$ maps topological
classes of (infinitely renormalizable) critical commuting pairs into
topological classes.

###  9.2. Exponential convergence of renormalizations

For critical circle maps with a single critical point of some odd integer
criticality, the renormalization operator $\mathscr{R}$ exhibits an infinite-
dimensional _horseshoe-like_ attractor. More precisely, there exists a pre-
compact $\mathscr{R}$-invariant set $\Lambda$, consisting of real-analytic
critical commuting pairs with irrational rotation number, such that the action
of $\mathscr{R}|_{\Lambda}$ is topologically conjugate to the two-sided shift
$\sigma$ acting on $(\mathbb{N}\cup\infty)^{\mathbb{Z}}$ (the action being
taken over the continued fraction expansion of the rotation number, as in
(9.1) above). Moreover, any given real-analytic pair with irrational rotation
number converges to $\Lambda$ geometrically. More precisely, the following
holds.

######  Theorem 9.3.

There exists a universal constant $\lambda$ in $(0,1)$ with the following
property: given two real-analytic unicritical commuting pairs $\zeta_{1}$ and
$\zeta_{2}$ with the same irrational rotation number and the same odd type at
the critical point, there exists a constant $C>0$ such that

$d_{r}\big{(}\mathscr{R}^{n}(\zeta_{1}),\mathscr{R}^{n}(\zeta_{2})\big{)}\leq
C\lambda^{n}$

for all $n\in\mathbb{N}$ and for any $0\leq r<\infty$, where $d_{r}$ denotes
the $C^{r}$ metric.

This theorem was obtained by the first named author and de Melo [36] for
rotation numbers of bounded type, and extended by Khmelev and Yampolsky [62]
to cover all irrational rotation numbers. As already mentioned, its proof
relies on holomorphic methods, to be discussed in Section 10.

The link between real-analytic critical circle maps and finitely smooth ones
is given by the following result, first obtained in [47, Theorem D] for the
$C^{0}$ metric, and then extended in [46, Theorem 11.1] for the $C^{r-1}$
metric.

######  Theorem 9.4 (Shadowing).

There exists a $C^{\omega}$-compact set $\mathscr{K}$ of real-analytic
critical commuting pairs with the following property: for any $r\geq 3$ there
exists a constant $\lambda=\lambda(r)\in(0,1)$ such that given a $C^{r}$
critical circle map $f$ with irrational rotation number there exist $C>0$ and
a sequence $\\{f_{n}\\}_{n\in\mathbb{N}}$ contained in $\mathscr{K}$ such that

$d_{r-1}\big{(}\mathscr{R}^{n}(f),f_{n}\big{)}\leq C\lambda^{n}\quad\mbox{for
all $n\in\mathbb{N}$,}$

and such that the pair $f_{n}$ has the same rotation number as the pair
$\mathscr{R}^{n}(f)$ for all $n\in\mathbb{N}$. Here $d_{r-1}$ denotes the
$C^{r-1}$ distance in the space of $C^{r-1}$ critical commuting pairs.

An important tool used in [47] in order to construct the compact set
$\mathscr{K}$ is the following notion. Let $I$ be a compact interval in the
real line, and let $f:I\to\mathbb{R}$ be a $C^{r}$ map. Then there exists a
$C^{r}$ map $F:\Omega\to\mathbb{C}$, where $\Omega$ is a neighbourhood of $I$
in the complex plane, with $F|_{I}=f$,
$\frac{\partial}{\partial\bar{z}}F(z)=0$ when $\mathrm{Im}\,z=0$ and moreover

$\frac{\frac{\partial}{\partial\bar{z}}F(z)}{(\mathrm{Im}\,z)^{r-1}}\to 0$

uniformly as $\mathrm{Im}\,z$ goes to zero (see [42, Lemma 2.1, p. 623] for
one possible way to construct such extension $F$). In this case, we say that
$F$ is _asymptotically holomorphic_ of order $r$ in $I$. To prove Theorem 9.4,
one first extends the initial critical commuting pair given by $f$ to a pair
of asymptotically holomorphic maps defined in an open complex neighbourhood of
each original interval. To the best of our knowledge, asymptotically
holomorphic maps were first used in one-dimensional dynamics by Lyubich in the
early nineties [67], and later by Graczyk, Sands and Świątek in [42]. One of
its fundamental properties, an _almost Schwarz inclusion_ , controls the
action of asymptotically holomorphic maps on some complex domains symmetric to
the real line (behaving, at small scale, almost as isometries of a suitable
hyperbolic metric, see [42, Proposition 2, p. 629] and also Lemma 10.8 below).
When combined with Theorem 5.2 (the real bounds), such control provides
geometric bounds on the quasiconformal distortion of the renormalizations of
the previously mentioned extensions (one does not study the dynamics of these
extensions, just their geometric behaviour). With these bounds at hand, the
deformations from $\mathscr{R}^{n}(f)$ to $f_{n}$ (in order to prove Theorem
9.4) are based on the following corollary of the classical _Measurable Riemann
Mapping Theorem_ (also known as Ahlfors-Bers theorem, see [1, Section V.C]),
whose proof can be found in [47, Proposition 5.5].

######  Proposition 9.5.

For any bounded domain $U$ in the complex plane there exists a number
$C(U)>0$, with $C(U)\leq C(W)$ if $U\subseteq W$, such that the following
holds: let $\big{\\{}G_{n}:U\to G_{n}(U)\big{\\}}_{n\in\mathbb{N}}$ be a
sequence of quasiconformal homeomorphisms555Let $\Omega\subset\mathbb{C}$ be a
domain and let $K\geq 1$. An orientation-preserving homeomorphism $f:\Omega\to
f(\Omega)$ is _$K$ -quasiconformal_ if it is absolutely continuous on lines
and satisfies
$\left|\overline{\partial}f(z)\right|\leq\left(\frac{K-1}{K+1}\right)\big{|}\partial
f(z)\big{|}\quad\mbox{for Lebesgue a.e. $z\in\Omega$\,.}$ The _Beltrami
coefficient_ of such homeomorphism $f$ is the measurable function
$\mu_{f}:\Omega\to\mathbb{D}$ given by
$\mu_{f}(z)=\frac{\overline{\partial}f(z)}{\partial f(z)}\quad\mbox{for
Lebesgue a.e. $z\in\Omega$\,.}$ such that

  * •

The domains $G_{n}(U)$ are uniformly bounded: there exists $R>0$ such that
$G_{n}(U)\subset B(0,R)$ for all $n\in\mathbb{N}$.

  * •

$\mu_{n}\to 0$ in the unit ball of $L^{\infty}$, where $\mu_{n}$ is the
Beltrami coefficient of $G_{n}$ in $U$.

Then given any domain $V$ such that $\overline{V}\subset U$ there exist
$n_{0}\in\mathbb{N}$ and a sequence $\big{\\{}H_{n}:V\to
H_{n}(V)\big{\\}}_{n\geq n_{0}}$ of biholomorphisms such that:

$\|H_{n}-G_{n}\|_{C^{0}(V)}\leq C(U)\left(\frac{R}{d\big{(}\partial V,\partial
U\big{)}}\right)\|\mu_{n}\|_{\infty}\quad\mbox{for all $n\geq n_{0}$,}$

where $d\big{(}\partial V,\partial U\big{)}$ denote the Euclidean distance
between the boundaries of $U$ and $V$.

This Proposition is applicable to many other situations – see for example [12,
Section 5.5].

Those were some of the main ideas in order to prove Theorem 9.4. With theorems
9.3 and 9.4 at hand, the following two results, which are [47, Theorem C] and
[46, Theorem B] respectively, can be obtained.

######  Theorem 9.6.

There exists a universal constant $\lambda\in(0,1)$ such that given two
$C^{3}$ critical circle maps $f$ and $g$ with the same irrational rotation
number of bounded type and the same odd integer criticality, there exists
$C=C(f,g)>0$ such that for all $n\in\mathbb{N}$ we have

$d_{0}\big{(}\mathscr{R}^{n}(f),\mathscr{R}^{n}(g)\big{)}\leq C\lambda^{n}\,.$

######  Theorem 9.7.

There exists a universal constant $\lambda\in(0,1)$ such that given two
$C^{4}$ critical circle maps $f$ and $g$ with the same irrational rotation
number and the same odd integer criticality, there exists $C=C(f,g)>0$ such
that for all $n\in\mathbb{N}$ we have

$d_{2}\big{(}\mathscr{R}^{n}(f),\mathscr{R}^{n}(g)\big{)}\leq C\lambda^{n}\,.$

By the already mentioned [35, First Main Theorem] and [59, Theorem 2],
theorems 9.6 and 9.7 imply theorems 8.2 and 8.1 respectively. We remark that
to obtain theorems 9.6 and 9.7 from theorems 9.3 and 9.4 some hard analysis is
involved, concerning a Lipschitz estimate for the renormalization operator
(when restricted to suitable bounded pieces of topological conjugacy classes).
We refer the reader to [46, Lemma 4.1] for more details.

Following the terminology from hyperbolic dynamics, the rotation number of a
critical circle map can be thought as an _unstable_ coordinate for the
renormalization operator (recall from (9.1) that $\mathscr{R}$ acts as the
Gauss map on the rotation number), while the criticality of its single
critical point (being preserved under renormalization) is a _central_ one. By
Theorem 9.3, _stable_ manifolds for renormalization are obtained by fixing
these two coordinates (see [102, Section 2.3] for a nice description of this
hyperbolic picture).

For multicritical circle maps with $N\geq 2$ critical points, a richer picture
emerges: the criticalities $d_{i}$ are again central directions, whereas the
parameters $\delta_{i}$ (the distance between consecutive critical points view
under the unique invariant measure, see Definition 3.3) are _new unstable
directions_ (such expanding behaviour comes from the associated rescaling
process explained in Section 9.1). This suggests that, inside topological
classes of multicritical commuting pairs, one should be able to see many
different horseshoe-like attractors for renormalization, each one containing
infinitely many periodic orbits. This hyperbolic picture has not been
established yet, which is why Question 8.3 remains open in its general form.

##  10\. Holomorphic methods

In this section we will survey some of the complex-analytic ideas that play a
decisive role in the theory of (multi)critical circle maps. Since these ideas
are quite deep, the narrative to follow is by necessity very sketchy. For the
general theory of complex dynamics we refer the reader to the books [10, 37,
74, 80].

The use of holomorphic methods in the study of renormalization and rigidity of
one-dimensional dynamical systems was started by Sullivan in the mid-eighties
(see [94]). Since the theory for circle maps follows in parallel with the
corresponding theory for unimodal maps, and borrows substantially from it, we
need to talk a bit about the latter first.

###  10.1. Sullivan’s program

We have already mentioned the general ansatz relating renormalization
convergence and rigidity. If we are given two topologically conjugate one-
dimensional maps $f$ and $g$ which are infinitely renormalizable (say with
some restrictions on their combinatorics), and if we know that the $C^{0}$
distances between their successive renormalizations contract to zero _at an
exponential rate_ , then the conjugacy between $f$ and $g$ should actually be
smooth. Hence the goal becomes to establish exponential contraction of
renormalizations. The strategy laid down by Sullivan in [94] (and explained in
greater detail in [79, Ch. VI]) to achieve this goal can be roughly described
as follows.

  1. (1)

First get geometric bounds on the orbits of the critical points of the (real)
one-dimensional systems. These so-called _real a-priori bounds_ should be
robust enough that, even if we start with maps which have only a mild, finite
degree of smoothness, their successive renormalizations will converge $C^{0}$
exponentially fast to the subspace consisting of real-analytic maps.

  2. (2)

Use such real a-priori bounds to show that the topological conjugacy between
the two systems has slightly more geometric regularity than being merely
continuous: it is actually _quasi-symmetric_ (at least when restricited to the
post-critical sets of both systems).

  3. (3)

Complexify the given real dynamical systems (when they are real-analytic), in
other words, find suitable complex-analytic extensions of these systems.

  4. (4)

Using the real bounds in (1) and the mild geometric control in (2), get
_complex a priori bounds_ for the complexified systems. These bounds are
usually bounds on the moduli of certain annuli (typically fundamental domains
for the complexified systems). Such bounds yield a strong form of compactness.

  5. (5)

Extend the renormalization operator to the complexified dynamical systems.
This operator will, in a suitable domain, be a compact operator due to step
(4).

  6. (6)

Use the bounds and compactness in (3) and some suitable infinite-dimensional
version of Schwarz’s lemma to establish the desired contraction property of
the underlying renormalization operator.

In the context of (real-analytic) unimodal maps of the interval, Sullivan
realized that the relevant complex-analytic dynamical systems are _quadratic-
like maps_ (or more generally polynomial-like maps), and was therefore able to
use the theory developed by Douady and Hubbard [22] for such maps. Recall that
a quadratic-like map is a proper, degree two holomorphic branched covering map
$F:U\to V$ between two topological disks $U,V\subset\mathbb{C}$ with $U$
compactly contained in $V$, branched at a unique critical point $c\in U$. The
_modulus_ of $F$ is by definition the conformal modulus of the annulus
$V\setminus U$. The set $\mathscr{K}_{F}=\bigcap_{n\geq 0}F^{-n}(V)$ is called
the _filled-in Julia set_ of $F$. It is a totally invariant set under the
dynamics, and it is compact due to the fact that $F$ is proper. Every point in
$U\setminus\mathscr{K}_{F}$ has a finite orbit that eventually lands in the
outer annulus $V\setminus U$. This annulus threfore works as a _fundamental
domain_ for the dynamics outside the filled-in Julia set. A central fact about
quadratic-like maps is the _straightening theorem_ of Douady and Hubbard [22]:
every quadratic like map is quasiconformally conjugate to an actual quadratic
polynomial map.

A quadratic-like map $F:U\to V$ is said to be _renormalizable_ if one can find
a sub-disk $D\subset U$ compactly contained in $U$ and containing $c$ and an
integer $p\geq 2$ such that $F^{p}|_{D}:D\to F^{p}(D)\subset V$ is well-
defined, and again a quadratic-like map. This new map, with $p$ smallest
possible and suitably rescaled (via a complex affine map), is called the
_first renormalization_ of $F$, and denoted $\mathscr{R}F$. The number $p$ is
called the _renormalization period_ of $F$, denoted $p(F)$. If all successive
renormalizations
$\mathscr{R}^{2}F=\mathscr{R}(\mathscr{R}F),\ldots,\mathscr{R}^{n}F=\mathscr{R}(\mathscr{R}^{n-1}F),\ldots$
are well-defined, then we say that $F$ is _infinitely renormalizable_. If in
addition all periods $p_{n}=p(\mathscr{R}^{n}F)$ form a bounded sequence, we
say that $F$ infinitely renormalizable of _bounded type_. The _complex bounds_
proved by Sullivan guarantee that if one starts with a real-analytic,
infinitely renormalizable quadratic unimodal map $f$ of bounded type on the
real line, then after a finite number $N$ of iterations, the renormalized
unimodal maps $\mathscr{R}^{n}f$ will be restrictions of quadratic-like maps
$F_{n}$ with $F_{n+1}=\mathscr{R}F_{n}$ for all $n\geq N$, and moreover the
moduli $\mathrm{mod}(\mathscr{R}^{n}F)$ ($n\geq N$) will be bounded from
below. In particular, the sequence $(\mathscr{R}^{n}F)_{n\geq N}$ will be a
pre-compact family (in the topology of uniform convergence on compacta), and
every limit of such renormalization sequence will be a quadratic-like map.
Here and throughout, all holomorphic maps considered commute with complex
conjugation, _i.e.,_ are symmetric about the real axis.

The crucial feature of quadratic-like maps in this theory, very closely
related to the straightening theorem, is that they are amenable to what
Sullivan calls a _pull-back_ argument. If $F_{i}:U_{i}\to V_{i}$, $i=0,1$, are
two symmetric, topologically conjugate quadratic-like maps, and if $h$ is a
quasisymmetric homeomorphism of the real line which sends the post-critical
set of $F_{0}$ to the post-critical set of $F_{1}$, then $F_{0}$ and $F_{1}$
are quasiconformally conjugate. More precisely, there exists a quasiconformal
homeomorphism $H:V_{0}\to V_{1}$ such that $H\circ F_{0}=F_{1}\circ H$; in
addition, the quasiconformal dilatation of $H$ depends only on the conformal
moduli $\mathrm{mod}(V_{i}\setminus U_{i})$ ($i=0,1$) and on the
quasisymmetric distortion of $h$.

The existence of such a conjugacy already allows us to speak of the
_quasiconformal_ or _Teichmüller_ distance between $F_{0}$ and $F_{1}$,
defined as

$d_{T}(F_{0},F_{1})\;=\;\inf_{\phi}\log{\frac{1+\|\mu_{\phi}\|_{\infty}}{1-\|\mu_{\phi}\|_{\infty}}}\
,$ (10.1)

the infimum being taken over all quasi-conformal conjugacies $\phi$ between
$F_{0}$ and $F_{1}$. This is in fact a pseudo-distance: its value will be zero
whenever the two maps are _conformally_ conjugate. It turns out that the Julia
set of an (symmetric) infinitely renormalizable quadratic-like map carries no
_invariant line fields_ (equivalently, no non-zero invariant Beltrami
differentials). This is another consequence of the straightening theorem.
Thus, for every quasiconformal conjugacy $\phi$ as above we have that
$\mu_{\phi}$ vanishes a.e. on the (filled-in) Julia set of $F_{0}$. In
particular, when calculating $\|\mu_{\phi}\|_{\infty}$ in the right-hand side
of (10.1), we only need to look at the values of $\mu_{\phi}(z)$ for $z\in
V_{0}$.

It is immediate from the definition that the Teichmüller distance is weakly
contracted under renormalization: any conjugacy between $F_{0}$ and $F_{1}$
restricts to a conjugacy between $\mathscr{R}(F_{0})$ and
$\mathscr{R}(F_{1})$.

Now, let $H$ be a quasiconformal conjugacy between $F_{0}$ and $F_{1}$, say
the one constructed via the pull-back argument. Its Beltrami differential
$\mu_{H}=\overline{\partial}H/\partial H$ is invariant under $F_{0}$, and
therefore it can be used to generate a _path_ of (pairwise qc-conjugate)
quadratic-like maps joining $F_{0}$ to $F_{1}$. To see this, define
$\mu_{t}=t\mu_{H}$ for all $t\in\mathbb{C}$ such that
$|t|<\|\mu_{H}\|_{\infty}^{-1}$ then integrate each $\mu_{t}$ using the
measurable Riemann mapping theorem to get a (normalized) quasiconformal
homeomorphism $H_{t}$, and then define $F_{t}=H_{t}\circ F_{0}\circ
H_{t}^{-1}$. Such a path is called a _Beltrami path_ joining $F_{0}$ to
$F_{1}$.

As one can see from the definitions given so far, renormalization maps
Beltrami paths to Beltrami paths. Some Beltrami paths are more efficient than
others, in the sense that they are close to being “geodesics” in the
Teichmüller metric. It will usually be the case that a very efficient Beltrami
path joining $F_{0}$ to $F_{1}$ will be mapped to an inefficient Beltrami path
joining $\mathscr{R}(F_{0})$ to $\mathscr{R}(F_{1})$: the image path “coils”.
It turns out that one can put this coiling property into more quantitative
terms, and the result is a form of Schwarz’s lemma in infinite
dimensions.666However, we warn the reader that the renormalization “operator”
is _not_ a complex-analytic operator.

There are some difficulties with carrying out the details of this approach.
One is the fact that the domain and range of a quadratic-like map vary with
the map itself, so it is hard to set up the renormalization procedure as an
actual operator on a space of maps defined over a fixed domain. Another
difficulty is the fact that, if we are given two quadratic-like maps and they
both restrict to the same quadratic unimodal map on the line, then they should
be regarded as essentially the same dynamical system; however, their
Teichmüller distance, according to the definition given above, will not be
zero! Sullivan soon realized that a way to circumvent these difficulties is to
take an inverse limit of the dynamics off the filled-in Julia set. To wit, if
$F:U\to V$ is the given quadratic-like map, one considers the inverse system

$\cdots\to F^{-(n+1)}(V\setminus\mathscr{K}_{F})\to
F^{-n}(V\setminus\mathscr{K}_{F})\to\cdots
F^{-1}(V\setminus\mathscr{K}_{F})\to V\setminus\mathscr{K}_{F}\ ,$

where each map, being a restriction of $F$, is an unbranched $2$-to-$1$
holomorphic covering. The inverse limit of this system, denoted
$\mathscr{L}(F)$, is a _Riemann surface lamination_ in a natural way. This
object is locally homeomorphic to the product of a disk by a Cantor set, and
the chart transitions are holomorphic on the leaves. The construction is
canonical in the sense that, if $F$ varies (but stays in the same topological
conjugacy class), then topologically $\mathscr{L}(F)$ does not change at all.
Only its conformal structure changes. Moreover, a quasiconformal conjugacy
between two such maps induces a homeomorphism between the two corresponding
laminations which is quasiconformal on each leaf. Hence, one can speak of the
(_moduli space_ or) _Teichmüller space_ of such lamination. It then follows
that renormalization induces an operator on such Teichmüller space.

Using these ideas, Sullivan was able to carry out the strategy outlined in
steps (1)-(6) above almost completely in the bounded-type case. We say
“almost” because in step (6) he was forced to settle for something less than
exponential contraction. Sullivan made an ingenious use of the theory of
Riemann surface laminations, and used the Teichmüller theory of such objects
(which he largely developed on the fly) to prove a (non-uniform) version of
Schwarz’s lemma in this context, which in turn allowed him to prove
renormalization convergence without a rate. The exponential convergence of
renormalizations for bounded type infinitely renormalizable maps was finally
achieved by McMullen [74, 75] by a different route, using his theory of
_rigidity of towers_.

######  Remark 10.1.

The theory of Riemann surface laminations is a beautiful subject in its own
right. See [40] for a nice exposition.

###  10.2. Holomorphic commuting pairs

In his PhD thesis [29], the first named author took up the task of carrying
out as much as possible of Sullivan’s program in the context of critical
circle maps with a single critical point of cubic type. Steps (1) and (2) of
Sullivan’s strategy were already in place due to the works of Herman and
Swiatek (Theorem 5.2) and Yoccoz (Theorem 7.8 in the unicritical case).

The key to the remaining steps is an analogue of the quadratic-like maps of
Douady and Hubbard, a holomorphic dynamical system that somehow extends the
real commuting pairs arising as successive renormalizations of a critical
circle map. This is the central contribution of [29] and of the subsequent
paper [31]. Here are the relevant definitions, taken almost verbatim from [36,
p. 346].

######  Definition 10.2.

By a bowtie we mean $4$-tuple
$({\mathscr{O}}_{\xi},{\mathscr{O}}_{\eta},{\mathscr{O}}_{\nu},{\mathscr{V}})$
of simply-connected domains in the complex plane such that:

  1. ($a$)

Each ${\mathscr{O}}_{\gamma}$ is a Jordan domain whose closure is contained in
${\mathscr{V}}$;

  2. ($b$)

We have
$\overline{{\mathscr{O}}}_{\xi}\cap\overline{{\mathscr{O}}}_{\eta}=\\{0\\}\subseteq{\mathscr{O}}_{\nu}$;

  3. ($c$)

The sets ${\mathscr{O}}_{\xi}\setminus{\mathscr{O}}_{\nu}$,
${\mathscr{O}}_{\eta}\setminus{\mathscr{O}}_{\nu}$,
${\mathscr{O}}_{\nu}\setminus{\mathscr{O}}_{\xi}$ and
${\mathscr{O}}_{\nu}\setminus{\mathscr{O}}_{\eta}$ are non-empty and
connected.

Figure 6.  A holomorphic commuting pair.

######  Definition 10.3.

Let
$({\mathscr{O}}_{\xi},{\mathscr{O}}_{\eta},{\mathscr{O}}_{\nu},{\mathscr{V}})$
be a bowtie. A holomorphic pair $\Gamma$ with domain
${\mathscr{U}}={\mathscr{O}}_{\xi}\cup{\mathscr{O}}_{\eta}\cup{\mathscr{O}}_{\nu}$
and co-domain ${\mathscr{V}}$ is the dynamical system generated by three
holomorphic maps $\xi:{\mathscr{O}}_{\xi}\to{\mathbb{C}}$,
$\eta:{\mathscr{O}}_{\eta}\to{\mathbb{C}}$ and
$\nu:{\mathscr{O}}_{\nu}\to{\mathbb{C}}$ satisfying the following conditions
(see Figure 6).

  1. H1

Both $\xi$ and $\eta$ are univalent onto
${\mathscr{V}}\cap{\mathbb{C}}(\xi(J_{\xi}))$ and
${\mathscr{V}}\cap{\mathbb{C}}(\eta(J_{\eta}))$ respectively, where
$J_{\xi}={\mathscr{O}}_{\xi}\cap{\mathbb{R}}$ and
$J_{\eta}={\mathscr{O}}_{\eta}\cap{\mathbb{R}}$. (Notation:
${\mathbb{C}}(I)=({\mathbb{C}}\setminus{\mathbb{R}})\cup I$.)

  2. H2

The map $\nu$ is a $3$-fold branched cover onto
${\mathscr{V}}\cap{\mathbb{C}}(\nu(J_{\nu}))$, where
$J_{\nu}={\mathscr{O}}_{\nu}\cap{\mathbb{R}}$, with a unique critical point at
$0$.

  3. H3

We have ${\mathscr{O}}_{\xi}\ni\eta(0)<0<\xi(0)\in{\mathscr{O}}_{\eta}$, and
the restrictions $\xi|[\eta(0),0]$ and $\eta|[0,\xi(0)]$ constitute a critical
commuting pair.

  4. H4

Both $\xi$ and $\eta$ extend holomorphically to a neighborhood of zero, and we
have $\xi\circ\eta(z)=\eta\circ\xi(z)=\nu(z)$ for all $z$ in that
neighborhood.

  5. H5

There exists an integer $m\geq 1$, called the height of $\Gamma$, such that
$\xi^{m}(a)=\eta(0)$, where $a$ is the left endpoint of $J_{\xi}$; moreover,
$\eta(b)=\xi(0)$, where $b$ is the right endpoint of $J_{\eta}$.

The relevant dynamical system here, which we will still denote by $\Gamma$, is
the pseudo-semigroup generated by the three maps $\xi,\eta,\nu$. The interval
$J=[a,b]$ is called the long dynamical interval of $\Gamma$, whereas
$\Delta=[\eta(0),\xi(0)]$ is the short dynamical interval of $\Gamma$. They
are both forward invariant under the dynamics, as the reader can easily check.
The rotation number of $\Gamma$ is by definition the rotation number of the
critical commuting pair of $\Gamma$ obtained by restriction to the real line
(condition H3). We say that the holomorphic commuting pair $\Gamma$ has
_geometric boundaries_ if $\partial\mathscr{U}$ and $\partial\mathscr{V}$ are
_quasi-circles_ 777A quasi-circle, we recall, is the image of a round disk
under a quasiconformal homeomorphism of the plane..

######  Remark 10.4.

Examples of holomorphic commuting pairs with arbitrary rotation number and
arbitrary heights can be constructed directly from the Arnold family. This is
carefully done in [31, §4]. We should also point out that there is nothing
special about cubic critical points. Holomorphic commuting pairs can be
defined so as to have a critical point with any odd-power criticality
whatever. To see how this is done, the reader should consult Arlane Vieira’s
thesis [97] (see also [103]).

It turns out that the holomorphic pair $\Gamma$ can be renormalized: the first
renormalization of the critical commuting pair of $\Gamma$ extends in a
natural way to a holomorphic pair ${\mathscr{R}}(\Gamma)$ with the same co-
domain ${\mathscr{V}}$. See Prop. 2.3 in [31] for the detailed construction of
${\mathscr{R}}(\Gamma)$. Renormalization is defined in such a way that the
restriction of the renormalized holomorphic pair $\mathscr{R}(\Gamma)$ to the
real line is the critical commuting pair that represents the renormalization
of the critical commuting pair $(\xi|_{[\eta(0),0]}\,,\,\eta|_{[0,\xi(0)]})$.

###  10.3. Pull-back argument

The first main result in [29] (or [31]) is the following analogue of
Sullivan’s _pull-back argument_.

######  Theorem 10.5 (Pull-back Argument).

Let $\Gamma$ and $\Gamma^{\prime}$ be holomorphic pairs with geometric
boundaries and let $h:J\to J^{\prime}$ be a quasisymmetric conjugacy between
the restrictions of $\Gamma$ and $\Gamma^{\prime}$ to their respective long
dynamical intervals $J$ and $J^{\prime}$. Then there exists a quasiconformal
conjugacy $H:{\mathscr{V}}\to{\mathscr{V}}^{\prime}$ between $\Gamma$ and
$\Gamma^{\prime}$ which is an extension of $h$.

The proof is more involved than that of the original pull-back argument, for
the following reason. In the quadratic-like case, we know by the straightening
theorem of Douady-Hubbard that every quadratic-like map is quasiconformally
conjugate to a quadratic polynomial, and the latter does not have wandering
domains (due to Sullivan’s no-wandering-domains theorem [92]). Hence
quadratic-like maps do not have wandering domains. By contrast, holomorphic
pairs could in principle have wandering domains. To deal with their putative
existence, one needs to use a form of quasiconformal surgery (something called
the _qc-sewing lemma_ of L. Bers, see [31, Lem. 3.2]). Wandering domains are
only ruled out _a posteriori_ , combining Theorem 10.5 with the fact that
holomorphic pairs constructed from the Arnold family do not carry such domains
(see [31, Th. 4.2]).

###  10.4. Complex bounds

Another important fact about holomorphic commuting pairs is that the class of
such objects contains all limits of successive renormalizations of a critical
circle map (or critical commuting pair). Moreover, we have _complex bounds_
for renormalization, in the following sense.

######  Theorem 10.6 (Complex Bounds).

Let $f:S^{1}\to S^{1}$ be a real-analytic critical circle map with arbitrary
irrational rotation number. Then there exists $n_{0}=n_{0}(f)$ such that for
all $n\geq n_{0}$ the $n$-th renormalization of $f$ extends to a holomorphic
pair with geometric boundaries whose fundamental annulus has conformal modulus
bounded from below by a universal constant.

We think of the unit circle $S^{1}=\mathbb{R}/\mathbb{Z}$ as embedded in the
infinite cylinder $\mathbb{C}/\mathbb{Z}$, and we use on latter the conformal
metric induced from the standard Euclidean metric $|dz|$ of the complex plane
via the exponential map $\exp(z)=e^{2\pi iz}$. Note that $\mathrm{Im}\,z$ is
well-defined for every $z\in\mathbb{C}/\mathbb{Z}$ (it is simply the imaginary
part of any one of its pre-images under the exponential). Note also that every
real-analytic circle map $f$ as above has a holomorphic extension to an
annular neighborhood of the unit circle inside $\mathbb{C}/\mathbb{Z}$.

The main step in the proof of Theorem 10.6 is to establish a geometric
estimate showing that, for all sufficiently large $n$, the appropriate inverse
branch of $f^{q_{n+1}}$ maps a sufficiently large disk around the $n$-th
renormalization domain $I_{n}\cup I_{n+1}$ well within itself. Here,
“sufficiently large” means large with respect to the size of $I_{n}\cup
I_{n+1}$. For each $m\geq 1$, let $D_{m}\subset\mathbb{C}/\mathbb{Z}$ denote
the disk having as one of its diameters the interval
$[f^{q_{m+1}}(c),f^{q_{m}-q_{m+1}}(c)]\subset S^{1}$ containing the critical
point $c$888It is easy to see that
$[f^{q_{m+1}}(c),f^{q_{m}-q_{m+1}}(c)]\supset I_{m}\cup I_{m+1}$. . The
geometric estimate is the following (once again, the statement is taken almost
verbatim from [36, Prop. 3.2]).

######  Proposition 10.7.

There exist universal constants $B_{0}$ and $B_{1}$ and for each $N\geq 1$
there exists $n(N)$ such that for all $n\geq n(N)$ the inverse branch
$f^{-q_{n+1}+1}$ taking $f^{q_{n+1}}(I_{n})$ back to $f(I_{n})$ is a well-
defined univalent map over $\Omega_{n,N}=(D_{n-N}\setminus S^{1})\cup
f^{q_{n+1}}(I_{n})$, and for all $z\in\Omega_{n,N}$ we have

$\frac{\mathrm{dist}\,\left(f^{-q_{n+1}+1}(z),f(I_{n})\right)}{|f(I_{n})|}\;\leq\;B_{0}\left(\frac{\mathrm{dist}\,(z,I_{n})}{|I_{n}|}\right)+B_{1}\
.$

As stated, Theorem 10.6 was proved in [36, §3]. But the story behind it is a
bit more involved. The first version of the complex bounds in the present
context was proved in [29] (also [31]) under two further assumptions on $f$,
namely

  1. (i)

the rotation number of $f$ is of bounded type;

  2. (ii)

$f$ is an _Epstein map_.

We say that a real analytic circle map is _Epstein_ if its lift to the real
line has a holomorphic extension $F$ to a neighborhood of the real axis in the
complex plane in such a way that $F$ has inverse branches which are globally
defined in the upper (or lower) half-plane. The main examples of Epstein
circle maps are the maps in the Arnold family introduced earlier (see §3.2).
The proof presented in [29, 31] makes use of the so-called _sector theorem_ of
Sullivan (see [94]; the version used in the circle case is in fact the one
proved in [30]). However, the sector theorem can only be used under the
bounded type assumption (i).

That assumption was removed by Yampolsky in [99], using a special case of
Proposition 10.7. Assuming that the map $f$ is Epstein, he exploits in full
the idea of _Poincaré neighborhood trapping_ , which we briefly explain. Let
$J\subseteq\mathbb{R}$ be a bounded open interval, and write
$\mathbb{C}(J)=\mathbb{C}\setminus(\mathbb{R}\setminus J)$. If
$\phi:\mathbb{C}(J)\to\mathbb{C}(\phi(J))$ is a symmetric holomorphic map,
then $\phi$ maps each Poincaré neighborhood
$\mathbb{P}_{\theta}(J)=\\{z:\,\mathrm{angle}(z,J)\geq\theta\\}$ into a
corresponding Poincaré neighborhood $\mathbb{P}_{\theta}(\phi(J))$ with the
same angle $\theta$. Here, $0<\theta<\pi$ and $\mathrm{angle}(z,J)$ denotes
the angle at $z$ under which $z$ views the interval $J$. This simple but
fundamental fact is easily seen to be a consequence of Schwarz’s lemma.

The Poincaré neighborhood trapping idea used in Yampolsky’s approach works
because he is assuming that $f$ is Epstein. But if we abandon the latter
hypothesis, then this tool is no longer directly applicable. In order to prove
Theorem 10.6, one needs the following “relaxed” version of Poincaré
neighborhood trapping (whose statement is taken verbatim from [36, Lem. 3.3]).

######  Lemma 10.8.

For every small $a>0$, there exists $\theta(a)>0$ satisfying $\theta(a)\to 0$
and $a/\theta(a)\to 0$ as $a\to 0$, such that the following holds. Let