# On the dimension theory of random walks and group actions by circle
diffeomorphisms

Weikun He, Yuxiang Jiao and Disheng Xu

###### Abstract

We establish new results on the dimensional properties of measures and
invariant sets associated to random walks and group actions by circle
diffeomorphisms. This leads to several dynamical applications. Among the
applications, we show, strengthening of a recent result of Deroin-Kleptsyn-
Navas [24], that the minimal set of a finitely generated group of real-
analytic circle diffeomorphisms, if exceptional, must have Hausdorff dimension
less than one. Moreover, if the minimal set contains a fixed point of
multiplicity $k+1$ of an diffeomorphism of the group, then its Hausdorff
dimension must be greater than $k/(k+1)$. These results generalize classical
results about Fuchsian group actions on the circle to non-linear settings.

This work is built on three novel components, each of which holds its own
interest: a structure theorem for smooth random walks on the circle, several
dimensional properties of smooth random walks on the circle and a dynamical
generalization of the critical exponent of Fuchsian groups.

###### Contents

  1. 1 Introduction
  2. 2 Statements of the main results
    1. 2.1 Dimension properties of stationary measures
    2. 2.2 Dimension properties of minimal sets
    3. 2.3 Applications of random walk structures and approximation results
  3. 3 Preliminaries
    1. 3.1 Group actions on the circle
    2. 3.2 Dimension and entropy
    3. 3.3 Distortion estimates
  4. 4 Random walks on $\mathrm{Diff}_{+}^{2}(\SS^{1})$
    1. 4.1 Preliminaries on random walks
    2. 4.2 Generalities on random transformations
    3. 4.3 The structure of random walks on $\mathrm{Diff}^{2}_{+}(\SS^{1})$
    4. 4.4 Basic properties of stationary measures
    5. 4.5 Uniform good words
    6. 4.6 Effective convergence to the Furstenberg boundary
  5. 5 Exact dimensionality of stationary measures
    1. 5.1 Preparation
    2. 5.2 Exact dimensionality (Proof Theorem 2.2)
  6. 6 Dimension Formulas
    1. 6.1 Dimension formula on the circle (Proof of Theorem 2.6)
    2. 6.2 Dimension formula on the interval (Proof of Theorem 2.7)
  7. 7 Approximation with a uniformly hyperbolic subsystem
    1. 7.1 Hyperbolic elements
    2. 7.2 Perfect pingpong pairs
    3. 7.3 Proof of Theorems 2.28 and 2.29
  8. 8 Variational principle for dimensions
    1. 8.1 Elements with controlled contracting rates
    2. 8.2 Proof of the variational principle (Theorems 2.9, 2.10 and 2.14)
  9. 9 The dynamical critical exponents
    1. 9.1 The $C^{1}$ dynamical critical exponent (Proof of Theorem 2.15)
    2. 9.2 The $C^{2}$ dynamical critical exponent (Proof of Theorem 2.21)
  10. 10 Groups with parabolic elements
    1. 10.1 Growth of derivatives around a parabolic fixed point
    2. 10.2 Boosting the dynamical critical exponent
  11. 11 The dynamical critical exponent and conformal measures
    1. 11.1 Conformal measures (Proof of Theorem 2.23)
    2. 11.2 Basic properties of the dynamical critical exponent on sets
    3. 11.3 The dynamical critical exponent for real analytic groups
    4. 11.4 Existence of atomless conformal measures
    5. 11.5 The pointwise dynamical critical exponent (Proof of Theorems 2.18, 2.24 and Corollary 2.19)
  12. 12 Conclusion and further discussions
    1. 12.1 Proofs of Main Theorem and other results
    2. 12.2 Comparison with the critical exponent of Fuchsian groups
    3. 12.3 Some counterexamples

##  1 Introduction

The dimension of fractal sets and associated measures defined through
dynamical systems has been extensively researched, including investigating
their relationships with other dynamically-defined objects and properties such
as entropies, Lyapunov exponents, critical exponents, and uniformity of
hyperbolicity. This topic has been explored in various areas of mathematics,
such as the dimension of Julia sets in holomorphic dynamics [6, 7, 80, 18, 60,
25, 1], etc.; the dimension of attractors and invariant measures of smooth
dynamical systems and their relationship with entropies, Lyapunov exponents
[48, 50, 55, 56, 86, 10], etc.; the study of self-affine sets/measures of
iterated function systems (IFS) in fractal geometry [39, 31, 84, 42, 27, 28,
9, 63, 71], etc.; the study of stationary measures for random walks [40, 53,
70, 49], etc.; and the dimension of limit sets of hyperbolic group actions
[83, 81, 16, 14, 68, 12], etc.

In this paper, we focus on the dimension theory of smooth group actions and
random walks on the circle. We establish several results concerning the
dimensions of invariant sets and stationary measures and derive several
applications.

To illustrate the power of our theory, in the following we show how our
results extend classical results on actions by Fuchsian groups on the circle
to the non-linear setting. Suppose $\Gamma\subset\mathrm{PSL}(2,{\mathbb{R}})$
is a finitely generated Fuchsian group. Then $\Gamma$ acts isometrically on
the Poincaré disk ${\mathbb{D}}$ which extends to the projective action on
$\SS^{1}=\partial{\mathbb{D}}$. Assuming that $\Gamma$ is non-elementary, then
its limit set $\Lambda_{\Gamma}\subset\partial{\mathbb{D}}$ is either the
circle or homeomorphic to a Cantor set. The group $\Gamma$ is then said to be
of the first kind in the former case and of the second kind in the latter
case. To each Fuchsian group is associated a quantity $\delta(\Gamma)$ called
its critical exponent, defined to be the abscissa of convergence of the
classical Poincaré series. The relationship between the critical exponent and
the dimension theory of the limit set has been systematically studied in works
of Beardon, Patterson and Sullivan. The following summarizes classical results
on dimension of limit sets of the second kind Fuchsian groups.

######  Theorem 1.1 (Beardon [12, 11], Patterson [68], Sullivan [81]).

Let $\Gamma$ be a non-elementary finitely generated Fuchsian group of the
second kind, then we have

  1. (1)

$0<\delta(\Gamma)<1.$

  2. (2)

$\dim_{\mathrm{H}}\Lambda_{\Gamma}=\delta(\Gamma).$

  3. (3)

If $\Gamma$ has parabolic elements then $\delta(\Gamma)>1/2.$

Now we turn to the non-linear setting. Let
$G\subset\mathrm{Diff}^{1}_{+}(\SS^{1})$ be a subgroup of orientation
preserving circle diffeomorphisms. The classification of the minimal set of
$G$-action is well-known (cf. [64, Theorem 2.1.1]) for such group $G$ (or more
generally groups formed by circle homeomorphisms): exactly one of the
following holds.

  1. (1)

$G$ has a finite orbit.

  2. (2)

$G$ acts minimally on the circle.

  3. (3)

$G$ has a unique $G$-invariant minimal set on the circle which is homeomorphic
to a Cantor set.

In the last case, we say $G$ has an _exceptional minimal set_. In particular,
for finitely generated Fuchsian groups, having an exceptional minimal set if
and only if the Fuchsian group is of the second kind.

In this paper, we introduce a dynamical analogue of the critical exponents for
groups of circle diffeomorphisms.

######  Definition 1.2.

Let $G$ be a subgroup of $\mathrm{Diff}^{1}_{+}(\SS^{1})$ without finite
orbits. Let $\Lambda$ be the unique minimal set of $G.$ The _dynamical
critical exponent_ of $G$ is defined as

$\delta(G)=\lim_{\varepsilon\to
0^{+}}\limsup_{n\to+\infty}\frac{1}{n}\log\\#\\{\,g\in G:\exists
x\in\Lambda,\,g^{\prime}|_{B(x,\varepsilon)}\geqslant 2^{-n}\,\\}.$ (1.1)

Here and throughout this paper, $\log$ is taken of base $2$.

It can be easily checked that this quantity coincides with the critical
exponent if $G$ is a Fuchsian group (see Section 12.2). The reason we refer to
$\delta(\,\bm{\cdot}\,)$ as the _dynamical_ critical exponent is because
unlike the classical critical exponent which is determined by geometric
information on hyperbolic spaces, the definition of the dynamical critical
exponent is based solely on the local contracting rate of the group elements
acting on the circle. This makes it easier to generalize to more general group
actions on manifolds.

One of our main results is the following theorem, which not only extends
Theorem 1.1 to the non-linear setting by considering the dynamical critical
exponent, but also provides a purely dynamical comprehension of Theorem 1.1,
see Section 12.2.

######  Main Theorem.

Let $G$ be a finitely generated subgroup of
$\mathrm{Diff}_{+}^{\omega}(\SS^{1})$ with an exceptional minimal set
$\Lambda$, then

  1. (1)

$0<\dim_{\mathrm{H}}\Lambda<1$.

  2. (2)

$\dim_{\mathrm{H}}\Lambda=\delta(G)$.

  3. (3)

If $G$ contains a nontrivial element having a fixed point in $\Lambda$ and of
multiplicity $(k+1)$ with $k\in{\mathbb{N}}$, then
$\dim_{\mathrm{H}}\Lambda>k/(k+1).$ Consequently, there is an upper bound for
the multiplicities of fixed points of elements in $G.$

Here a fixed point $x_{0}$ of a diffeomorphism
$f\in\mathrm{Diff}_{+}^{k+1}(\SS^{1})$ is said to be of _multiplicity_ $(k+1)$
if $x_{0}$ is a $(k+1)$-multiple zero of the function $\varphi(x)=f(x)-x$.

######  Remark 1.3.

  1. (1)

Our investigation also applies to settings with less regularity (but requires
some technical assumptions). See the next section for details.

  2. (2)

The conditions “$x\in\Lambda$” in (1.1) and “the fixed point is in $\Lambda$”
in the third item are both necessary, as will be shown in Section 12.3.

  3. (3)

The finitely generated condition of $G$ cannot be relaxed. It is easy to
construct an infinitely generated Fuchsian group for which
$\dim_{\mathrm{H}}\Lambda=1$.

  4. (4)

In a Fuchsian group, a parabolic element has a fixed point of multiplicity $2$
and there is no fixed point of higher multiplicity. Higher multiplicity could
appear in general diffeomorphisms, thanks to additional flexibility, which
results in a stronger lower bound for the dimension of the minimal set.

Under the same assumption as in the above theorem, Deroin, Kleptsyn and Navas
[24] showed that $\Lambda$ has zero Lebesgue measure, which made an important
progress towards a problem raised by Ghys and Sullivan asking whether the
exceptional minimal set of a finitely-generated subgroup of
$\mathrm{Diff}^{2}(\SS^{1})$ has zero Lebesgue measure. See also [65, Question
14] and [37, 41] corresponding results in foliation theory. Thus, our main
theorem strengthens their result from $\mathrm{Leb}(\Lambda)=0$ to
$\dim_{\mathrm{H}}\Lambda<1$.

One of the applications of our main result is the classification of orbit
closures. The orbit closures of group actions is an important object in the
study of homogeneous dynamics, for instance, Ratner’s proof of Raghunathan
conjecture [72]. Recent developments in this area include [13, 15, 26, 17],
among others. The following is what we can say in our setting.

######  Corollary 1.4.

Let $G\subset\mathrm{Diff}^{\omega}_{+}(\SS^{1})$ be a finitely generated
subgroup. Then any $G$-orbit closure is

  1. (1)

an infinite countable set, in which case $G$ is isomorphic to either
${\mathbb{Z}}\times{\mathbb{Z}}/k{\mathbb{Z}}$ or a semi-direct product
${\mathbb{Z}}\rtimes{\mathbb{Z}}/(2k{\mathbb{Z}})$ with the presentation
$\left\langle a,b~{}|~{}bab^{-1}=a^{-1},~{}b^{2k}=1\right\rangle,$ for some
positive integer $k$;

  2. (2)

or a submanifold, i.e. a finite set, a finite union of closed intervals or the
whole circle;

  3. (3)

or the union of a countable set with the unique $G$-invariant exceptional
minimal set $\Lambda$ which has Hausdorff dimension $\delta(G)$.

Consequently, if $G$ is a nonabelian free group freely generated by
$f_{1},\cdots,f_{n}\in\mathrm{Diff}_{+}^{\omega}(\SS^{1})$ satisfying
$\max_{1\leqslant i\leqslant
n}\left\\{\|f_{i}^{\prime}\|_{C^{0}},\|(f_{i}^{-1})^{\prime}\|_{C^{0}}\right\\}\leqslant
2n-1,$ then every $G$-orbit closure is a submanifold, and the bound $2n-1$ is
sharp.

######  Remark 1.5.

In Remark 12.2, we will see that the group $G\cong\left\langle
a,b~{}|~{}bab^{-1}=a^{-1},~{}b^{2k}=1\right\rangle$ can be realized as a
subgroup of $\mathrm{Diff}^{\omega}_{+}(\SS^{1})$. The group is solvable but
not nilpotent and satisfies that $\\{g^{2}:g\in G\\}$ is an abelian subgroup.
Thus it falls into the first case of the classification of the solvable group
actions on the circle in [19, Theorem 1.9]. For $k=1$, it is $D_{\infty}$, the
infinite dihedral group. In general, every solvable subgroup of
$\mathrm{Diff}_{+}^{\omega}(\SS^{1})$ is metabelian [35], while every
nilpotent subgroup of $\mathrm{Diff}_{+}^{2}(\SS^{1})$ is abelian according to
[69], see also [30].

This work grew out of an investigation into the dimension theory of smooth
random walks on the circle, which is intimately connected to that of group
actions on the circle. Main Theorem turns out to be the most interesting
consequence of our study. We will state other meaningful results, including a
general version of the Main Theorem in the $C^{2}$-regularity, in the next
section. Now we highlight several concepts developed in our theory, each of
them leads to several applications and other results which are interesting on
their own.

##### Structure theorem of smooth random walks on the circle.

The first key ingredient is a thorough study of structures and symmetries of
smooth random walks on the circle.

Thanks to a result due to Furstenberg [33], we know that, under mild
assumptions the action of a typical random product of matrices in
$\mathrm{SL}(2,{\mathbb{R}})$ on the ${\mathbb{R}}{\mathbb{P}}^{1}$ displays
the (non-stationary) _North-South dynamics_. We generalize this to the smooth
setting and establish a structure theorem for $C^{2}$ random walks on the
circle (Theorems 4.12, 4.15 and 4.16). Roughly speaking, the dynamics of a
typical random iteration by circle diffeomorphisms looks like a finite copies
of North-South dynamics, with certain _rotation symmetry_. The structure
theorem gives a clear description of the global dynamics for smooth random
walks and provides a bridge to extend results for
$\mathrm{SL}(2,{\mathbb{R}})$ action on ${\mathbb{R}}{\mathbb{P}}^{1}$ to the
non-linear case.

In this paper, we also present two direct consequences of the structure
theorem, which are of independent interests. The first one reveals a _time
reverse symmetry_ for semigroup actions on the circle. That is, every sub-
semigroup of $\mathrm{Diff}_{+}^{2}(\SS^{1})$ admits the same number of
minimal sets to its inverse semigroup (Theorem 2.26). Another application is a
variant of Margulis’s theorem ([58], also known as Ghys’s conjecture) in the
$C^{2}$ case, i.e. the existence of a _perfect ping-pong pair_. More
precisely, under a mild assumption, within a sub-semigroup of
$\mathrm{Diff}_{+}^{2}(\SS^{1})$, we show that there exists a pair of elements
which freely generates a free group. Furthermore, this pair generates the
simplest dynamic as possible, a finite copies of “pingpong-dynamics” on the
circle, see Theorem 2.27.

##### Dimension theory of stationary measures on the circle (I): exact
dimensionality and a dimension formula.

Our estimate of the Hausdorff dimension of minimal sets is based on an
estimation of the dimension of measures supported on the set. One of the key
steps is to establish the exact dimensionality and the dimension formula for
stationary measures.

In the context of $\mathrm{SL}(2,{\mathbb{R}})$ random walks, Ledrappier [49]
first showed a Ledrappier-Young type formula for the Furstenberg measures. The
exact dimensionality of these measures is known since then, see Hochman-
Solomyak [40]. Furthermore, under certain Diophantine condition, they
established the dimension formula
$\dim\nu=\min\left\\{1,\frac{h_{\mathrm{RW}}(\mu)}{2\chi(\mu)}\right\\},$
where $\nu$ is the Furstenberg measure of $\mu,$ $h_{\mathrm{RW}}(\mu)$ is the
random walk entropy and $\chi(\mu)$ is the Lyapunov exponent. This formula
provides a tool to estimate the dimension of stationary measure.

In our setting, namely smooth random walks on the circle, we can also show the
exact dimensionality of stationary measures, Theorem 2.2. Moreover, under a
discreteness condition, we establish a dimension formula using an entropy
argument, see Theorem 2.6.

##### Dimension theory of stationary measures on the circle (II):
approximation techniques and variational principles.

Another key point to estimate the Hausdorff dimension of minimal sets is to
approximate it by the dimension of stationary measures supported on minimal
sets. This problem is often known as the variational principle for dimensions.
For conformal IFSs (hence semigroup actions), Feng and Hu [31] systemically
studied the variational principle for the associated attractor. In this paper,
we obtain the variational principle for the minimal set of smooth group
actions on the circle (Theorem 2.9) by using a covering argument and results
about group actions on the circle developed in [23, 24]. Moreover, the
covering argument helps us obtain an approximation result of smooth random
walks on the circle. Namely, under mild assumptions, we show that smooth
random walks on the circle can be approximated by uniform hyperbolic pingpong
dynamics with approximately the same entropy, Lyapunov exponents and dimension
of stationary measures simultaneously (Theorem 2.28). This can be viewed as a
variant of the approximation result of Avila-Crovisier-Wilkinson in [5] and a
non-linear extension of [63] on the circle.

##### The study of dynamical critical exponents.

As previously mentioned, we introduce the concept of the dynamical critical
exponent (DCE), which gives rise to several applications. The first goal is
show that it is equal to the Hausdorff dimension of the exceptional minimal
set, namely the item (2) of Main Theorem. The proof combines the structural
theorem of random walks above, the dimensional tools associated with the
stationary measure of random walks, and the combinatorics of the free group.
The main idea behind the proof is the variational principle of the critical
exponent (Proposition 9.1). Some related results in the linear/homogeneous
setting occurred in [63, 62, 43]. In our setting, the nonlinear and non-
uniform hyperbolic nature of our problem complicates the proof. The novelty of
our argument is the use of a combinatorial argument to control simultaneously
the group structure and the dynamics on the circle.

Next, to obtain the item (3) of Main Theorem, we use the equality between the
DCE and the Hausdorff dimension combined with an estimate of the DCE. The
latter is partially inspired by the methods in [12] and [68].

Finally, to establish that the dimension of the exceptional minimal set is
less than $1$, i.e. the item (1) of Main Theorem, we further localize the DCE,
leading to the definition of pointwise dynamical critical exponents (PDCE). We
then connect the PDCE with the existence of conformal measures and with the
dimension of the exceptional minimal set. Utilizing a delicate proof by
contradiction, we show the dimension is less than 1.

### Organization

We will state in full generality the main results in Section 2. In Section 3,
we recall some notations and results. We then establish the structure of
random walks on $\mathrm{Diff}_{+}^{2}(\SS^{1})$ in Section 4. Sections 5-7
are devoted to study the dimension theory of stationary measures. We will
prove the exact dimensionality, the dimension formula and an approximation
result for stationary measures in these sections. We study the variational
principle and the dynamical critical exponent in Sections 8-11. In Section 12,
we complete the proof of the main theorem and proofs of remaining results.
Section 12 also contains some (counter-)examples and further discussions.

Here is a diagram illustrating the logical dependency between the main
sections.

§3§8§10§4§5§6§9§11Main Theorem§7.1, §7.2§7.3

### Acknowledgement

W. H. is supported by the National Key R&D Program of China (No.
2022YFA1007500) and the National Natural Science Foundation of China (No.
12288201). D. X. is supported by NSFC grant 12090015. Part of this work has
been done during D. X. visiting AMSS and the University of Chicago, D. X.
thanks AMSS and the University of Chicago for hospitality during the visiting.
We thank A. Avila for useful discussions and the suggestions to study Theorem
2.28. We thank A. Wilkinson for useful suggestions and pointing out an earlier
version of Theorem 1.4 can be improved. We thank De-Jun Feng, Shaobo Gan,
François Ledrappier, Pablo Lessa, Jialun Li, Minghui Ouyang, Wenyu Pan, Yi
Shi, Wenyuan Yang for useful discussions.

### Notation

We summarize here our notation and conventions.

$\log$ |  Logarithm with base $2.$
---|---
$d(\,\bm{\cdot}\,,\,\bm{\cdot}\,)$ |  The metric on $\SS^{1}.$
$B(x,\rho)$ |  The open ball with center $x$ and radius $\rho.$
$A^{(\rho)}$ |  The $\rho$-neighborhood of $A,$ i.e., $\bigcup_{x\in A}B(x,\rho)$.
$|I|$ |  Lebesgue measure of the interval $I.$
$tI$ |  The interval with the same center as $I$ and $t$-times its length, $t>0.$
$\mathrm{Homeo}_{+}(\SS^{1})$ | The group of orientation preserving circle homeomorphisms.
$\mathrm{Diff}_{+}^{k}(\SS^{1})$ |  The group of $C^{k}$ orientation preserving circle diffeomorphisms.
$C_{+}^{k}(I,I)$ |  The semigroup of $C^{k}$ orientation preserving maps on $I$ without critical points.
$\Lambda$ |  The minimal set of a subgroup of $\mathrm{Homeo}_{+}(\SS^{1})$ that has no finite orbits.
$\mu$ |  A finitely supported probability measure on $\mathrm{Diff}_{+}^{2}(\SS^{1})$ or $C_{+}^{2}(I,I).$
$\nu$ |  A probability measure on $\SS^{1}$ or $I.$ Usually take $\nu$ to be a stationary measure.
$\mathcal{S}$ |  A finite subset of $\mathrm{Diff}_{+}^{2}(\SS^{1})$ or $C_{+}^{2}(I,I),$ usually denotes the support of $\mu.$
$\mu^{*n}$ |  $n$-fold convolution of a measure $\mu$ with itself in a (semi)group.
$\mathcal{A}^{*n},\mathcal{A}^{*\leqslant n}$ |  Set of products of $n$ (resp. at most $n$) elements of a subset $\mathcal{A}$ in a (semi)group.
$T_{\mu}$ |  The semigroup generated by $\operatorname{\mathrm{supp}}\mu.$
$\Sigma,\Sigma^{+},\Sigma^{-}$ |  The underlying space $\Sigma=\mathcal{S}^{\mathbb{Z}}$ and $\Sigma^{+}=\mathcal{S}^{{\mathbb{Z}}_{\geqslant 0}},$ $\Sigma^{-}=\mathcal{S}^{{\mathbb{Z}}_{<0}},$ see Section 4.1.
$\omega,\omega^{+},\omega^{-}$ |  Elements in $\Sigma,\Sigma^{+},\Sigma^{-},$ respectively.
$\sigma$ |  The left shift map on $\Sigma$ or $\Sigma^{+}.$
$\bm{\mathrm{P}},\bm{\mathrm{P}}^{+},\bm{\mathrm{P}}^{-}$ |  The probability measure on the underlying spaces with $\bm{\mathrm{P}}=\mu^{\mathbb{Z}}$, see Section 4.1.
$F,F^{+},f_{\omega}^{n},f_{\omega^{+}}^{n}$ |  The cocycle over $(\Sigma,\sigma)$ or $(\Sigma^{+},\sigma),$ see Section 4.1.
$\pi^{\pm}$ |  Projections of $\Sigma$ down to $\Sigma^{\pm}.$
$P,Q$ |  Projections of $\Sigma\times\SS^{1}$ down to $\Sigma$ and $\SS^{1},$ respectively.
$\varkappa,\widetilde{\varkappa}$ |  The distortion coefficient and distortion norm on an interval, see Section 3.3.
$\mu^{+},\mu^{-}$ |  The probability measure $\mu$ and $\mu^{*(-1)},$ see Section 4.2.
$d,r$ |  The constants of a random walk induced by $\mu,$ see Theorem 4.12, 4.16.
$[d]$ |  The set $\left\\{0,\dotsc,d-1\right\\}$ which is equipped with addition modulo $d$.
$\nu_{i}^{\pm},i\in[d].$ |  The ergodic $\mu^{\pm}$-stationary measures, see Theorem 4.12.
$m_{i}^{\pm},i\in[d].$ |  The ergodci $u/s$-states on $\Sigma\times\SS^{1},$ see Theorem 4.12.
$\lambda_{i}^{\pm},i\in[d].$ |  The Lyapunov exponents correspond to $\nu_{i}^{\pm},$ see Theorem 4.16.
$\SS_{k}$ |  Family of $k$-element subsets of $\SS_{1}$ equipped with a metric, see Section 4.2.
$\Pi(\omega,i),\Xi(\omega,i)$ |  The maps from $\Sigma$ to $\SS_{r},$ see Theorem 4.12.
$\Pi(\omega),\Xi(\omega)$ |  The maps from $\Sigma$ to $\SS_{dr},$ see Theorem 4.12.
$W^{s}(\omega,i),W^{u}(\omega,i)$ |  The $s$-manifolds and $u$-manifolds, see (4.4) and Theorem 4.16.
$\Sigma_{\varepsilon}$ |  A set of uniform good words with parameter $\varepsilon>0,$ see Proposition 4.24.
$\underline{x}$ |  An element in $\SS_{k},$ which is a $k$-element subset of $\SS^{1}.$
$u_{\underline{x}}$ |  Uniform measure on $\underline{x}\subset\SS_{1},$ see Section 4.3.
$\underline{d}(\,\bm{\cdot}\,,\,\bm{\cdot}\,)$ |  The metric on $\SS_{k}$ induced by $d(\,\bm{\cdot}\,,\,\bm{\cdot}\,),$ see Section 4.3.
$\underline{\nu}$ |  A probability measure on $\SS_{k}.$
$\underline{\nu}_{i}^{\pm},i\in[d]$ |  The probability measures on $\SS_{r}$ corresponds to $\nu_{i}^{\pm},$ see Section 4.6.
$H(\mu)$ |  The discrete entropy of a finitely supported probability measure $\mu.$
$h_{\mathrm{RW}}(\mu),h_{\mathrm{F}}(\mu,\nu)$ |  The random walk entropy and Furstenberg entropy, see Definitions 2.3 and 2.1.
$H(\nu,\mathcal{A}),H(\nu,\mathcal{A}|\mathcal{B})$ | The Shannon entropy and conditional Shannon entropy.
$\mathcal{D}_{n}$ |  The level-$n$ dyadic partition of $\SS^{1}.$
$\dim\nu$ |  The exact dimension of a probability measure (if exists), see (2.1).
$H^{\alpha}(\,\bm{\cdot}\,)$ |  $\alpha$-Hausdorff outer measure.
$\dim_{\mathrm{H}}E,\dim_{\mathrm{H}}\nu$ |  Hausdorff dimension of a set $E$ or a measure $\nu.$
$\delta(G),\delta_{2}(G)$ |  The $C^{1},C^{2}$ dynamical critical exponents of $G,$ see Definition 1.2 and Section 2.2.
$\delta(G,\Delta)$ |  The dynamical critical exponent of $G$ on $\Delta$, see Section 2.2.
$\phi\ll\psi,\phi\gg\psi$ |  $\phi\leqslant C\psi$ (resp. $\psi\leqslant C\phi$) where $C>0$ is an absolute constant.
$\phi\ll_{\square}\psi,\phi\gg_{\square}\psi$ |  $\phi\leqslant C\psi$ (resp. $\psi\leqslant C\phi$) where the constant $C>0$ only depends on $\square$.
$\phi\asymp_{\square}\psi$ |  $\phi\ll_{\square}\psi$ and $\psi\ll_{\square}\phi.$

##  2 Statements of the main results

###  2.1 Dimension properties of stationary measures

##### Exact dimensionality of stationary measures.

In this paper, a probability measure on an underlying topological space always
refers to a Borel probability measure. Let $\nu$ be a probability measure on a
metric space. Recall that $\nu$ is said to be _exact dimensional_ if there
exists a constant $\alpha$ such that

$\lim_{\rho\to 0^{+}}\frac{\log\nu(B(x,\rho))}{\log\rho}=\alpha,\quad\
\nu\mathrm{-}\mathrm{a.e.}\ x.$ (2.1)

In this case, we call $\alpha$ the _exact dimension_ of $\nu$ and denote it by
$\dim\nu.$ In many situations, dynamically-defined measures (e.g. self-similar
measures of IFSs and stationary measures of random walks) are shown/believed
to be exact-dimensional. Works on this topic includes [49, 56, 10, 31, 40, 70,
53, 54].

Let $\mu$ be a finitely supported probability measure on
$\mathrm{Diff}_{+}^{1}(\SS^{1})$. A Borel probability measure $\nu$ on
$\SS^{1}$ is said to be _$\mu$ -stationary_ if

$\nu=\mu*\nu=\int f_{*}\nu\,\mathrm{d}\mu(f),$

where $f_{*}\nu$ is the pushforward of $\nu$ by the diffeomorphism $f$. We
call a $\mu$-stationary measure _ergodic_ if it cannot be written as a
nontrivial convex combination of two $\mu$-stationary measures. The _Lyapunov
exponent_ of an ergodic $\mu$-stationary measure $\nu$ is given by

$\lambda(\mu,\nu)=\iint\log
f^{\prime}(x)\,\mathrm{d}\mu(f)\,\mathrm{d}\nu(x).$ (2.2)

######  Definition 2.1.

Let $\nu$ be a $\mu$-stationary measure, the _Furstenberg entropy_ of
$(\mu,\nu)$ is defined by

$h_{\mathrm{F}}(\mu,\nu)\coloneqq\iint\log\frac{\mathrm{d}f_{*}\nu}{\mathrm{d}\nu}(x)\,\mathrm{d}f_{*}\nu(x)\mathrm{d}\mu(f).$

In this paper, we show the exact dimensionality of stationary measures for
general $C^{2}$ random walks on circle, generalizing the results in [40, 49]
to the smooth setting.

######  Theorem 2.2.

Let $\mu$ be a finitely supported probability measure on
$\mathrm{Diff}_{+}^{2}(\SS^{1})$ such that $\operatorname{\mathrm{supp}}\mu$
does not preserve any probability measure on $\SS^{1}.$ Let $\nu$ be an
ergodic $\mu$-stationary measure on $\SS^{1},$ then $\nu$ is exact dimensional
and

$\dim\nu=\frac{h_{\mathrm{F}}(\mu,\nu)}{|\lambda(\mu,\nu)|}.$

Here we make the assumption that the support of $\mu$ does not preserve any
invariant measure. While this assumption may appear restrictive, it is
actually quite mild. If it were not satisfied, then either the support of
$\mu$ would preserve a finite orbit, or it would lie within an abelian group.
In the latter case, the associated stationary measure could be non-exact
dimensional if its rotation number is Liouville, as shown in [78].

##### Dimension formulas for smooth actions on the circle and the interval.

In general, computing the Furstenberg entropy mentioned in Theorem 2.2 seems
to be difficult. In practice, it may be necessary to use alternative
quantities to determine the exact dimension of the stationary measure. Under
certain discreteness or separation assumptions, the random walk entropy can be
a viable candidate to substitute for $h_{\mathrm{F}}$ in Theorem 2.2.

######  Definition 2.3.

For a finitely supported measure $\mu$ in a group $G$, the _Shannon entropy_
$H(\mu)$ of $\mu$ is given by
$-\sum_{f\in\operatorname{\mathrm{supp}}\mu}\mu(f)\log\mu(f)$. Let $\mu^{*n}$
denote the $n$-fold convolution of $\mu$ in $G$. The _random walk entropy_ of
$\mu$ is then defined as

$h_{\rm{RW}}(\mu)\coloneqq\lim_{n\to\infty}\frac{1}{n}H(\mu^{\ast n}).$

The limit is guaranteed to exist by sub-additivity, as shown in [46]. The
random walk entropy can capture to which extend the semigroup generated by
$\operatorname{\mathrm{supp}}\mu$ is free. Specifically, we have
$h_{\mathrm{RW}}(\mu)\leqslant H(\mu)$, and the two are equal if and only if
the semigroup generated by $\operatorname{\mathrm{supp}}\mu$ is generated
freely by it.

In general, $h_{\mathrm{RW}}$ provides an upper bound for $h_{\mathrm{F}}$, as
is shown, for example, in [40]. Moreover, two quantities coincide for random
walks on flag varieties by discrete subgroups of linear groups [45, 46, 51],
see also [32]. Under a local discreteness condition, we extend this equality
to groups of circle diffeomorphisms. Let us first recall the definition of
local discreteness.

######  Definition 2.4.

A group $G\subset\mathrm{Diff}^{1}_{+}(\SS^{1})$ is called _$C^{1}$ -locally
discrete_ (abbreviate to _locally discrete_) if for any interval
$I\subset\SS^{1}$, there does not exists sequence $(g_{n})\subset G$ of
distinct elements such that $g_{n}|_{I}\to\rm{id}$ in the $C^{1}$-topology.

The local discreteness condition is implicitly or explicitly used in [34, 35,
73, 74, 23, 24, 2, 3], particularly in the case of actions of subgroups of
$\mathrm{Diff}^{\omega}(\SS^{1})$. In the analytic setting, local non-
discreteness usually implies the existence of a local flow in the $C^{1}$
local closure of the group action [73, 74], and it often implies that the
group acts minimally on $\SS^{1}$ [59, Proposition 3.2].

######  Remark 2.5.

The authors of [2, 3] considered a slightly different definition of local
discreteness. They only consider intervals $I$ that have non-empty
intersection with the group invariant minimal set. Basically, all of the
results in our paper that involve the discreteness assumption can be
strengthened to the case in [2, 3] without much difficulty, after some
necessary modifications of the statements.

######  Theorem 2.6.

Let $\mu$ be a finitely supported probability measure on
$\mathrm{Diff}^{2}_{+}(\SS^{1}).$ Assume that the group generated by
$\operatorname{\mathrm{supp}}\mu$ is locally discrete and has no finite
orbits. Then
$h_{\mathrm{F}}(\mu,\nu)=h_{\mathrm{RW}}(\mu)\leqslant|\lambda(\mu,\nu)|$ for
every $\mu$-stationary measure $\nu.$ Consequently, every ergodic
$\mu$-stationary measure is of exact dimension
$\frac{h_{\mathrm{RW}}(\mu)}{|\lambda(\mu,\nu)|}.$

This follows from a stronger version as below. For a closed interval $I,$ we
denote $C_{+}^{2}(I,I)$ to be the semigroup of $C^{2}$ orientation preserving
maps on $I$ that have no critical point. For a probability measure $\mu$
supported on a semigroup, we denote by $T_{\mu}$ the semigroup generated by
$\operatorname{\mathrm{supp}}\mu$.

######  Theorem 2.7.

Let $\mu$ be a finitely supported probability measure on
$\mathrm{Diff}^{2}_{+}(\mathbb{S}^{1})$ such that
$\operatorname{\mathrm{supp}}\mu$ does not preserve any probability measure on
$\SS^{1}$. Let $\nu$ be an ergodic $\mu$-stationary measure on $\SS^{1}$. Then
$\nu$ is exact dimensional and

  1. (1)

either $\dim\nu=\frac{h_{\mathrm{RW}}(\mu)}{|\lambda(\mu,\nu)|},$

  2. (2)

or there exists a closed interval $J\subset\SS^{1}$ and two sequence of
elements $\\{g_{n}\\},\\{f_{n}\\}\subset T_{\mu}$ with $g_{n}\neq f_{n},$ such
that $g_{n}^{-1}f_{n}$ tends to $\mathrm{id}$ on $J$ in the $C^{1}$-topology.

Moreover, the conclusion holds when replacing $\SS^{1}$ with a closed interval
$I$ and $\mathrm{Diff}_{+}^{2}(\SS^{1})$ with $C_{+}^{2}(I,I),$ where the
elements $g_{n}$ and $f_{n}$ found in the second case additionally satisfy
$f_{n}(J)\subset g_{n}(I),$ which ensures that $g_{n}^{-1}f_{n}$ is well-
defined on $J.$

######  Remark 2.8.

The item (2) is similar in spirit with the notion of weak separation condition
defined in [47] in the context of self-similar IFSs.

Theorem 2.7 specialized to the setting of Furstenberg measures on
${\mathbb{R}}{\mathbb{P}}^{1}$ is weaker than the main result of Hochman-
Solomyak [40]. They showed that the dimension formula holds unless there
exists a sequence of maps $g_{n}^{-1}\circ f_{n}$ tending to $\mathrm{id}$
super-exponentially fast, similarly to the result of [39].

###  2.2 Dimension properties of minimal sets

##### Variational principle for dimensions.

For a Borel probability measure $\nu$ on a metric space, recall that its
Hausdorff dimension is defined by

$\dim_{\mathrm{H}}\nu=\inf\left\\{\,\dim_{\mathrm{H}}E:\nu(E)>0\,\right\\},$

where $\dim_{\mathrm{H}}E$ denotes the Hausdorff dimension of the Borel set
$E.$ A well-known result of Young [86] says that whenever $\nu$ is exact
dimensional, we have $\dim_{\mathrm{H}}\nu=\dim\nu$.

By definition, we clearly have
$\dim_{\mathrm{H}}\nu\leqslant\dim_{\mathrm{H}}\operatorname{\mathrm{supp}}\nu$.
Now, when dealing with the minimal set of a semigroup action, a natural
question arises: can we find a random walk on this semigroup whose stationary
measure approximates the minimal set in terms of Hausdorff dimensions? This
type of variational principle for contracting IFSs has been studied in [31].
We consider this variational principle of dimensions for smooth group actions
on the circle.

######  Theorem 2.9.

Let $G\subset\mathrm{Diff}_{+}^{\omega}(\SS^{1})$ be a finitely generated
subgroup which preserves an exceptional minimal set $\Lambda.$ Then

$\dim_{\mathrm{H}}\Lambda=\sup\left\\{\dim_{\mathrm{H}}\nu:\begin{aligned} &\
\nu\text{ is an ergodic $\mu$-stationary measure on }\Lambda,\\\ &\
\text{where }\mu\text{ is a finitely supported probability measure on
}G\end{aligned}\right\\}.$

As a byproduct of the proof of Theorem 2.9, we show the following result,
which asserts that we can always approximate the Hausdorff dimension of
$\Lambda$ by attractors of contracting IFSs.

######  Theorem 2.10.

Let $G\subset\mathrm{Diff}_{+}^{\omega}(\SS^{1})$ be a finitely generated
subgroup which preserves an exceptional minimal set $\Lambda.$ For every
$\varepsilon>0,$ there exists a finite subset
$\left\\{g_{i}\right\\}_{i=1}^{\ell}\subset G$ and a closed interval
$I\subset\SS^{1}$ such that

  1. (1)

$\left\\{g_{i}\right\\}_{i=1}^{\ell}$ defines a contracting IFS on $I,$ namely
for every $1\leqslant i\leqslant\ell,$ $g_{i}(I)\subset I$ and
$(g_{i})^{\prime}|_{I}<1;$

  2. (2)

$\left\\{g_{i}|_{I}\right\\}_{i=1}^{\ell}$ satisfies the open set condition,
namely $g_{i}(I)\cap g_{j}(I)=\varnothing$ for every $i\neq j;$

  3. (3)

the attractor $\Lambda^{\prime}$ of
$\left\\{g_{i}|_{I}\right\\}_{i=1}^{\ell},$ which is given by

$\Lambda^{\prime}=\bigcap_{n=1}^{\infty}\bigcup_{1\leqslant
i_{1},\cdots,i_{n}\leqslant\ell}g_{i_{n}}\circ\cdots\circ g_{i_{1}}(I),$

is contained in $\Lambda$ and
$\dim_{\mathrm{H}}\Lambda^{\prime}\geqslant\dim_{\mathrm{H}}\Lambda-\varepsilon.$

The case for groups of real analytic circle diffeomorphisms is derived from a
more general version applicable to $C^{2}$ diffeomorphisms. The general
version requires some technical assumptions : properties $(\star)$ and
$(\Lambda\star)$, as introduced by Deroin-Kleptsyn-Navas [23]. These
properties are expected to hold for most finitely generated subgroups
$G\subset\mathrm{Diff}_{+}^{2}(\SS^{1})$ without finite orbits and is verified
in the real analytic cases in [24]. Further discussions will be presented in
Section 3.1.

######  Definition 2.11.

A point $x\in\SS^{1}$ is _non-expandable_ for the action of
$G\subset\mathrm{Diff}_{+}^{1}(\SS^{1})$ if $g^{\prime}(x)\leqslant 1$ for
every $g\in G.$ We denote by $\operatorname{NE}=\operatorname{NE}(G)$ the set
of non-expandable points.

######  Definition 2.12.

We say $G\subset\mathrm{Diff}^{1}(\SS^{1})$ satisfies _property $(\star)$_ if
it acts minimally and for every $x\in\operatorname{NE},$ there exist
$g_{+},g_{-}\in G$ such that $g_{+}(x)=g_{-}(x)=x$ and $x$ is an isolated-
from-the-right (resp. isolated-from-the-left) point of the set of fixed points
$\operatorname{Fix}(g_{+})$ (resp. $\operatorname{Fix}(g_{-})$).

######  Definition 2.13.

We say $G\subset\mathrm{Diff}^{1}(\SS^{1})$ satisfies _property
$(\Lambda\star)$_ if it admits an exceptional minimal set $\Lambda$ for every
$x\in\operatorname{NE}\cap\Lambda,$ there exists $g_{+},g_{-}\in G$ such that
$g_{+}(x)=g_{-}(x)=x$ and $x$ is an isolated-from-the-right (resp. isolated-
from-the-left) point of the set of fixed points $\operatorname{Fix}(g_{+})$
(resp. $\operatorname{Fix}(g_{-})$).

######  Theorem 2.14.

Let $G\subset\mathrm{Diff}_{+}^{2}(\SS^{1})$ be a finitely generated subgroup
which does not preserve any probability measure and satisfies property
$(\star)$ or $(\Lambda\star)$. Let $\Lambda$ be the unique minimal set of $G$,
then

$\dim_{\mathrm{H}}\Lambda=\sup\left\\{\dim_{\mathrm{H}}\nu:\begin{aligned} &\
\nu\text{ is an ergodic $\mu$-stationary measure on }\Lambda,\\\ &\
\text{where }\mu\text{ is a finitely supported probability measure on
}G\end{aligned}\right\\}.$ (2.3)

##### Dynamical critical exponents.

Recall the notion of the dynamical critical exponent introduced in Definition
1.2. The following result reveals the relation between the dynamical critical
exponent and the Hausdorff dimension of the minimal set for general group
actions on the circle by $C^{2}$ diffeomorphisms.

######  Theorem 2.15.

Let $G\subset\mathrm{Diff}_{+}^{2}(\SS^{1})$ be a finitely generated subgroup
without finite orbits and let $\Lambda$ be its unique minimal set. We have the
following:

  1. (1)

If $G$ satisfies property $(\star)$ or $(\Lambda\star)$, then
$\delta(G)\geqslant\dim_{\rm H}\Lambda.$

  2. (2)

If $G$ is locally discrete and virtually free, then
$\delta(G)\leqslant\dim_{\rm H}\Lambda.$

In particular, if both assumptions hold, we have
$\delta(G)=\dim_{\mathrm{H}}\Lambda.$

Recall that in the definition of a dynamical critical exponent (Definition
1.2), the point $x$ is required to be in the set $\Lambda$. This may not
always be desirable, as it restricts the applicability of the concept. This
raises the question of whether there exists an a priori criterion to determine
if a group action is minimal via a variant of the dynamical critical exponent.
To address this issue, we generalize the concept of dynamical critical
exponents to arbitrary subsets of $\SS^{1},$ thus allowing a more flexible and
useful tool for analyzing the dynamics of group actions.

######  Definition 2.16.

Let $G\subset\mathrm{Diff}_{+}^{1}(\SS^{1})$ be a subgroup. For a subset
$\Delta\subset\SS^{1},$ we define the _dynamical critical exponent of $G$ on
$\Delta$_ as

$\delta(G,\Delta)\coloneqq\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log\\#\left\\{g\in G:\exists
x\in\Delta,g^{\prime}|_{B(x,\varepsilon)}\geqslant 2^{-n}\right\\}.$

######  Remark 2.17.

The discussions in Section 11 reveal that for the real analytic case, the
dynamical critical exponent at a point outside of $\Lambda$ can be viewed as
the exponent of convergence for certain Poincaré series. Likewise, the
dynamical critical exponent $\delta(G)$ can be interpreted as the exponent of
convergence for certain series. This interpretation is utilized in the proof
of the third item in our main theorem, as described in Section 10.2.

For a subgroup $G\subset\mathrm{Diff}^{1}_{+}(\SS^{1})$ without finite orbits,
we note that $\delta(G,\Lambda)=\delta(G)$ where $\Lambda$ is its unique
minimal set. Besides, if we take $\Delta=\SS^{1}$, the dynamical critical
exponent $\delta(G,\SS^{1})$ is a quantity independent with $\Lambda.$ The
following theorem is a stronger version of the first item in our main theorem,
which also provides a criterion for the $G$-action being minimal by asking
whether $\delta(G,\SS^{1})\geqslant 1.$

######  Theorem 2.18.

Let $G\subset\mathrm{Diff}_{+}^{\omega}(\SS^{1})$ be a finitely generated
subgroup with an exceptional minimal set $\Lambda.$ Then

$\dim_{\mathrm{H}}\Lambda=\delta(G)\leqslant\delta(G,\SS^{1})<1.$

More precisely, we have

$\delta(G,\SS^{1})=\max\left\\{\dim_{\mathrm{H}}\Lambda,~{}\sup_{x\in\SS^{1}}\frac{k(x)}{k(x)+1}\right\\}=\max\left\\{\dim_{\mathrm{H}}\Lambda,~{}\max_{x\in\SS^{1}\setminus\Lambda}\frac{k(x)}{k(x)+1}\right\\},$

where $k(x)$ is defined in Definition 11.8, which relates to the multiplicity
at $x.$

######  Corollary 2.19.

Let $G\subset\mathrm{Diff}_{+}^{\omega}(\SS^{1})$ be a finitely generated
subgroup without finite orbits. Assume that $\delta(G,\SS^{1})\geqslant 1,$
then $G$ acts minimally on the circle.

To relax assumptions on the group structure as in Theorem 2.15, we introduce
another notion of dynamical critical exponent, the $C^{2}$-dynamical critical
exponent.

######  Definition 2.20.

Let $G\subset\mathrm{Diff}_{+}^{2}(\SS^{1})$ be a finitely generated subgroup
without finite orbits. Let $\Lambda$ be its unique minimal set. We define the
_$C^{2}$ -dynamical critical exponent_ of $G$ as

$\delta_{2}(G)\coloneqq\lim_{\varepsilon\to
0^{+}}\lim_{C\to+\infty}\limsup_{n\to+\infty}\frac{1}{n}\log\\#\left\\{\,g\in
G:\exists x\in\Lambda,g^{\prime}|_{B(x,\varepsilon)}\geqslant
2^{-n},\widetilde{\varkappa}(g,B(x,\varepsilon))\leqslant C\,\right\\},$

where $\widetilde{\varkappa}(g,I)\coloneqq\sup_{x\in I}(\log
g^{\prime}(x))^{\prime}$ controls the distortion, as explained in Section 3.3.

######  Theorem 2.21.

Let $G\subset\mathrm{Diff}_{+}^{2}(\SS^{1})$ be a finitely generated, locally
discrete subgroup without finite orbits. Let $\Lambda$ be the unique minimal
set and assume that $G$ satisfies property $(\star)$ or $(\Lambda\star).$ Then

$\dim_{\mathrm{H}}\Lambda=\delta_{2}(G).$

##### Conformal measures.

The notion of conformal measure is a powerful tool for studying conformal
dynamics, particularly in one-dimensional dynamics. This concept was initially
explored in the context of hyperbolic geometries. Patterson introduced a
measure that satisfies a specific homogeneous condition to study limit sets of
Fuchsian groups, now known as the Patterson-Sullivan measure, in [68]. Later,
Sullivan provided a general definition of conformal measures to investigate
the fractal geometry of limit sets in conformal dynamics, as detailed in [82].
He also constructed a conformal measure for rational maps on the Julia set.
Let us first review the definition of conformal measures.

######  Definition 2.22.

Let $G$ be a group of conformal transformations. A measure $\nu$ on the
underlying space is said to be _conformal_ with exponent $\delta$ (or simply
$\delta$-conformal), if for every Borel set $A$ and for every map $g\in G$ one
has

$\nu(gA)=\int_{A}|g^{\prime}(x)|^{\delta}\mathrm{d}\nu(x).$

We focus on actions of subgroups $G\subset\mathrm{Diff}_{+}^{2}(\SS^{1})$
having an exceptional minimal set $\Lambda.$ Conformal measures are useful to
understand the fractal $\Lambda.$ To do so, it is important to know whether
there exist atomless conformal measures supported on $\Lambda$. In the special
case where $\operatorname{NE}(G)=\varnothing,$ the techniques from [82] allows
to show the existence of atomless $\delta$-conformal measures $\Lambda$.
Furthermore, $\delta$ is equal to the Hausdorff dimension of $\Lambda$ and
$0<\delta=\dim_{\mathrm{H}}\Lambda<1$.

Dropping the condition of $\operatorname{NE}(G)=\varnothing$, it is unknown
whether atomless conformal measures supported on $\Lambda$ exist.
Nevertheless, it is shown in [23] that if that is the case then the value of
the exponent $\delta$ is unique and strictly less than $1$.

In this paper, we establish the following.

######  Theorem 2.23.

Let $G\subset\mathrm{Diff}_{+}^{2}(\SS^{1})$ be a finitely generated subgroup
with an exceptional minimal set $\Lambda.$ Assume that $G$ satisfies property
$(\Lambda\star).$ If there exists an atomless $\delta$-conformal measure
supported on $\Lambda$, then $\delta=\dim_{\mathrm{H}}\Lambda.$

This result extends Sullivan’s result [82] to the case where
$\operatorname{NE}(G)\neq\varnothing$, and also directly implies the
uniqueness of $\delta$ which was shown in [23].

Moving forward, we are able to construct atomless conformal measures on
exceptional minimal sets in certain cases, thanks to the notion of pointwise
dynamical critical exponents.

######  Theorem 2.24.

Let $G\subset\mathrm{Diff}_{+}^{\omega}(\SS^{1})$ be a finitely generated
subgroup with an exceptional minimal set $\Lambda.$ Assume that the stabilizer
of any point $x\in\SS^{1}\setminus\Lambda$ in $G$ is trivial. Then there
exists an atomless $\delta$-conformal measure supported on $\Lambda$ with
$\delta=\dim_{\mathrm{H}}\Lambda.$

######  Remark 2.25.

Notice that the triviality condition of the stabilizers in Theorem 2.24 is
rather mild, see Remark 10.5. For example it holds for any non elementary
Fuchsian group.

A general version of Theorem 2.24 combined with [23, Theorem F] allows us to
obtain a precise estimate of the Hausdorff dimension of exceptional minimal
sets, which implies the first item in the main theorem.

###  2.3 Applications of random walk structures and approximation results

As we have mentioned in the introduction, we establish a structure theorem for
smooth random walks. We list two statements of its applications.

######  Theorem 2.26.

Let $T\subset\mathrm{Diff}^{2}_{+}(\SS^{1})$ be a semigroup without finite
orbits. Then $T^{-1}\coloneqq\left\\{f^{-1}:f\in T\right\\}$ has the same
number of minimal sets with $T$.

######  Theorem 2.27.

111Note that the $C^{2}$ assumption of this theorem can be relaxed to $C^{1}$.
The $C^{1}$ case is more complicated and will be proved in a forthcoming paper
of the second and third author.

Let $T\subset\mathrm{Diff}^{2}(\SS^{1})$ be a semigroup with no invariant
probability measure on $\SS^{1}.$ Then there exists a perfect pingpong pair
$(h_{1},h_{2})\subset T.$

Here, the perfect pingpong pair is defined in Definition 7.3, which can be
viewed as a subsystem with the strongest hyperbolicity. A perfect pingpong
pair has clear dynamics, which allow us to predict the behavior for every
point. However, the subsystem generated by this pair of elements may not be
_large_ enough comparing to the whole system.

Another problem worth considered is how large a uniformly hyperbolic subsystem
could be. A famous theorem by A. Katok [44] asserts that any ergodic
hyperbolic measure of a $C^{1+}$ diffeomorphism can be approximated by a
horseshoe with approximately the same entropy. In [5, Theorem 3.3], Avila,
Crovisier and Wilkinson further showed that it is possible to let the
horseshoe have a dominated splitting, with approximately the same Lyapunov
exponents. In the case of $2$D matrix valued cocycle over full shift of finite
type, Morris and Shmerkin [63] showed that any cocycle with distinct Lyapunov
exponents can be approximated by a subsystem with approximately the same
entropy and Lyapunov exponents and having additionally a dominated splitting.
Notice that dominated splitting can be translated into the cone field
conditions for dynamics on projective spaces (cf. [4]).

We show an analogue of these results in the setting of random walks by circle
diffeomorphisms. There are three fundamental quantities characterizing a
random walk: the entropy, the Lyapunov exponent, and the dimension of the
stationary measure. In this paper, we aim to approximate all three quantities
simultaneously using a uniformly hyperbolic subsystem. We present two versions
here.

For a positive integer $n$ and a subset $\mathcal{S}$ of a (semi)group, we
write $\mathcal{S}^{*n}=\left\\{g_{1}\cdots
g_{n}:g_{i}\in\mathcal{S}\right\\}.$

######  Theorem 2.28.

Let $\mathcal{S}\subset\mathrm{Diff}_{+}^{2}(\SS^{1})$ be a finite set without
common invariant probability measures, then there exists positive integers
$d,r>0$ such that the following holds. For every nondegenerate probability
measure $\mu$ supported on $\mathcal{S}$ and $\varepsilon>0$, there exists
arbitrarily large $N$ and a subset $\Gamma\subset\mathcal{S}^{*N}$ such that:

  1. (1)

There are exactly $d$ ergodic $\mu$-stationary measures $\nu_{i}$. The
Lyapunov exponent $\lambda_{i}$ of each $\nu_{i}$ is negative.

  2. (2)

$\\#\\{\,(f_{1},\cdots,f_{N})\in\mathcal{S}^{N}:f_{N}\cdots
f_{1}\in\Gamma\,\\}\geqslant 2^{N(H(\mu)-\varepsilon)}$.

  3. (3)

There exists $dr$ disjoint open intervals
$\left\\{U_{i,j}\right\\}_{1\leqslant i\leqslant d,\\\ 1\leqslant j\leqslant
r}$ such that for every $(i,j)$, for all $f\in\Gamma$, we have
$f\overline{U_{i,j}}\subset U_{i,j}$ and
$f^{\prime}(x)\in[2^{N(\lambda_{i}-\varepsilon)},2^{N(\lambda_{i}+\varepsilon)}]$
for all $x\in U_{i,j}$,

  4. (4)

The semigroup generated by $\Gamma$ has a unique minimal set $K_{i,j}$ in each
$U_{i,j}$ with $\dim_{\mathrm{H}}K_{i,j}\geqslant\dim\nu_{i}-\varepsilon.$

It is worth noting that the numbers $d$ and $r$ depend only on the set
$\mathcal{S}$. More precisely, $d$ is equal to the number of minimal set of
the semigroup generated by $\mathcal{S}$. The number $r$ has also a dynamical
meaning, see Theorems 4.12, 4.16 and 4.15 for more information.

######  Theorem 2.29.

Let $\mu$ be finitely supported probability measure on
$\mathrm{Diff}_{+}^{2}(\SS^{1})$ without invariant probability measures on
$\SS^{1}.$ Let $\mathcal{S}=\operatorname{\mathrm{supp}}\mu$ and assume that
the group generated by $\mathcal{S}$ is locally discrete. Let $\nu$ be an
ergodic $\mu$-stationary measure and $\lambda=\lambda(\mu,\nu)<0.$ Then for
every $\varepsilon>0$ sufficiently small, there exists an arbitrarily large
integer $N$ and a subset $\Gamma\subset\mathcal{S}^{*N}$ such that:

  1. (1)

The cardinality of $\Gamma$ is at least
$2^{N(h_{\mathrm{RW}}(\mu)-\varepsilon)}.$

  2. (2)

There exists an open interval $U\subset\SS^{1}$ which is strictly preserved by
every $f\in\Gamma.$

  3. (3)

The closures of $f(U)$ are pairwise disjoint for $f\in\Gamma.$

  4. (4)

For every $f\in\Gamma,$ $x\in U,$
$f^{\prime}(x)\in[2^{N(\lambda-\varepsilon)},2^{N(\lambda+\varepsilon)}].$

  5. (5)

The semigroup generated by $\Gamma$ has a unique minimal set $K\subset U$ with
a Hausdorff dimension at least $(\dim\nu-\varepsilon).$

The discreteness condition is necessary because conditions (1), (3), and (4)
imply that $-\lambda\leqslant h_{\mathrm{RW}}(\mu)$. Furthermore,
$h_{\mathrm{RW}}(\mu)$ cannot be replaced by $H(\mu)$ because it is possible
for $\\#\mathcal{S}^{*N}$ to be much less than $2^{NH(\mu)}$.

##  3 Preliminaries

###  3.1 Group actions on the circle

We list some useful results about groups acting on the circle by
diffeomorphisms in this subsection. The following lemma is well-known. For a
proof, see for example [2, Lemma 7.12].

######  Lemma 3.1.

Let $G\subset\mathrm{Homeo}(\SS^{1})$ be a subgroup without finite orbits on
$\SS^{1}$. If $G_{1}$ is a subgroup of finite index in $G$, then the unique
minimal set of $G$ is also the unique minimal set of $G_{1}.$

The following is also a well-known lemma. We include a proof for the reader’s
convenience.

######  Lemma 3.2.

Let $G\subset\mathrm{Diff}^{2}_{+}(\SS^{1})$ be a finitely generated subgroup.
If $G$ preserves a probability measure $\nu$ then either it acts minimally or
it has a finite orbit on $\SS^{1}$.

###### Proof.

Note that for all $x\in\SS^{1}$, we have $\nu([x,f(x)[)=\rho(f),$ where
$\rho(f)$ is the rotation number. By Denjoy’s theorem, if there exists $f\in
G$ with irrational $\rho(f)$ then $G$ acts minimally. Hence we may assume that
every element in $G$ has a rational rotation number. Choose a point
$x_{0}\in\operatorname{\mathrm{supp}}\nu$. Note that if
$N\rho(f)\in{\mathbb{Z}}$ for some integer $N>0$ then $f$ preserves the set
$\left\\{\,y\in\operatorname{\mathrm{supp}}\nu:\nu([x_{0},y[)\in\frac{1}{N}{\mathbb{Z}}\,\right\\}$,
which is finite. Taking $N$ to be a common multiple of the denominators of the
rotation numbers of a finite generating set of $G$, we obtain a finite set
which is preserved by $G$. ∎

We recall the concepts of non-expandable points, properties $(\star)$ and
$(\Lambda\star)$ mentioned in Section 2.2. We present some related results
shown by Deroin-Navas-Kleptsyn [23, 24].

######  Theorem 3.3 ([23, Theorems A, D]).

Let $G\subset\mathrm{Diff}_{+}^{2}(\SS^{1})$ be a finitely generated subgroup
with the minimal set $\Lambda$. Assume that $G$ satisfies property $(\star)$
or $(\Lambda\star).$ Then

  1. (1)

The set $\Lambda\cap\operatorname{NE}$ is finite.

  2. (2)

For each $x\in\Lambda\setminus G(\operatorname{NE}),$ the set of derivatives
$\left\\{g^{\prime}(x):g\in G\right\\}$ is unbounded.

######  Theorem 3.4 ([24, Main Theorem]).

Let $G$ be a finitely generated subgroup of
$\mathrm{Diff}_{+}^{\omega}(\SS^{1}).$

  1. (1)

If $G$ is free of rank $\geqslant 2$ and acts minimally on $\SS^{1},$ then it
satisfies property $(\star).$

  2. (2)

If $G$ acts on the circle with an exceptional minimal set $\Lambda,$ then it
satisfies property $(\Lambda\star).$

We will also make use of the following theorem due to Hector [37] (see also
[64]).

######  Theorem 3.5 ([37]).

If $G$ is a subgroup of $\mathrm{Diff}_{+}^{\omega}(\SS^{1})$ having an
exceptional minimal set, then the stabilizer of any point in $\SS^{1}$ is
either trivial or infinite cyclic.

The following is a consequence of Hector’s theorem. A proof can be found in
[59, Proposition 3.2].

######  Corollary 3.6.

Let $G\subset\mathrm{Diff}_{+}^{\omega}(\SS^{1})$ be a finitely generated,
locally nondiscrete group without finite orbits, then $G$ acts minimally on
$\SS^{1}.$

###  3.2 Dimension and entropy

Let $E$ be a subset of a metric space $X$, for $\alpha>0,$ the
$\alpha$-Hausdorff outer measure is defined as

$H^{\alpha}(E)\coloneqq\lim_{\rho\to 0^{+}}H^{\alpha}_{\rho}(E),$

where

$H^{\alpha}_{\rho}(E)\coloneqq\inf\left\\{\sum_{n=1}^{\infty}(\operatorname{\mathrm{diam}}U_{n})^{\alpha}:(U_{n})\text{
is a countable cover of }E,\ \operatorname{\mathrm{diam}}U_{n}<\rho\right\\}.$

Then there exists a unique constant $\alpha_{0}\geqslant 0$ such that
$H^{\alpha}(E)=\infty$ for every $\alpha<\alpha_{0}$ and $H^{\alpha}(E)=0$ for
every $\alpha>\alpha_{0}.$ The constant $\alpha_{0}$ is called the Hausdorff
dimension of $E,$ denoted by $\dim_{\mathrm{H}}E.$

The following lemma is a crucial technical tool used throughout this paper. It
will be applied to several different dynamically defined covers.

######  Lemma 3.7.

Let $C>0$ and $\lambda>0$ be parameters. For $n\in{\mathbb{N}}$, let
$\mathcal{E}_{n}$ be a collection of intervals in $\SS^{1}$ of length at most
$C2^{-\lambda n}$. Let
$E=\limsup_{n\to+\infty}\bigcup_{J\in\mathcal{E}_{n}}J$. Let
$\widetilde{\mathcal{E}}_{n}\subset\mathcal{E}_{n}$ be a maximal subcollection
consisting of pairwise disjoint intervals. Then

$\limsup_{n\to+\infty}\frac{1}{n}\log\\#\widetilde{\mathcal{E}}_{n}\geqslant\lambda\dim_{\mathrm{H}}E.$

###### Proof.

For every $J\in\mathcal{E}_{n}\setminus\widetilde{\mathcal{E}}_{n}$, by the
maximality of $\widetilde{\mathcal{E}}_{n}$, there is
$I\in\widetilde{\mathcal{E}}_{n}$ such that $J\cap I\neq\varnothing$. Then
$J\subset\widetilde{I}$ where $\widetilde{I}$ denotes the interval of the same
center as $I$ and of length $C2^{-\lambda n+2}$. It follows that

$\bigcup_{J\in\mathcal{E}_{n}}J\subset\bigcup_{I\in\widetilde{\mathcal{E}}_{n}}\widetilde{I}.$

Given $\rho>0$, for any $N\geqslant\lambda^{-1}(|\log\rho|+\log C+2)$, we have

$E\subset\bigcup_{n\geqslant
N}\bigcup_{I\in\widetilde{\mathcal{E}}_{n}}\widetilde{I},$

which is a cover of $E$ by intervals of length at most $\rho$. Thus for any
$s>0$ and $\rho>0$,

$H_{\rho}^{s}(E)\leqslant\sum_{n\geqslant
N}\sum_{I\in\widetilde{\mathcal{E}}_{n}}|\widetilde{I}|^{s}\\\
\ll_{C,s}\sum_{n\geqslant N}2^{-s\lambda n}\\#\widetilde{\mathcal{E}}_{n}.$

The right-hand side is the tail of a convergent series whenever
$-s\lambda+\beta<0$ where
$\beta=\limsup_{n\to+\infty}\frac{1}{n}\log\\#\widetilde{\mathcal{E}}_{n}$. In
this case, $H^{s}(E)=0$. Hence
$\dim_{\mathrm{H}}E\leqslant\frac{\beta}{\lambda}$. ∎

Let $\nu$ be a Borel probability measure on $X$ and $\mathcal{A}$ a finite
measurable partition of $X.$ The Shannon entropy of $\nu$ with respect to
$\mathcal{A}$ is

$H(\nu,\mathcal{A})\coloneqq-\sum_{A\in\mathcal{A}}\nu(A)\log\nu(A).$

Note that $H(\nu,\mathcal{A})\leqslant\log\\#\mathcal{A}.$ Let $\mathcal{B}$
be another finite measurable partition on $X,$ the conditional entropy is

$H(\nu,\mathcal{A}|\mathcal{B})\coloneqq-\sum_{B\in\mathcal{B}}\sum_{A\in\mathcal{A}}\nu(A\cap
B)\log\frac{\nu(A\cap B)}{\nu(B)}.$

Similarly, an upper bound for the conditional entropy is given by

$H(\nu,\mathcal{A}|\mathcal{B})\leqslant\max_{B\in\mathcal{B}}\log\\#\left\\{A\in\mathcal{A}:\nu(A\cap
B)>0\right\\}.$ (3.1)

If $\mu$ is a probability measure with finite support, we use $H(\mu)$ and
$H(\mu|\mathcal{B})$ to denote $H(\mu,\mathcal{A})$ and
$H(\mu,\mathcal{A}|\mathcal{B})$, respectively, where $\mathcal{A}$ is the
discrete partition. This notation is used only for finitely supported
probability measures on $\mathrm{Diff}_{+}^{1}(\SS^{1})$ in this paper.

Fix a finite measurable partition $\mathcal{A}$, the function $\nu\mapsto
H(\nu,\mathcal{A})$ is concave and almost convex: for any probability vector
$\bm{p}=(p_{1},\cdots,p_{k})$ and any Borel probability measures
$\nu_{1},\dotsc,\nu_{k}$, we have

$\sum_{i=1}^{k}p_{i}H(\nu_{i},\mathcal{A})\leqslant
H\left(\sum_{i=1}^{k}p_{i}\nu_{i},\mathcal{A}\right)\leqslant\sum_{i=1}^{k}p_{i}H(\nu_{i},\mathcal{A})+H(\bm{p}).$
(3.2)

where $H(\bm{p})=-\sum p_{i}\log p_{i}$.

We often consider the entropy with respect to dyadic partitions on $\SS^{1}$.
We identify $\SS^{1}$ with $[0,1[$. For a positive integer $n$, let

$\mathcal{D}_{n}\coloneqq\left\\{{\left[\frac{k}{2^{n}},\frac{k+1}{2^{n}}\right[}:0\leqslant
k\leqslant 2^{n}-1\right\\}.$

Moreover, for a positive number $t,$ let $\mathcal{D}_{t}=\mathcal{D}_{\lfloor
t\rfloor}.$ If the sequence $(\frac{1}{n}H(\nu,\mathcal{D}_{n}))_{n}$
converges and

$\lim_{n\to+\infty}\frac{1}{n}H(\nu,\mathcal{D}_{n})=\alpha,$

we say that $\nu$ has entropy dimension $\alpha$. It is shown in [86, Theorem
4.4] (see also [29, Theorem 1.3]) that if $\nu$ is exact dimensional of
dimension $\alpha$ then it has entropy dimension $\alpha$.

###  3.3 Distortion estimates

This subsection is a collection of useful tools of distortion controls that
enable controlling the iterations of $C^{2}$ maps on a small piece using only
the information at one point. These techniques date back to Denjoy [21],
Schwartz [79], and Sacksteder [77], and later a more concise view was proposed
by Sullivan [82]. Further discussions on this topic can be found in [22, 23,
24].

Let $f\in\mathrm{Diff}^{2}_{+}(\SS^{1})$ and $I\subset\SS^{1}$ be an interval.
Denote

$\varkappa(f,I)\coloneqq\sup_{x,y\in I}|\log f^{\prime}(x)-\log
f^{\prime}(y)|.$

We also consider the Lipschitz norm of $\log f^{\prime}$ on $I$, denoted by

$\widetilde{\varkappa}(f,I)=\left\|\log
f^{\prime}\right\|_{\mathrm{Lip}(I)}\coloneqq\sup_{x\neq y\in I}\frac{|\log
f^{\prime}(x)-\log f^{\prime}(y)|}{d(x,y)}.$

In [23], the quantity $\varkappa(f,I)$ is called distortion coefficient and
$\widetilde{\varkappa}(f^{-1},fI)$ is called distortion norm. Obviously,
$\varkappa(f,I)\leqslant\widetilde{\varkappa}(f,I)|I|.$ Besides, there are
some basic estimate of the distortion of compositions, that is

$\varkappa(fg,I)\leqslant\varkappa(g,I)+\varkappa(f,g(I)),$
$\widetilde{\varkappa}(fg,I)\leqslant\widetilde{\varkappa}(g,I)+\|g^{\prime}\|_{C^{0}}\cdot\widetilde{\varkappa}(f,g(I)).$
(3.3)

Now we fix a finite subset $\mathcal{S}\subset\mathrm{Diff}_{+}^{2}(\SS^{1}),$
let

$M=M(\mathcal{S})\coloneqq\max\left\\{\max_{g\in\mathcal{S}}\|g^{\prime}\|_{C^{0}},\max_{g\in\mathcal{S}}\|\log
g^{\prime}\|_{\mathrm{Lip}}\right\\}.$

Take $f\in\mathcal{S}^{*n}$ and write $f$ as $g_{n}\cdots g_{2}g_{1}$ where
$g_{i}\in\mathcal{S}.$ Let

$f_{0}=\mathrm{id},\quad f_{k}=g_{k}\cdots g_{2}g_{1},\ \forall 1\leqslant
k\leqslant n.$

Then for every interval $I\subset\SS^{1},$ we have

$\varkappa(f,I)\leqslant\sum_{k=0}^{n-1}\varkappa(g_{k+1},f_{k}(I))\leqslant
M\sum_{k=0}^{n-1}|f_{k}(I)|,$ (3.4)

and

$\widetilde{\varkappa}(f,I)\leqslant\sum_{k=0}^{n-1}\|f_{k}^{\prime}\|_{C^{0}}\cdot\widetilde{\varkappa}(g_{k+1},f_{k}(I))\leqslant
M^{n-1}\sum_{k=0}^{n-1}\widetilde{\varkappa}(g_{k+1},f_{k}(I))\leqslant
M^{n}\sum_{k=0}^{n-1}|f_{k}(I)|.$ (3.5)

Let $x_{0}\in I$ and let $I_{0}=I$ and $I_{k}=f_{k}(I_{0})$ for
$k=1,\dotsc,n$. We have the following inequality [24, Corollary 2.3]

$\forall
k=1,\dotsc,n,\quad\left|\log\frac{f_{k}^{\prime}(x_{0})|I_{0}|}{|I_{k}|}\right|\leqslant\varkappa(f_{k},I_{0})\leqslant
M\sum_{i=0}^{k-1}|I_{i}|,$ (3.6)

and summing over the exponentiation of (3.6),

$\log\left(\sum_{k=0}^{n-1}|I_{k}|\right)\leqslant\log|I|+\log\left(\sum_{k=0}^{n-1}f_{k}^{\prime}(x_{0})\right)+M\sum_{k=0}^{n-2}|I_{k}|.$

######  Proposition 3.8.

Fix $x_{0}\in\SS^{1}$ and $f\in\mathcal{S}^{*n}.$ Define $M=M(\mathcal{S})$
and $(f_{k})_{1\leqslant k\leqslant n}$ as above and let
$S=\sum_{k=0}^{n-1}f_{k}^{\prime}(x_{0}).$ Then for every
$\delta\leqslant(2MS)^{-1},$ we have

$\varkappa(f,B(x_{0},\delta))\leqslant 2MS\delta,$ (3.7)
$\widetilde{\varkappa}(f^{-1},fB(x_{0},\delta))\leqslant
4MS/f^{\prime}(x_{0}).$ (3.8)

###### Proof.

The first inequality (3.7) is [24, Proposition 2.4]. It remains to prove
(3.8).

We first upgrade (3.7) so that we can replace $B(x_{0},\delta)$ in (3.7) by
any subinterval. Let $I\subset B(x_{0},\delta)$ be an interval. Since
$\varkappa(f_{k},B(x_{0},\delta))\leqslant 1$ by (3.7), we have
$|f_{k}I|\leqslant 2|I|f_{k}^{\prime}(x_{0}).$ In view of (3.4),

$\varkappa(f,I)\leqslant M\sum_{k=0}^{n-1}|f_{k}I|\leqslant 2MS|I|\leqslant
1.$

Combining with $\varkappa(f,B(x_{0},\delta))\leqslant 2MS\delta\leqslant 1,$
we have

$\widetilde{\varkappa}(f^{-1},fB(x_{0},\delta))=\sup_{I\subset
B(x_{0},\delta)}\frac{\varkappa(f^{-1},fI)}{|fI|}\leqslant\sup_{I\subset
B(x_{0},\delta)}\frac{2\varkappa(f,I)}{f^{\prime}(x_{0})|I|}\leqslant\frac{4MS}{f^{\prime}(x_{0})}.\qed$

##  4 Random walks on $\mathrm{Diff}_{+}^{2}(\SS^{1})$

###  4.1 Preliminaries on random walks

##### Cocycles.

Let $\mathcal{S}\subset\mathrm{Diff}_{+}^{1}(\SS^{1})$ be a finite set. We
equip $\mathcal{S}$ with the discrete topology and let
$\Sigma=\mathcal{S}^{\mathbb{Z}}$ be the product space which is compact. We
also consider the spaces $\Sigma^{+}=\mathcal{S}^{{\mathbb{Z}}_{\geqslant 0}}$
and $\Sigma^{-}=\mathcal{S}^{{\mathbb{Z}}_{<0}}.$ Elements in
$\Sigma,\Sigma^{+},\Sigma^{-}$ are usually denoted by
$\omega,\omega^{+},\omega^{-},$ respectively. The maps
$\pi^{+}:\Sigma\to\Sigma^{+}$ and $\pi^{-}:\Sigma\to\Sigma^{-}$ refer to
natural projections. We use $\sigma$ to denote the left shift map on $\Sigma$
or $\Sigma^{+}.$

For an element $\omega\in\Sigma,$ we write
$\omega=(\cdots,f_{-2},f_{-1},f_{0},f_{1},\cdots)$ where
$f_{n}\in\mathcal{S},n\in{\mathbb{Z}}.$ Denote

$f_{\omega}^{n}\coloneqq\begin{cases}f_{n-1}\cdots f_{1}f_{0}&\text{if
}n\geqslant 0;\\\ f_{n}^{-1}\cdots f_{-2}^{-1}f_{-1}^{-1}&\text{if
}n<0.\end{cases}$

We abbreviate $f_{\omega}^{1}$ to $f_{\omega}.$ It induces an invertible
cocycle over $\sigma:\Sigma\to\Sigma$ defined as

$F:\Sigma\times\SS^{1}\to\Sigma\times\SS^{1},\quad(\omega,x)\mapsto(\sigma\omega,f_{\omega}x).$

Then the map $F$ satisfies
$F^{n}(\omega,x)=(\sigma^{n}\omega,f_{\omega}^{n}x)$ for every
$n\in{\mathbb{Z}}.$

We also use the notation $f_{\omega^{+}}$ and $f_{\omega^{+}}^{n}$ defined
similarly for $\omega^{+}\in\Sigma^{+}$ and $n\geqslant 0,$ which induces a
forward cocycle as

$F^{+}:\Sigma^{+}\times\SS^{1}\to\Sigma^{+}\times\SS^{1},\quad(\omega^{+},x)\mapsto(\sigma\omega^{+},f_{\omega^{+}}x).$

Then $F$ is semi-conjugate to $F^{+}$ via the natural projection,
$(\pi^{+},\mathrm{id})\circ F=F^{+}\circ(\pi^{+},\mathrm{id}).$

##### Random walks.

Let $\mu$ be a finitely supported probability measure on
$\mathrm{Diff}_{+}^{1}(\SS^{1})$. We take
$\mathcal{S}=\operatorname{\mathrm{supp}}\mu$ which induces a cocycle
$F:\Sigma\times\SS^{1}\to\Sigma\times\SS^{1}$ as above. Equip $\Sigma$ with
its Borel $\sigma$-algebra. Let $\bm{\mathrm{P}}=\mu^{\mathbb{Z}}$ be the
product probability measure on $\Sigma.$ Similarly, the spaces $\Sigma^{+}$
and $\Sigma^{-}$ are equipped with probability measures
$\bm{\mathrm{P}}^{+}=\mu^{{\mathbb{Z}}_{\geqslant 0}}$ and
$\bm{\mathrm{P}}^{-}=\mu^{{\mathbb{Z}}_{<0}}$ respectively. In this sense, the
cocycle $F:\Sigma\times\SS^{1}\to\Sigma\times\SS^{1}$ equipped with a
probability measure $\bm{\mathrm{P}}$ on $\Sigma$ is called the random walk
induced by $\mu.$

Now we recall the definition of stationary measures mentioned in Section 2.1.
The stationary measures correspond to the invariant measures of the forward
cocycle $F^{+}$ in the following way.

######  Proposition 4.1 ([85, Propositions 5.5 and 5.13]).

Let $\nu$ be a probability measure on $\SS^{1},$ then

  1. (1)

$\nu$ is $\mu$-stationary if and only if $\bm{\mathrm{P}}^{+}\times\nu$ is
$F^{+}$-invariant.

  2. (2)

$\nu$ is an ergodic $\mu$-stationary measure if and only if
$\bm{\mathrm{P}}^{+}\times\nu$ is ergodic $F^{+}$-invariant.

##### Lyapunov Exponents.

For an element $(\omega^{+},x)\in\Sigma^{+}\times\SS^{1},$ recall the Lyapunov
exponent at $(\omega^{+},x)$ is defined as (if exists)

$\lambda(\omega^{+},x)\coloneqq\lim_{n\to+\infty}\frac{1}{n}\log(f_{\omega^{+}}^{n})^{\prime}(x).$

If $\nu$ is an ergodic $\mu$-stationary measure then
$\bm{\mathrm{P}}^{+}\times\nu$ is an ergodic $F^{+}$-invariant measure. By
Birkhoff’s ergodic theorem,

$\lambda(\omega^{+},x)=\iint\log
f_{\omega^{+}}^{\prime}(y)\,\mathrm{d}\bm{\mathrm{P}}^{+}(\omega^{+})\,\mathrm{d}\nu(y)=\iint\log
f^{\prime}(y)\,\mathrm{d}\mu(f)\,\mathrm{d}\nu(y)$

for $\bm{\mathrm{P}}^{+}\times\nu$ almost every $(\omega^{+},x).$ This
corresponds to the Lyapunov exponent $\lambda(\mu,\nu)$ given in (2.2).

In general, the Lyapunov exponent can be defined similarly for every ergodic
$F^{+}$-invariant probability measure on $\Sigma^{+}\times\SS^{1}$, or every
ergodic $F$-invariant probability measure on $\Sigma\times\SS^{1}.$
Specifically, let $m$ be an $F$-invariant probability measure on
$\Sigma\times\SS^{1},$ the Lyapunov exponent of $(\omega,x)$ is given by

$\lambda(\omega,x)\coloneqq\lim_{n\to+\infty}\frac{1}{n}\log(f_{\omega}^{n})^{\prime}(x).$

Then for $m$-almost every $(\omega,x),$ the limit exists and coincides with
the backward limit

$\lim_{n\to-\infty}\frac{1}{n}\log(f_{\omega}^{n})^{\prime}(x).$

Moreover, if $m$ is ergodic, then $\lambda(\omega,x)$ is constant almost
everywhere, denoted by $\lambda(m)$. We also remark that if $m$ is an ergodic
$F$-invariant probability measure and $m^{+}=(\pi^{+},\mathrm{id})_{*}m$ which
is an ergodic $F^{+}$-invariant probability measure on $\Sigma\times\SS^{1},$
then $\lambda(m)=\lambda(m^{+}).$

###  4.2 Generalities on random transformations

We denote by $P$ and $Q$ to be the natural projections of
$\Sigma\times\SS^{1}$ to $\Sigma$ and $\SS^{1},$ respectively. For a Borel
probability measure $m$ on $\Sigma\times\SS^{1}$ with
$P_{*}m=\bm{\mathrm{P}}$, write

$\mathrm{d}m(\omega,x)=\mathrm{d}\bm{\mathrm{P}}(\omega)\,\mathrm{d}m_{\omega}(x)$

for its disintegration along $P:\Sigma\times\SS^{1}\to\Sigma$ in the sense of
Rokhlin. The measure $m$ being $F$-invariant translates to the following
equivariant property: for $\bm{\mathrm{P}}$-almost every $\omega\in\Sigma$,

$m_{\sigma(\omega)}=(f_{\omega})_{*}m_{\omega}.$ (4.1)

Let $\mathcal{P}=\mathcal{P}_{F,\bm{\mathrm{P}}}$ denote the set of
$F$-invariant probability measures $m$ on $\Sigma\times\SS^{1}$ such that
$P_{*}m=\bm{\mathrm{P}}$. Let $\mathcal{P}^{u}\subset\mathcal{P}$ (resp.
$\mathcal{P}^{s}$) denote the subset of all $u$-states (resp. $s$-states).
Recall that $m\in\mathcal{P}$ is a _$u$ -state_ (resp. _$s$ -state_) if its
disintegration $\omega\mapsto m_{\omega}$ factors through
$\pi^{-}:\Sigma\to\Sigma^{-}$ (resp. through $\pi^{+}:\Sigma\to\Sigma^{+}$).
For convenience later on, we denote $\mu^{+}$ to be $\mu$ and $\mu^{-}$ to be
the pushforward of $\mu$ under the map $f\mapsto f^{-1}$.

######  Proposition 4.2 ([85, Proposition 5.17]).

The map $m\mapsto Q_{*}m$ is a bijection between the convex set
$\mathcal{P}^{u}$ (resp. $\mathcal{P}^{s}$) and the convex set of
$\mu^{+}$-stationary (resp. $\mu^{-}$-stationary) measures.

Note that this bijection is clearly linear, and hence preserves ergodicity.

The following fact is a special case of Avila-Viana invariance principle [8],
which is a generalization of an earlier result in the linear case by
Ledrappier [52].

######  Theorem 4.3 ([8, Theorem B]).

Let $m$ be an $F$-invariant probability measure on $\Sigma\times\SS^{1}$ with
$P_{*}m=\bm{\mathrm{P}}$. If $\lambda(\omega,x)\leqslant 0$ holds for
$m$-almost every $(\omega,x)$ then $m\in\mathcal{P}^{u}$. Dually, if
$\lambda(\omega,x)\geqslant 0$ holds for $m$-almost every $(\omega,x)$ then
$m\in\mathcal{P}^{s}$.

Consequently, every ergodic $m\in\mathcal{P}$ is either a $u$-state or an
$s$-state. It follows that $\mathcal{P}$ is the convex hull of
$\mathcal{P}^{u}\cup\mathcal{P}^{s}$. We also have the following.

######  Corollary 4.4.

Assume that $\operatorname{\mathrm{supp}}\mu$ does not preserve any
probability measure on $\SS^{1}.$ Then for every ergodic $u$-state $m$,
$\lambda(m)<0.$ For every ergodic $s$-state $m,$ $\lambda(m)>0.$

###### Proof.

The equality $\lambda(m)=0$ holds if and only if $m$ is both a $u$-state and
an $s$-state. Then the condition measure $m_{\omega}$ is thus constant, which
is a Borel probability measure on $\SS^{1}$ invariant for every
$f\in\operatorname{\mathrm{supp}}\mu.$ ∎

We will also need a result of Ruelle-Wilkinson [76] about invertible cocycles
with negative Lyapunov exponents.

######  Theorem 4.5 ([76, Theorem II]).

Let $m$ be an ergodic $F$-invariant probability measure on
$\Sigma\times\SS^{1}$. Assume that $\lambda(m)<0.$ Then there exists a subset
$X\subset\Sigma\times\SS^{1}$ and a positive integer $k$ such that

  1. (1)

$m(X)=1$ and

  2. (2)

for every $(\omega,x)\in X,$
$\\#(X\cap\left\\{\omega\right\\}\times\SS^{1})=k.$

By considering the inverse cocycle, we see that the same holds if we assume
$\lambda(m)>0$. Using ergodicity and equivariance (4.1), we have the following
fact about the disintegration of $m$.

######  Corollary 4.6.

If $m\in\mathcal{P}$ is ergodic and $\lambda(m)\neq 0$, then there exists a
positive integer $k$ such that for $\bm{\mathrm{P}}$-almost every
$\omega\in\Sigma$, $m_{\omega}$ is a uniform probability measure on a set of
$k$ elements.

The next result is due to Malicet [57]. Recall $T_{\mu}$ is the semigroup
generated by $\operatorname{\mathrm{supp}}\mu.$

######  Theorem 4.7 ([57, Theorem B]).

Let $\mu$ be a finitely supported probability measure on
$\mathrm{Homeo}(\SS^{1})$ such that $\operatorname{\mathrm{supp}}\mu$ does not
preserve any Borel probability measure on $\SS^{1}.$ Then there are only
finitely many ergodic $\mu$-stationary measures on $\SS^{1}.$ Their
topological supports are pairwise disjoint and are exactly the
$T_{\mu}$-minimal sets.

###  4.3 The structure of random walks on $\mathrm{Diff}^{2}_{+}(\SS^{1})$

Let $\mu$ be a finitely supported probability measure on
$\mathrm{Diff}_{+}^{2}(\SS^{1})$ without common invariant probability measures
on $\SS^{1}.$ In order to study the structure of the random walk induced by
$\mu$, we first establish some relations among all the ergodic
$\mu^{\pm}$-stationary measures. One basic question is whether the number $k$
in Corollary 4.6 is the same among different $\mu^{\pm}$-stationary measures.

The questions of this type can be answered by constructing a dynamically
defined transitive permutation among all the ergodic $\mu^{\pm}$-stationary
measure, see Lemma 4.11. The construction is partially inspired by Hertz-
Hertz-Tahzibi-Ures [75], where they associate each invariant measure of
positive center exponent to one of negative center exponent by considering the
extremal points of the Pesin center manifold. Using this permutation, we can
show that these stationary measures share similar properties. Thus they form a
structural dynamic on $\SS^{1}.$

Let $\Theta^{s}\subset\Sigma\times\SS^{1}$ denote

$\Theta^{s}=\left\\{\,(\omega,x)\in\Sigma\times\SS^{1}:\limsup_{n\to+\infty}\frac{1}{n}\log(f_{\omega}^{n})^{\prime}(x)<0\,\right\\}.$

It is clear that $\Theta^{s}$ is $F$-invariant. For any
$(\omega,x)\in\Theta^{s}$, $(f_{\omega}^{n})^{\prime}(x)\to 0$ exponentially
fast. Hence $\sum_{n\geqslant 0}(f_{\omega}^{n})^{\prime}(x)<+\infty$. In view
of the distortion estimate (Proposition 3.8), there are constants
$\delta,c,C>0$ such that

$\forall y\in B(x,\delta),\,\forall n\geqslant
0,\quad(f_{\omega}^{n})^{\prime}(y)\leqslant C2^{-cn}.$ (4.2)

Therefore, for any $\omega\in\Sigma$, the slice
$W^{s}(\omega)=\left\\{x\in\SS^{1}:(\omega,x)\in\Theta^{s}\right\\}$ is open
in $\SS^{1}$. Moreover $W^{s}(\omega)\neq\SS^{1}$ since otherwise, using
compactness, we could cover $\SS^{1}$ by finitely many open balls with the
property of (4.2) leading to $(f_{\omega}^{n})^{\prime}(y)\leqslant C2^{-cn}$
uniformly in $y\in\SS^{1}$, which is absurd.

For $(\omega,x)\in\Theta^{s}$, let $W^{s}(\omega,x)$ denote the connected
component of $W^{s}(\omega)$ containing $x$. It is an open interval of
$\SS^{1}$. Using (4.2), we see that

$W^{s}(\omega,x)=\left\\{\,y\in\SS^{1}:\limsup_{n\to+\infty}\frac{1}{n}\log
d(f_{\omega}^{n}(y),f_{\omega}^{n}(x))<0\,\right\\}.$

Let $R^{s}(\omega,x)=(\omega,y)$ where $y$ is the right end-point of
$W^{s}(\omega,x)$. Clearly $R^{s}(\omega,x)\not\in\Theta^{s}$. Thus, we get a
map $R^{s}\colon\Theta^{s}\to\Sigma\times\SS^{1}\setminus\Theta^{s}$. From the
$F$-invariance of $\Theta^{s}$, we find that $R^{s}$ commutes with $F$.

Let $m\in\mathcal{P}^{u}$. By Corollary 4.4, we have $m(\Theta^{s})=1$. Thus,
$R^{s}_{*}m$ is a well defined Borel probability measure on
$\Sigma\times\SS^{1}$. Moreover, $R^{s}_{*}m\in\mathcal{P}$ since $R^{s}\circ
F=F\circ R^{s}$ and $P\circ R^{s}=P$. Note that,
$R^{s}_{*}m(\Sigma\times\SS^{1}\setminus\Theta^{s})=1$, implying that for
$R^{s}_{*}m$-almost every $(\omega,x)$, $\lambda(\omega,x)\geqslant 0$. By
Theorem 4.3, $R^{s}_{*}m\in\mathcal{P}^{s}$. To summarize, $m\mapsto
R^{s}_{*}m$ is a map $R^{s}_{*}\colon\mathcal{P}^{u}\to\mathcal{P}^{s}$.
Similarly, we define the left-end point map
$L^{s}\colon\Theta^{s}\to\Sigma\times\SS^{1}\setminus\Theta^{s}$.

In a dual manner (considering the inverse cocyle $F^{-1}$), we define
$\Theta^{u}$, $W^{u}(\omega,x)$ for $(\omega,x)\in\Theta^{u}$. We consider
$R^{u},L^{u}\colon\Theta^{u}\to\Sigma\times\SS^{1}\setminus\Theta^{u}$ and
their induced maps
$R^{u}_{*},L^{u}_{*}\colon\mathcal{P}^{s}\to\mathcal{P}^{u}.$

######  Lemma 4.8.

Let $m\in\mathcal{P}^{u}$. For $\bm{\mathrm{P}}$-almost every
$\omega\in\Sigma$, the map $x\mapsto Q\circ R^{s}(\omega,x)$ is injective on
$\operatorname{\mathrm{supp}}m_{\omega}$.

###### Proof.

By Corollary 4.6, for $\bm{\mathrm{P}}$-almost every $\omega\in\Sigma$,
$\operatorname{\mathrm{supp}}m_{\omega}$ is finite. It follows that there
exists $c>0$ such that the set

$\Sigma^{\prime}=\left\\{\omega:d(x,x^{\prime})\geqslant c,\;\forall x\neq
x^{\prime}\in\operatorname{\mathrm{supp}}m_{\omega}\right\\}$

has a positive $\bm{\mathrm{P}}$ measure. By the ergodicity of $\sigma$, for
$\bm{\mathrm{P}}$-almost every $\omega\in\Sigma,$ there are infinitely many
$n\in{\mathbb{N}}$ such that $\sigma^{n}\omega\in\Sigma^{\prime}$. For such
$\omega$, for every $x\neq
x^{\prime}\in\operatorname{\mathrm{supp}}m_{\omega}$, we have
$d\bigl{(}f_{\omega}^{n}(x),f_{\omega}^{n}(x^{\prime})\bigr{)}\not\to 0$. In
particular, $W^{s}(\omega,x)\neq W^{s}(\omega,x^{\prime})$. ∎

For $\omega\in\Sigma$, let

$\Pi(\omega)=\bigcup_{m\in\mathcal{P}^{u}}\operatorname{\mathrm{supp}}m_{\omega}\quad\text{and}\quad\Xi(\omega)=\bigcup_{m\in\mathcal{P}^{s}}\operatorname{\mathrm{supp}}m_{\omega}.$

Clearly, the union can be taken over ergodic $u$-states (resp. $s$-states) and
it still defines the same set. Then for $\bm{\mathrm{P}}$-almost every
$\omega\in\Sigma$, $\Pi(\omega)$ and $\Xi(\omega)$ are finite sets by Theorem
4.7.

######  Lemma 4.9.

For $\bm{\mathrm{P}}$-almost every $\omega\in\Sigma$, we have
$\\#\Pi(\omega)=\\#\Xi(\omega)$ and

$\Pi(\omega)=\SS^{1}\setminus
W^{u}(\omega)\quad\text{and}\quad\Xi(\omega)=\SS^{1}\setminus W^{s}(\omega).$

###### Proof.

Applying Lemma 4.8 to a suitable convex combination of ergodic $u$-states and
remembering $R^{s}_{*}m\in\mathcal{P}^{s}$, we see that for
$\bm{\mathrm{P}}$-almost every $\omega\in\Sigma$, the map $x\mapsto Q\circ
R^{s}(\omega,x)$ is injective from $\Pi(\omega)$ to $\Xi(\omega)$. Similarly,
$x\mapsto Q\circ L^{s}(\omega,x)$ is injective from $\Pi(\omega)$ to
$\Xi(\omega)$ and $x\mapsto Q\circ R^{u}(\omega,x)$ and $x\mapsto Q\circ
L^{u}(\omega,x)$ are injective maps from $\Xi(\omega)$ to $\Pi(\omega)$,
proving the claim about cardinality.

Moreover, this shows that, $W^{s}(\omega,x)$, $x\in\Pi(\omega)$ are disjoint
open intervals with endpoints in $\Xi(\omega)$. The total number of endpoints
being equal to the number of intervals, forces these intervals and these point
to partition the circle. Hence $\Xi(\omega)=\SS^{1}\setminus W^{s}(\omega)$. ∎

######  Lemma 4.10.

The numbers of ergodic $u$-states and $s$-states are equal. If
$m\in\mathcal{P}^{u}$ is ergodic then so is $R^{s}_{*}m\in\mathcal{P}^{s}$.

###### Proof.

Let $\\{m_{1},\cdots,m_{d}\\}\in\mathcal{P}^{u}$ be the set of ergodic
$u$-states. Since they are singular to each other, for
$\bm{\mathrm{P}}$-almost every $\omega\in\Sigma$,
$(\operatorname{\mathrm{supp}}\,(m_{i})_{\omega})_{1\leqslant i\leqslant d}$
form a partition of $\Pi(\omega)$. Set $m_{i}^{\prime}=R^{s}_{*}m_{i}.$ Then,
by Lemma 4.8,
$(\operatorname{\mathrm{supp}}\,(m^{\prime}_{i})_{\omega})_{1\leqslant
i\leqslant d}$ are pairwise disjoint. It follows that the number of ergodic
$s$-states, denoted as $d^{\prime}$, is no less than $d$. The equality
$d^{\prime}=d$ holds if and only if every $m_{i}^{\prime}$ is ergodic. But,
using $R^{u}$, we see that indeed $d\geqslant d^{\prime}.$ Hence,
$d=d^{\prime}$ and the lemma is proved. ∎

######  Lemma 4.11.

The maps $R^{s}_{*}$ and $R^{u}_{*}$ together induces a transitive permutation
of ergodic measures in $\mathcal{P}$. In particular, there is an integer
$r\geqslant 1$ such that for every ergodic $m\in\mathcal{P}$,
$\\#\operatorname{\mathrm{supp}}m_{\omega}=r$ for $\bm{\mathrm{P}}$-almost
every $\omega\in\Sigma$.

###### Proof.

The proof of Lemma 4.9 shows also the following for $\bm{\mathrm{P}}$-almost
every $\omega\in\Sigma$. Write $\Pi(\omega)=\\{x_{1},\dotsc,x_{k}\\}$ with
$x_{1},\dotsc,x_{k}$ arranged in cyclic order and set $y_{j}=Q\circ
R^{s}(\omega,x_{j})$ so that $y_{j}\in\Xi(\omega)$ for each $j$. Then
$x_{1},y_{1},\dotsc,x_{k},y_{k}$ are arranged in cyclic order. Moreover
$R^{u}(\omega,y_{j})=(\omega,x_{j+1\ \mathrm{mod}\ k})$. In particular,
$R^{s}$ and $R^{u}$ together induce a transitive permutation of points in
$\Pi(\omega)\cup\Xi(\omega).$

Then the claim of the lemma follows from Lemma 4.10 together with the
observation that two ergodic measures $m,m^{\prime}$ in $\mathcal{P}$ are
equal if and only if $m_{\omega}$ and $m^{\prime}_{\omega}$ have common atoms
for a set of $\omega$ with positive $\bm{\mathrm{P}}$ measure. The “in
particular” part follows from the fact that $R^{s}_{*}$ and $R^{u}_{*}$
preserve $\\#\operatorname{\mathrm{supp}}m_{\omega}$, a direct consequence of
Lemma 4.8. ∎

Let $d$ denote the number of ergodic $u$-states. We write
$[d]=\\{0,\dotsc,d-1\\}$ and the addition in $[d]$ is to be understood
reduction modulo $d$. Fix an arbitrary ergodic $u$-state
$m^{+}_{0}\in\mathcal{P}^{u}$. By Lemma 4.11, applying alternatively
$R^{u}_{*}$ and $R^{s}_{*}$, we find in the sequence

$m_{0}^{+}\mapsto m_{0}^{-}\mapsto m_{1}^{+}\mapsto\cdots m_{d-1}^{+}\mapsto
m_{d-1}^{-}\mapsto m_{0}^{+}$

all ergodic $u$-states $m_{i}^{+}$ and all ergodic $s$-states $m_{i}^{-}$,
$i\in[d]$. For $i\in[d]$ and $\omega\in\Sigma$, define

$\Pi(\omega,i)=\operatorname{\mathrm{supp}}(m^{+}_{i})_{\omega}\quad\text{and}\quad\Xi(\omega,i)=\operatorname{\mathrm{supp}}(m^{-}_{i})_{\omega}.$

Then for each $i\in[d]$, we have that $Q\circ(R^{u}\circ R^{s})^{d}$ preserves
$\Pi(\omega,i)$ and maps every element of $\Pi(\omega,i)$ to the element on
its right.

##### The space $\SS_{k}$.

For a positive integer $k,$ let $\SS_{k}$ denote the family of $k$-element
subsets of $\SS^{1}.$ For later convenience, we equip $\SS_{k}$ with a natural
metric, making it a metric space. This metric is defined as

$\underline{d}(\underline{x},\underline{y})=\inf_{\tau\in\mathfrak{S}_{k}}\sum_{i=1}^{k}d(x_{i},y_{\tau(i)})$

where
$\underline{x}=\left\\{x_{1},\cdots,x_{k}\right\\},\underline{y}=\left\\{y_{1},\cdots,y_{k}\right\\}\in\SS_{k}$
and $\mathfrak{S}_{k}$ is the symmetric group of $[k].$ Therefore, every
element in $\mathrm{Homeo}(\SS^{1})$ can be naturally regarded as an element
in $\mathrm{Homeo}(\SS_{k}).$

For every $\underline{x}\in\SS_{k},$ we define the probability measure
$u_{\underline{x}}$ to be the uniform measure on $\underline{x}.$ Let
$\mathcal{M}(\SS^{1})$ be the space of Radon measures on $\SS^{1}$ equipped
with the weak* topology. Then $u_{\bullet}:\underline{x}\mapsto
u_{\underline{x}}$ is a topological embedding. In fact, it is a homothety onto
its image if we equip $\mathcal{M}(\SS^{1})$ with the Wasserstein metric

$\mathrm{dist}(\eta,\zeta)=\sup_{\phi}\left|\int\phi\,\mathrm{d}\eta-\int\phi\,\mathrm{d}\zeta\right|,$

where the supreme is taken over all $1$-Lipschitz functions, where
$\eta,\zeta\in\mathcal{M}(\SS^{1}).$

##### Structure of random walks on $\mathrm{Diff}_{+}^{2}(\SS^{1})$.

The following is immediate from our discussion.

######  Theorem 4.12 (Structure of random walks I: construction).

Assume that $\operatorname{\mathrm{supp}}\mu$ does not preserve any
probability measure on $\SS^{1}.$ Then there exists two positive integers
$d,r$ and two measurable maps $\Pi:\Sigma\times[d]\to\SS_{r}$ and
$\Xi:\Sigma\times[d]\to\SS_{r}$ such that

  1. (1)

Let $m_{i}^{\pm}$ be probability measures on $\Sigma\times\SS^{1}$ defined by

$\mathrm{d}m_{i}^{+}=\mathrm{d}\bm{\mathrm{P}}(\omega)\mathrm{d}u_{\Pi(\omega,i)},\quad\mathrm{d}m_{i}^{-}=\mathrm{d}\bm{\mathrm{P}}(\omega)\mathrm{d}u_{\Xi(\omega,i)},$

then $m_{i}^{+}$’s (resp. $m_{i}^{-}$’s) are exactly the ergodic $u$-states
(resp. $s$-states) that projects to $\bm{\mathrm{P}}.$

  2. (2)

Let $\nu_{i}^{\pm}$ be probability measures on $\SS^{1}$ defined by
$\nu_{i}^{+}=Q_{*}m_{i}^{+}$ and $\nu_{i}^{-}=Q_{*}m_{i}^{-}$. They can also
be expressed as

$\nu_{i}^{+}=\int
u_{\Pi(\omega,i)}\,\mathrm{d}\bm{\mathrm{P}}(\omega),\quad\nu_{i}^{-}=\int
u_{\Xi(\omega,i)}\,\mathrm{d}\bm{\mathrm{P}}(\omega).$

Then $\nu_{i}^{+}$’s (resp. $\nu_{i}^{-}$’s) are exactly the ergodic
$\mu^{+}$-stationary (resp. $\mu^{-}$-stationary) measures.

Moreover, for $\bm{\mathrm{P}}$ almost every $\omega,$ the following holds

  1. (3)

$\Pi(\omega,i)$ only depends on $(\pi^{-}\omega,i)$ and $\Xi(\omega,i)$ only
depends on $(\pi^{+}\omega,i).$

  2. (4)

Cocycle invariance: $\forall n\in{\mathbb{Z}},$ we have
$f_{\omega}^{n}\Pi(\omega,i)=\Pi(\sigma^{n}\omega,i),$
$f_{\omega}^{n}\Xi(\omega,i)=\Xi(\sigma^{n}\omega,i).$

  3. (5)

Define the sets on $\SS^{1}$ as

$\Pi(\omega)=\bigcup_{i\in[d]}\Pi(\omega,i),\quad\Xi(\omega)=\bigcup_{i\in[d]}\Xi(\omega,i).$

Then $\Pi(\omega)\cup\Xi(\omega)$ is made up of $2dr$ different points. Denote
$x_{0},x_{1},\cdots,x_{2dr-1}$ to be these points arranged in cyclic order on
$\SS^{1}$ such that $x_{0}\in\Pi(\omega,0).$ Then

$x_{2j}\in\Pi(\omega,j\ \mathrm{mod}\ d),\ x_{2j+1}\in\Xi(\omega,j\
\mathrm{mod}\ d),\quad\forall 0\leqslant j\leqslant dr-1.$

######  Remark 4.13.

Since $\Pi,\Xi$ only depends on $\pi^{-}\omega,\pi^{+}\omega,$ we will
sometimes use the notation $\Pi(\omega^{-})$ and $\Xi(\omega^{+})$ for
$\omega^{-}\in\Sigma^{-}$ and $\omega^{+}\in\Sigma^{+},$ respectively.

######  Remark 4.14.

The map $\omega\mapsto\Pi(\omega,i)$ plays the role of the Furstenberg
boundary map. It can also be defined as follows (see [85, Lemma 5.22]). For
$\bm{\mathrm{P}}$-almost every $\omega\in\Sigma$, we have

$(f_{\sigma^{-n}\omega}^{n})_{*}\nu_{i}^{+}\stackrel{{\scriptstyle\text{weak-*}}}{{\longrightarrow}}u_{\Pi(\omega,i)},$
(4.3)

as $n\to+\infty$. An effective version of (4.3) will be shown in Section 4.6.

Moreover, we can show the constants $d,r$ given in this theorem are
topological invariants. But the proof will be left to Section 4.4, where we
will show that $dr$ is indeed the least number of pairs of topologically
hyperbolic fixed points in the semigroup $T_{\mu}.$ In which we need to use
the construction of hyperbolic elements in Section 7.1.

######  Theorem 4.15.

Let $\mu$ be as in the previous theorem and $d,r$ be the constants given
there. Then $d,r$ are topological invariants of the semigroup $T_{\mu}$.
Specifically, if $\mu,\mu^{\prime}$ are two probability measures on
$\mathrm{Diff}^{2}_{+}(\SS^{1})$ satisfying the assumption of the previous
theorem and that there exists $h\in\mathrm{Homeo}(\SS^{1})$ with
$T_{\mu}=hT_{\mu^{\prime}}h^{-1},$ then the constants $d,r$ are the same for
$\mu$ and $\mu^{\prime}.$

The next statement supplements Theorem 4.12 with more dynamical information,
that is, how $f^{n}_{\omega}$ behaves when $n$ is large.

######  Theorem 4.16 (Structure of random walks II: dynamics).

Assume that $\operatorname{\mathrm{supp}}\mu$ does not preserve any
probability measure on $\SS^{1}.$ Let $\nu_{i}^{+}$ and $\nu_{i}^{-}$,
$i\in[d]$ be the ergodic stationary measures defined in the previous theorem
corresponding to $\Pi(\,\bm{\cdot}\,,i)$ and $\Xi(\,\bm{\cdot}\,,i).$ Then for
$\bm{\mathrm{P}}$-almost every $\omega$ and $i\in[d],$ there exists subsets
$W^{s}(\omega,i)\subset\SS^{1},W^{u}(\omega,i)\subset\SS^{1}$ satisfying

  1. (1)

Each $W^{s}(\omega,i),W^{u}(\omega,i)$ is a disjoint union of $r$ open
intervals.

  2. (2)

For every $i\in[d],$ $\Pi(\omega,i)\subset
W^{s}(\omega,i)\subset\SS^{1}\setminus\Xi(\omega)$ and $\Xi(\omega,i)\subset
W^{u}(\omega,i)\subset\SS^{1}\setminus\Pi(\omega).$

  3. (3)

the sets $W^{s}(\omega,i)$, $i\in[d]$ are pairwise disjoint and we have
$\bigcup_{i\in[d]}W^{s}(\omega,i)=\SS^{1}\setminus\Xi(\omega).$ The same holds
for $\bigcup_{i\in[d]}W^{u}(\omega,i)=\SS^{1}\setminus\Pi(\omega).$

  4. (4)

Cocycle invariance: $\forall n\in{\mathbb{Z}},$ we have
$f_{\omega}^{n}W^{s}(\omega,i)=W^{s}(\sigma^{n}\omega,i)$ and
$f_{\omega}^{n}W^{u}(\omega,i)=W^{u}(\sigma^{n}\omega,i).$

  5. (5)

For every connected component $I$ of $W^{s}(\omega,i)$ and $J$ of
$W^{u}(\omega,i)$, we have $\nu_{i}^{+}(I)=1/r$ and $\nu_{i}^{-}(J)=1/r.$ In
particular,

$\nu_{i}^{+}(W^{s}(\omega,i))=1,\quad\nu_{i}^{-}(W^{u}(\omega,i))=1,\quad\forall
i\in[d].$

  6. (6)

For every $x\in\Pi(\omega,i)$ and $y\in\Xi(\omega,i),$

$\lim_{n\to\pm\infty}\frac{1}{n}\log(f^{n}_{\omega})^{\prime}(x)=\lambda_{i}^{+},\quad\lim_{n\to\pm\infty}\frac{1}{n}\log(f^{n}_{\omega})^{\prime}(y)=\lambda_{i}^{-}.$

Where $\lambda_{i}^{+}<0$ and $\lambda_{i}^{-}>0$ are the Lyapunov exponents
corresponding to $\nu_{i}^{+}$ and $\nu_{i}^{-}.$

  7. (7)

For every closed subintervals $I\subset W^{s}(\omega),J\subset W^{u}(\omega),$
as $n\to+\infty,$ we have

$|f_{\omega}^{n}I|\to 0,\quad|f^{-n}_{\omega}J|\to 0,\quad\text{exponentially
fast.}$

###### Proof.

For $\omega\in\Sigma$ and $i\in[d],$ define

$W^{s}(\omega,i)=\bigcup_{x\in\Pi(\omega,i)}W^{s}(\omega,x)\quad\text{and}\quad
W^{u}(\omega,i)=\bigcup_{x\in\Xi(\omega,i)}W^{u}(\omega,x),$ (4.4)

so that $W^{s}(\omega)=\bigcup_{i\in[d]}W^{s}(\omega,i)$ and
$W^{u}(\omega)=\bigcup_{i\in[d]}W^{u}(\omega,i)$. All items except (5) follow
immediately from the discussion above. In particular, (3) is Lemma 4.9 and (7)
follows from (4.2) and compactness.

It remains to prove (5). Without loss of generality, we show the argument for
$\nu_{i}^{+}.$ Then the boundary points of $W^{s}(\omega,i)$ only depends on
$\pi^{+}(\omega)$ and hence $W^{s}(\omega,i)$ only depends on
$\pi^{+}(\omega).$ Fix an $\omega^{+}\in\Sigma^{+}$ and take a connected
component of $I$ of $W^{s}(\omega^{+},i).$ Then for every $\omega$ with
$\pi^{+}\omega=\omega^{+},$ we have $\\#(I\cap\Pi(\omega,i))=1.$ Since
$\Pi(\omega,i)$ only depends on $\pi^{-}\omega,$ it follows that

$\nu_{i}^{+}(I)=\frac{1}{r}\int_{\Sigma^{-}}\\#\left\\{I\cap\Pi(\omega^{-},i)\right\\}\mathrm{d}\bm{\mathrm{P}}^{-}(\omega^{-})=\frac{1}{r}.\qed$

######  Example 4.17.

Figure 4.18 illustrates the dynamics of a random walk in the case $d=r=2$.
Here $n$ is a large positive integer and $\omega^{\prime}=\sigma^{n}\omega$.
The red points denote the points in $\Pi(\,\bm{\cdot}\,)$, and the blue points
denote the points in $\Xi(\,\bm{\cdot}\,)$. The black arcs map to the black
arcs with some contraction.

$\Pi(\omega,0)$$\Pi(\omega,1)$$\Pi(\omega,0)$$\Pi(\omega,1)$$\Xi(\omega,0)$$\Xi(\omega,1)$$\Xi(\omega,0)$$\Xi(\omega,1)$$\Pi(\omega^{\prime},0)$$\Pi(\omega^{\prime},1)$$\Pi(\omega^{\prime},0)$$\Pi(\omega^{\prime},1)$$\Xi(\omega^{\prime},0)$$\Xi(\omega^{\prime},1)$$\Xi(\omega^{\prime},0)$$\Xi(\omega^{\prime},1)$$f_{\omega}^{n}$
Figure 4.18: A typical map $f^{n}_{\omega}:\SS^{1}\to\SS^{1}$.

For the convenience in later discussions, we define “typical” words in this
system.

######  Definition 4.19.

We say $\omega\in\Sigma$ is regular for the random walk if it

  1. (1)

satisfies all the conditions in the previous two theorems, and

  2. (2)

$(\omega,x)$ is Birkhoff regular for every $m_{i}^{+}$ for every
$x\in\Pi(\omega,i)$ and $(\omega,y)$ is Birkhoff regular for every $m_{i}^{-}$
for every $y\in\Xi(\omega,i).$

The previous two theorems show that the regular words $\omega$ for the random
walk form a full $\bm{\mathrm{P}}$-measure subset of $\Sigma.$

###  4.4 Basic properties of stationary measures

In the rest of this whole section, we focus on the random walk induced by a
finitely supported probability measure $\mu$ on
$\mathrm{Diff}_{+}^{2}(\SS^{1})$ without common invariant probability measures
on $\SS^{1}.$ We follow the notation in last subsection. Recall that $T_{\mu}$
denotes the semigroup generated by $\operatorname{\mathrm{supp}}\mu.$ There
are some basic properties about $\mu$-stationary measures.

######  Lemma 4.20.

Stationary measures are atomless.

###### Proof.

Otherwise, for some $c>0,$ the atoms of $\nu$ with weights at least $c$ form a
finite orbit of $T_{\mu}.$ Then, the uniform measure on this finite orbit is
an invariant probability measure for $\operatorname{\mathrm{supp}}\mu.$ ∎

For $c>0$, a subset of $\SS^{1}$ is said to be $c$-separated if its elements
are of distance strictly larger than $c$ one from each other.

######  Lemma 4.21.

There exists a constant $c>0$ depending only on $\mu$, such that for every
$\omega$ regular for the random walk, $\Pi(\omega)$ and $\Xi(\omega)$ are
$c$-separated.

###### Proof.

By Lemma 4.20 and the compactness of $\SS^{1},$ we can take $c>0$ such that
for every interval $I$ on $\SS^{1}$ with length $|I|\leqslant c,$ it holds
that

$\forall i\in[d],\quad\nu_{i}^{+}(I)<\frac{1}{r}.$

Two consecutive points in $\Xi(\omega)$ bound an open interval $I$, which is a
connected component of $W^{s}(\omega,i)$ for some $i\in[d]$. By Theorem
4.16(5), $\nu_{i}^{+}(I)=1/r$. Hence $\left|I\right|>c$. We deduce that
$\Xi(\omega)$ is $c$-separated. The proof for $\Pi(\omega)$ is similar. ∎

As a corollary of this lemma, the number of fixed points of a hyperbolic
element in the semigroup $T_{\mu}$ is bounded from below. This will be helpful
in showing that $d,r$ are topological invariants.

######  Definition 4.22.

Let $f\in\mathrm{Homeo}(\SS^{1})$. A fixed point $x$ of $f$ is called a
topologically hyperbolic if there exists $\varepsilon>0$ such that

  1. (1)

either for every $y\in B(x,\varepsilon),$ $f^{n}(y)\to x(n\to+\infty),$

  2. (2)

or for every $y\in B(x,\varepsilon),$ $f^{-n}(y)\to x(n\to+\infty).$

We call $x$ an attractor of $f$ in the first case and repellor otherwise.

Observe that if every fixed point of $f\in\mathrm{Homeo}(\SS^{1})$ is
topologically hyperbolic, then necessarily, $f$ has finitely many fixed points
and its attractors and repellors are arranged alternatively.

######  Corollary 4.23.

Let $f\in T_{\mu}$ be a diffeomorphism such that every fixed point is
topologically hyperbolic, then it has at least $dr$ attractors and $dr$
repellors.

###### Proof.

Assume for a contradiction that there is $f\in T_{\mu}$ has $q$ attractors and
$q$ repellors with $q<dr.$ Denote these fixed points by
$a_{1},r_{1},a_{2},r_{2},\cdots,a_{q},r_{q}$ ordered in the cyclic order and
where $a_{i}$’s are the attractors and $r_{i}$’s are the repellors. Take
$\varepsilon^{\prime}>0$ such that for every interval $I\subset\SS^{1}$ of
length $|I|\leqslant 2\varepsilon^{\prime},$ $\nu_{i}^{+}(I)<1/qdr$ for every
ergodic $\mu^{+}$-stationary measure. Consider the set
$\Sigma_{1}=\left\\{\omega^{-}\in\Sigma^{-}:\Pi(\omega^{-})\cap(\bigcup_{j}B(r_{j},\varepsilon^{\prime}))\neq\varnothing\right\\}.$
Then from Theorem 4.12(2),

$\bm{\mathrm{P}}^{-}(\Sigma_{1})\leqslant
r\sum_{i\in[d]}\sum_{j=1}^{q}\nu_{i}^{+}(B(r_{j},\varepsilon^{\prime}))<1.$

Let $\Sigma_{2}=\Sigma^{-}\setminus\Sigma_{1}$ with positive
$\bm{\mathrm{P}}^{-}$ measure.

Assume that $f\in\mathcal{S}^{*n}$ where
$\mathcal{S}=\operatorname{\mathrm{supp}}\mu.$ Let $m$ be a positive integer
large enough so that

$f^{m}\bigl{(}\SS^{1}\setminus\bigcup_{j=1}^{q}B(r_{j},\varepsilon^{\prime})\bigr{)}\subset\bigcup_{j=1}^{q}B(a_{j},c/2),$

where $c$ is the constant in Lemma 4.21. Consider the set

$\Sigma^{\prime}=\left\\{\omega\in\Sigma:\pi^{-}\omega\in\Sigma_{2},f_{\omega}^{nm}=f^{m}\right\\},$

then
$\bm{\mathrm{P}}(\Sigma^{\prime})=\bm{\mathrm{P}}^{-}(\Sigma_{2})\mu^{*nm}(f^{m})>0.$
But by the definition of $\Sigma_{2}$ and the cocycle invariance of
$\Pi(\,\bm{\cdot}\,),$ for $\bm{\mathrm{P}}$-almost every
$\omega\in\Sigma^{\prime},$ we have

$\Pi(\sigma^{nm}\omega)=f^{m}\Pi(\omega)\subset\bigcup_{j=1}^{q}B(a_{j},c/2).$

Since $q<dr$, $\Pi(\sigma^{nm}\omega)$ has two points within a distance of at
most $c$. This contradicts Lemma 4.21. ∎

Now we prove Theorem 4.15 assuming Lemma 7.1.

###### Proof of Theorem 4.15.

By Theorem 4.7, the constant $d$ is precisely the number of $T_{\mu}$ minimal
sets, which is obviously a topological invariant. By Lemma 7.1 and Corollary
4.23, $2dr$ is the least number of fixed points of elements in $T_{\mu}$
having only topologically hyperbolic fixed points. Thus, $dr$ is also
invariant under $C^{0}$ conjugates. Hence, so is $r.$ ∎

###  4.5 Uniform good words

In this subsection, we will introduce a powerful tool: the set of uniform good
words. Roughly speaking, it is a set of words with large probability and
uniform control, and it has global contraction properties on each connected
component of the $s$-manifolds $W^{s}(\omega)$. It enables us to handle smooth
actions in a manner similar to linear actions, except that in the smooth case
there are $dr$ cones instead of one cone in the linear case.

######  Proposition 4.24.

For any $\varepsilon>0,$ there exists a subset
$\Sigma_{\varepsilon}\subset\Sigma$ satisfying the following.

  1. (1)

$\bm{\mathrm{P}}(\Sigma_{\varepsilon})>1-\varepsilon$.

  2. (2)

For every $(\omega,i)\in\Sigma_{\varepsilon}\times[d]$ and every
$x\in\Pi(\omega,i)$, we have
$\lim_{n\to+\infty}\frac{1}{n}\log(f_{\omega}^{n})^{\prime}(x)=\lambda_{i}^{+}$.
The convergence is uniform in $\omega$,$i$ and $x$.

  3. (3)

there exists $C=C(\varepsilon)>0$ and an open set
$U(\omega)=U(\omega,\varepsilon)>0$ for every $\omega\in\Sigma_{\varepsilon},$
such that

     * •

$\Pi(\omega)\subset U(\omega)\subset W^{s}(\omega),$

     * •

$\SS^{1}\setminus U(\omega)\subset\Xi(\omega)^{(\varepsilon)},$ where
$\Xi(\omega)^{(\varepsilon)}=\bigcup_{x\in\Xi(\omega)}B(x,\varepsilon).$

     * •

for all $n\geqslant 0$, $\widetilde{\varkappa}(f_{\omega}^{n},I)\leqslant
C(\varepsilon)$ for every connected component $I$ of $U(\varepsilon,\omega).$

We call the set $\Sigma_{\varepsilon}\subset\Sigma$ constructed above is a set
of uniform good words.

######  Lemma 4.25.

Let $\Sigma^{\prime}\subset\Sigma$ be a subset and $x\in\SS^{1}.$ Assume there
exists $C>0,c>0$ such that for every $\omega\in\Sigma^{\prime},$
$(f_{\omega}^{n})^{\prime}(x)\leqslant C2^{{-c}n}.$ Then there exists
$C^{\prime},\rho>0$ only depends on $C$ and $c$ such that

$\widetilde{\varkappa}(f_{\omega}^{n},B(x,\rho))\leqslant
C^{\prime},\quad\forall\omega\in\Sigma^{\prime},\ n\in{\mathbb{N}}.$

###### Proof.

Let

$M=\max\left\\{\max_{f\in\mathcal{S}}\|f^{\prime}\|_{C^{0}},\max_{f\in\mathcal{S}}\|\log
f^{\prime}\|_{\mathrm{Lip}}\right\\}.$

Since $(f_{\omega}^{n})^{\prime}(x)\leqslant C2^{-cn},$ then

$\sum_{n=0}^{+\infty}(f_{\omega}^{n})^{\prime}(x)\leqslant\frac{C}{1-2^{-c}}$

is a uniform bound. Let

$\rho=\frac{1-2^{-c}}{2MC},\quad C^{\prime}=2M\frac{C}{1-2^{-c}}.$

By Proposition 3.8, $\widetilde{\varkappa}(f_{\omega}^{n},B(x,\rho))\leqslant
C^{\prime}$ for every $\omega\in\Sigma^{\prime},n\in{\mathbb{N}}.$ ∎

###### Proof of Proposition 4.24.

First, we can take $\Sigma^{\prime}\subset\Sigma$ with
$\bm{\mathrm{P}}(\Sigma^{\prime})>1-\varepsilon/2$ such that

$\forall\omega\in\Sigma^{\prime},\,\forall i\in[d],\,\forall
x\in\Pi(\omega,i),\quad\lim_{n\to+\infty}\frac{1}{n}\log(f_{\omega}^{n})^{\prime}(x)=\lambda_{i}^{+},$

where the convergence is uniform. Let

$c=\frac{1}{2}\inf_{i\in[d]}-\lambda_{i}^{+}>0,$

then there exists $C>0$ such that $\forall\omega\in\Sigma^{\prime},$

$(f_{\omega}^{n})^{\prime}(x)\leqslant C2^{-cn},\quad\forall
x\in\Pi(\omega),n\in{\mathbb{N}}.$

By Lemma 4.25, there exists $C^{\prime},\rho>0$ such that for every
$\omega\in\Sigma^{\prime},$

$\widetilde{\varkappa}(f_{\omega}^{n},B(x,\rho))\leqslant
C^{\prime},\quad\forall x\in\Pi(\omega),n\in{\mathbb{N}}.$

In particular, for every $y\in B(x,\rho),$
$\frac{1}{n}\log(f_{\omega}^{n})^{\prime}(y)\to\lambda_{i}^{+}<0,$ it follows
that $y\neq\Xi(\omega).$

For every $\omega\in\Sigma^{\prime},$ let

$K(\omega)=\SS^{1}\setminus\left(\bigcup_{x\in\Pi(\omega)}B(x,\rho)\right).$

Then $\Xi(\omega)\subset K(\omega)\subset W^{u}(\omega)$ and $K(\omega)$ is a
finite union of closed intervals. By (7) of Theorem 4.16, for every connected
component $J$ of $K(\omega),$ $|f_{\omega}^{-n}J|\to 0.$ Now we can take
$\Sigma^{\prime\prime}\subset\Sigma^{\prime}$ be a subset with
$\bm{\mathrm{P}}(\Sigma^{\prime\prime})>1-\varepsilon$ and $N$ sufficiently
large satisfying

$|f_{\omega}^{-N}J|<\varepsilon$

for every $\omega\in\Sigma^{\prime\prime}$ and every connected component $J$
of $K(\omega).$

Let $\Sigma_{\varepsilon}=\sigma^{-N}\Sigma^{\prime\prime},$ then
$\bm{\mathrm{P}}(\Sigma_{\varepsilon})=\bm{\mathrm{P}}(\Sigma^{\prime\prime})>1-\varepsilon$
and the condition (2) holds. For every $\omega\in\Sigma_{\varepsilon},$ set

$U(\omega)=f_{\sigma^{N}\omega}^{-N}\left(\bigcup_{x\in\Pi(\sigma^{N}\omega)}B(x,\rho)\right),$

so that $\Pi(\omega)\subset U(\omega)\subset
W^{s}(\omega)=\SS^{1}\setminus\Xi(\omega).$ The complement of $U(\omega)$ is
exactly

$f_{\sigma^{N}\omega}^{-N}(K(\sigma^{N}\omega)),$

whose connected components have length less than $\varepsilon$ by the choice
of $N.$ It remains to estimate the distortion on
$f_{\sigma^{N}\omega}^{-N}B(x,\rho).$ Let

$M=\max\left\\{\sup_{f\in\mathcal{S}}\|f^{\prime}\|,\sup_{f\in\mathcal{S}}\|\log
f^{\prime}\|_{\mathrm{Lip}},2\right\\},$

and take $C=C(\varepsilon)=M^{N}C^{\prime}.$ Since
$\sigma^{N}\omega\in\Sigma^{\prime\prime},$ combining (3.5), we have

$\widetilde{\varkappa}(f_{\omega}^{n},f_{\sigma^{N}\omega}^{-N}B(x,\rho))\leqslant
C(\varepsilon),\quad\forall x\in\Pi(\sigma^{N}\omega),\forall n\geqslant 0.$

This finishes the proof of the proposition. ∎

######  Corollary 4.26.

For every $\varepsilon>0,$ let $\Sigma_{\varepsilon}\subset\Sigma$ be the set
defined in Proposition 4.24 with the corresponding
$U(\omega)=U(\omega,\varepsilon).$ Then there exists positive numbers
$\varepsilon_{n}\to 0$ such that

$|f_{\omega}^{n}I|<2^{(\lambda_{i}^{+}+\varepsilon_{n})n}$

for every $\omega\in\Sigma_{\varepsilon},n\in{\mathbb{N}}$ and a connected
component $I$ of $U(\omega)\cap W^{s}(\omega,i).$

Combined with Theorem 4.16(5) and Lemma 4.20, the following is immediate.

######  Corollary 4.27.

There exists $\Sigma^{\prime}\subset\Sigma$, such that
$\bm{\mathrm{P}}(\Sigma^{\prime})>0$ and the following holds for every
$\omega\in\Sigma^{\prime}$. For every $i\in[d]$ and every $x\in\Pi(\omega,i)$
there is an open interval $I$ containing $x$ satisfying
$\nu_{i}(I)\geqslant\frac{1}{2r}$ and such that

$\forall
n\in{\mathbb{N}},\quad\lambda^{+}_{i}-\varepsilon_{n}\leqslant\frac{1}{n}\log|f_{\omega}^{n}I|\leqslant\lambda^{+}_{i}+\varepsilon_{n}$

where $\varepsilon_{n}\to 0^{+}$ is a sequence independent of $(\omega,x)$.

###  4.6 Effective convergence to the Furstenberg boundary

We may consider the distribution of $\Pi(\omega,i)$ as a probability measure
on $\SS_{r}.$ For every $i\in[d],$ consider the measurable maps

$\Pi(\,\bm{\cdot}\,,i):\Sigma\to\SS_{r},\quad\Xi(\,\bm{\cdot}\,,i):\Sigma\to\SS_{r}$

Let $\underline{\nu}_{i}^{+},\underline{\nu}_{i}^{-}$ be the probability
measures on $\SS_{r}$ which are the push forward of $\bm{\mathrm{P}}$ by
$\Pi(\,\bm{\cdot}\,,i),\Xi(\,\bm{\cdot}\,,i),$ respectively. For every
$\underline{x}\in\SS_{r},$ recall $u_{\underline{x}}$ denotes the uniform
probability measure on $\underline{x}\subset\SS^{1}.$ Combining Theorem
4.12(2) and the definition of $\underline{\nu}_{i}^{\pm},$ we have

$\nu_{i}^{+}=\int_{\SS_{r}}u_{\underline{x}}\mathrm{d}\underline{\nu}_{i}^{+}(\underline{x}),\quad\nu_{i}^{-}=\int_{\SS_{r}}u_{\underline{x}}\mathrm{d}\underline{\nu}_{i}^{-}(\underline{x}).$

######  Proposition 4.28.

Given $\varepsilon>0$, there exists $N>0$ such that for every $i\in[d]$, for
all $\underline{x}\in\operatorname{\mathrm{supp}}\underline{\nu}_{i}^{+}$ and
all $n\geqslant N$,

$\bm{\mathrm{P}}\left\\{\omega:\underline{d}(f_{\sigma^{-n}\omega}^{n}\underline{x},\Pi(\omega,i))<2^{(\lambda_{i}^{+}+\varepsilon)n}\right\\}\geqslant
1-\varepsilon.$

This Proposition can be interpreted as an effective version of the the
convergence (4.3).

For $\omega\in\Sigma$ and $i\in[d]$, define

$\underline{W}^{s}(\omega,i)=\Bigl{\\{}\,\underline{x}\in\SS_{r}:\bigcup_{x\in\underline{x}}W^{s}(\omega,x)=W^{s}(\omega,i)\,\Bigr{\\}}.$

In other words, $\underline{W}^{s}(\omega,i)$ is the set of
$\underline{x}\in\SS_{r}$ whose elements fall in different connected
components of $W^{s}(\omega,i)$. Note that $\underline{W}^{s}(\omega,i)$
depends only on $(\omega^{+},i)$ and that $\Pi(\omega,i)\in W^{s}(\omega,i)$
for $\bm{\mathrm{P}}$-almost every $\omega\in\Sigma$ and hence
$\operatorname{\mathrm{supp}}\underline{\nu}_{i}^{+}\subset\underline{W}^{s}(\omega,i)$
for $\bm{\mathrm{P}}$-almost every $\omega\in\Sigma$.

We will show Proposition 4.28 for all $\underline{x}\in\SS_{r}$ such that
$\underline{x}\in\underline{W}^{s}(\omega,i)$ for $\bm{\mathrm{P}}$-almost
every $\omega\in\Sigma$. Such $\underline{x}$ form a larger set than
$\operatorname{\mathrm{supp}}\underline{\nu}_{i}^{+}$ in some situations. For
instance, if $d=r=1$, then the proposition holds for all $x\in\SS^{1}$.

###### Proof.

In view of Lemma 4.20, let $\varepsilon^{\prime}\in(0,\varepsilon/2)$ be such
that

$\forall j\in[d],\,\forall
x\in\SS^{1},\quad\nu_{j}^{-}(B(x,\varepsilon^{\prime}))\leqslant\frac{\varepsilon}{2dr^{2}}.$

Let $\Sigma_{\varepsilon^{\prime}}\subset\Sigma$ be the set of uniform good
words with the corresponding constant $C=C(\varepsilon^{\prime})$ and open
sets $U(\omega)=U(\omega,\varepsilon^{\prime})$ given by Proposition 4.24.
Consider

$\Sigma^{\prime}=\left\\{\,\omega\in\Sigma_{\varepsilon^{\prime}}:\underline{x}\subset
U(\omega)\,\right\\}.$

We claim that $\bm{\mathrm{P}}(\Sigma^{\prime})\geqslant 1-\varepsilon.$
Indeed, if $\omega\in\Sigma_{\varepsilon^{\prime}}$ and
$\omega\not\in\Sigma^{\prime}$, then there exists $x\in\underline{x}$ such
that $x\in\subset\SS^{1}\setminus
U(\omega)\subset\bigcup_{y\in\Xi(\omega)}B(y,\varepsilon^{\prime})$. It
follows that $B(x,\varepsilon^{\prime})\cap\Pi(\omega)\neq\varnothing.$
Recalling (1) and (2) in Theorem 4.12, we have

$\bm{\mathrm{P}}(\Sigma_{\varepsilon^{\prime}}\setminus\Sigma^{\prime})\leqslant\int
r\sum_{j\in[d]}\sum_{x\in\underline{x}}u_{\Xi(\omega,i)}(B(x,\varepsilon^{\prime}))\,\mathrm{d}\bm{\mathrm{P}}(\omega)=r\sum_{j\in[d]}\sum_{x\in\underline{x}}\nu_{j}^{-}(B(x,\varepsilon^{\prime}))\leqslant
r^{2}d\cdot\frac{\varepsilon}{2dr^{2}}=\frac{\varepsilon}{2},$

Showing $\bm{\mathrm{P}}(\Sigma^{\prime})\geqslant 1-\varepsilon.$

Moreover, for $\bm{\mathrm{P}}$-almost every $\omega\in\Sigma^{\prime}$, since
$x\in\underline{W}^{s}(\omega,i)$, each element of $x$ falls exactly in one
the $r$ connected components of $U(\omega)$. By Corollary 4.26, there exists
$N$ depending only on $\mu$ and $\varepsilon$ such that for all $n\geqslant
N$, all $\omega\in\Sigma^{\prime}$ and every connected component $I$ of
$U(\omega),$
$|f_{\omega}^{n}I|<\frac{1}{r}2^{(\lambda_{i}^{+}+\varepsilon)n}.$

Now for every $n\geqslant N$, if $\sigma^{-n}\omega\in\Sigma^{\prime},$ then
elements of $\underline{x}$ can be paired with those of
$\Pi(\sigma^{-n}\omega,i)$ such that each pair belong to the same connected
component of $U(\sigma^{-n}\omega)$. Hence

$\underline{d}(f_{\sigma^{-n}\omega}^{n}\underline{x},\Pi(\omega,i))<r\cdot\frac{1}{r}2^{(\lambda_{i}^{+}+\varepsilon)n}=2^{(\lambda_{i}^{+}+\varepsilon)n}.$

Since
$\bm{\mathrm{P}}(\sigma^{-n}\Sigma^{\prime})=\bm{\mathrm{P}}(\Sigma^{\prime})\geqslant
1-\varepsilon,$ the conclusion follows. ∎

##  5 Exact dimensionality of stationary measures

In this section, we will prove Theorem 2.2, which gives the exact
dimensionality of the ergodic stationary measures. Our strategy is similar to
that used in [40, Theorem 3.4] for $\mathrm{PSL}(2,{\mathbb{R}})$ actions, but
general circle diffeomorphisms are not as rigid as Möbius transformations.
Therefore, we will make use of Pesin theory, distortion controls, and a
hyperbolic time argument (the set given by (5.5) corresponds to hyperbolic
times) to obtain our result.

###  5.1 Preparation

Here, we list some useful results taken from [40].

Let $\mu$ be a finitely supported probability measure on
$\mathrm{Diff}_{+}^{2}(\SS^{1})$ without common invariant probability
measures. Let $\nu$ be a $\mu$-stationary measure on $\SS^{1}$ and let $m$ be
the corresponding ergodic $u$-state (recall Proposition 4.2). For every
$f\in\mathcal{S}$ and interval $I\subset\SS^{1}$, define

$\varphi(f,I)\coloneqq\frac{f_{*}\nu(I)}{\nu(I)}.$ (5.1)

Besicovitch’s derivation theorem states that

$\frac{\mathrm{d}f_{*}\nu}{\mathrm{d}\nu}(x)=\lim_{I\ni x,\ |I|\to
0}\varphi(f,I),\quad\ \nu\text{-}\mathrm{a.e.}\ x.$

######  Lemma 5.1 ([40, Lemma 3.5]).

For every $f\in\mathcal{S}$ and $t>0,$

$\nu\left\\{x:\sup\nolimits_{I\ni x}\varphi(f,I)>t\right\\}\leqslant
2t^{-1},\quad f_{*}\nu\left\\{x:\inf\nolimits_{I\ni
x}\varphi(f,I)<t^{-1}\right\\}\leqslant 2t^{-1}.$

Using the relation $Q_{*}m=\nu$ and the fact that $m$ is a $u$-state, we
obtain the following consequence, which is essentially [40, Corollary 3.6].

######  Lemma 5.2.

For every $t>0,$

$m\left\\{(\omega,x)\in\Sigma\times\SS^{1}:\sup\nolimits_{I\ni
x}\varphi(f_{\sigma^{-1}\omega},I)\geqslant t\right\\}\ll_{r}t^{-1},$

and

$m\left\\{(\omega,x)\in\Sigma\times\SS^{1}:\inf\nolimits_{I\ni
x}\varphi(f_{\sigma^{-1}\omega},I)\leqslant t^{-1}\right\\}\ll_{r}t^{-1}.$

Define
$\mathcal{O}:\Sigma\times\SS^{1}\times{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ as

$\mathcal{O}(\omega,x,\rho)\coloneqq\sup_{I\ni
x,|I|\leqslant\rho}\log\varphi(f_{\sigma^{-1}\omega},I)-\inf_{I\ni
x,|I|\leqslant\rho}\log\varphi(f_{\sigma^{-1}\omega},I).$

Besicovitch’s derivation theorem can be restated as $\lim_{\rho\to
0^{+}}\mathcal{O}(\omega,x,\rho)=0$ for $m$-almost every
$(\omega,x)\in\Sigma\times\SS^{1}$. By Lemma 5.2 and the layer cake
representation, the following is immediate.

######  Lemma 5.3 (see [40, Corollary 3.7]).

The function $(\omega,x)\mapsto\sup_{\rho>0}\mathcal{O}(\omega,x,\rho)$ is in
$L^{1}(\Sigma\times\SS^{1},m).$

The following is a variant of Maker’s Theorem, as stated in [40].

######  Theorem 5.4 ([40, Theorem 3.8]).

Let $(X,\mathcal{F},\theta,T)$ be an ergodic probability measure preserving
system. Let $G_{t}:X\to{\mathbb{R}}$ be a measurable $1$-parameter family of
measurable functions such that $\sup_{t}|G_{t}|\in L^{1}.$ Suppose that

$G=\lim_{t\to 0}G_{t}$

exists almost everywhere. Let $t_{N,n}:X\to{\mathbb{R}}$ be functions with the
property that for $\theta$-a.e. $x$ and every $\varepsilon>0,$ for large
enough $N,$

$|t_{N,n}|<\varepsilon,\quad\text{for }1\leqslant n\leqslant(1-\varepsilon)N.$

Then

$\lim_{N\to+\infty}\frac{1}{N}\sum_{n=1}^{N}G_{t_{N,n}(x)}(T^{n}x)=\int
G\,\mathrm{d}\theta,\quad\ \theta\text{-}\mathrm{a.e.}\ x.$

###  5.2 Exact dimensionality (Proof Theorem 2.2)

Let $\nu$ be an ergodic $\mu$-stationary measure and let $m\in\mathcal{P}^{u}$
be the corresponding ergodic $u$-state. Let
$\lambda=\lambda(\mu,\nu)=\lambda(m)$ denote the Lyapunov exponent, which is
negative under our assumption. Let $h=h_{\mathrm{F}}(\mu,\nu)$ be the
Furstenberg entropy. Note that using the relation between $\nu$ and $m$ and a
change of variable, the Furstenberg entropy can be expressed as

$h=\int\log\frac{\mathrm{d}(f_{\sigma^{-1}\omega})_{*}\nu}{\mathrm{d}\nu}(x)\,\mathrm{d}m(\omega,x).$

###### Proof of Theorem 2.2.

We aim to show that for $\nu$-almost every $x\in\SS^{1}$, or, in other words,
for $m$-almost every $(\omega,x)\in\Sigma\times\SS^{1}$,

$\lim_{\rho\to 0^{+}}\frac{\log\nu(B(x,\rho))}{\log\rho}=-\frac{h}{\lambda}.$
(5.2)

Fix an arbitrary element $\omega\in\Sigma$ regular for the random walk
(Definition 4.19) and a point $x\in\operatorname{\mathrm{supp}}m_{\omega}.$
Let $I_{0}=B(x,\rho)$ and for integers $n\geqslant 0$, set

$x_{n}=f_{\omega}^{-n}x\quad\text{and}\quad I_{n}=f_{\omega}^{-n}I_{0}.$

For any integer $N\geqslant 1$, we can write

$\nu(I_{0})=\nu(I_{N})\prod_{n=0}^{N-1}\frac{\nu(I_{n})}{\nu(I_{n+1})}=\nu(I_{N})\prod_{n=0}^{N-1}\frac{\nu(I_{n})}{(f_{\sigma^{-(n+1)}\omega})_{*}\nu(I_{n})}.$

Recall the definition of $\varphi(f,I)$ in (5.1),

$-\log\nu(I_{0})=-\log\nu(I_{N})+\sum_{n=0}^{N-1}\log\varphi(f_{\sigma^{-(n+1)}\omega},I_{n}).$

Since

$\left|\log\frac{\mathrm{d}(f_{\sigma^{-(n+1)}\omega})_{*}\nu}{\mathrm{d}\nu}(x_{n})-\log\varphi(f_{\sigma^{-(n+1)}\omega},I_{n})\right|\leqslant\mathcal{O}(\sigma^{-n}\omega,|I_{n}|),$

we have

$\left|-\log\nu(I_{0})-\sum_{n=0}^{N-1}\log\frac{\mathrm{d}(f_{\sigma^{-(n+1)}\omega})_{*}\nu}{\mathrm{d}\nu}(x_{n})\right|\leqslant-\log\nu(I_{N})+\sum_{n=0}^{N-1}\mathcal{O}(\sigma^{-n}\omega,|I_{n}|).$
(5.3)

Note that the sum in the left-hand side is independent of $\rho$ and it is a
Birkhoff sum of the function
$(\omega,x)\mapsto\log\frac{\mathrm{d}(f_{\sigma^{-1}\omega})_{*}\nu}{\mathrm{d}\nu}(x)$
for the transformation $F^{-1}$. Hence, by Birkhoff’s ergodic theorem, outside
a null set, as $N\to+\infty$,

$\frac{1}{N}\sum_{n=0}^{N-1}\log\frac{\mathrm{d}(f_{\sigma^{-(n+1)}\omega})_{*}\nu}{\mathrm{d}\nu}(x_{n})\to
h.$ (5.4)

Recall that our objective is to estimate $\lim_{\rho\to
0^{+}}\frac{\log\nu(I_{0})}{\log\rho}$. To make use of (5.3), we still have
the freedom to choose $N=N(\omega,\rho)$ according to $\rho>0$. This choice
will guarantee $N(\omega,\rho)\sim\frac{\log\rho}{\lambda}$ and that in the
same time, the right-hand side of (5.3) is relatively small compared to
$N(\omega,\rho)$ as $\rho\to 0^{+}$.

By Corollary 4.27, we can fix a subset $\Sigma^{\prime}\subset\Sigma$ and a
sequence $\varepsilon_{n}\to 0^{+}$ at first, such that
$\bm{\mathrm{P}}(\Sigma^{\prime})>0$ and that for $m$-almost every
$(\omega^{\prime},y)\in\Sigma^{\prime}\times\SS^{1}$, there is an interval
$I=I(\omega^{\prime},y)$ containing $y$ and satisfying
$\nu(I)\geqslant\frac{1}{2r}$ and for any $n\in{\mathbb{N}}$,
$\log|f_{\omega^{\prime}}^{n}y|\leqslant(\lambda+\varepsilon_{n})n$.

For every $\omega\in\Sigma,$ we define

$A_{\omega}\coloneqq\left\\{N:\sigma^{-N}\omega\in\Sigma^{\prime}\right\\},$
(5.5)

which is the family of hyperbolic times that $f_{\sigma^{-N}\omega}^{N}$
possesses a significant hyperbolicity for $N\in A_{\omega}$. For
$\bm{\mathrm{P}}$-almost every $\omega\in\Sigma$, the set $A_{\omega}$ is
infinite and thus,

$N(\omega,\rho)\coloneqq\min\left\\{N\in
A_{\omega}:2^{(\lambda+\varepsilon_{N})N}\leqslant\rho\right\\}$

is well defined for all $\rho>0$. We claim that

$\frac{\log\rho}{N(\omega,\rho)}\to\lambda\quad\text{as }\rho\to 0^{+}.$ (5.6)

Indeed, if we list elements of $A_{\omega}=\\{N_{1},N_{2},\dotsc\\}$ in
increasing order, then by Birkhoff’s ergodic theorem, for
$\bm{\mathrm{P}}$-almost every $\omega$,
$N_{k}/k\to\bm{\mathrm{P}}(\Sigma^{\prime})^{-1}$ and hence $N_{k-1}/N_{k}\to
1$ as $k\to+\infty$. By definition, $N(\omega,\rho)$ is some $N_{k}$ such that
$(\lambda+\varepsilon_{N_{k}})N_{k}\leqslant\log\rho<(\lambda+\varepsilon_{N_{k-1}})N_{k-1}$.
Dividing by $N_{k}$ and taking limit shows (5.6).

We write $N=N(\omega,\rho)$ as a short hand. From $N\in A_{\omega}$ and the
point $x\in\operatorname{\mathrm{supp}}m_{\omega}$, we know that there is an
interval $I=I(F^{-N}(\omega,x))$ containing $f_{\omega}^{-N}x$ and satisfying
$\nu(I)\geqslant\frac{1}{2r}$ and

$|f_{\sigma^{-N}\omega}^{N}I|\leqslant
2^{(\lambda+\varepsilon_{N})N}\leqslant\rho.$

Hence $f_{\sigma^{-N}\omega}^{N}I\subset B(x,\rho)=I_{0}$ and
$I_{N}=f_{\omega}^{-N}I_{0}\supset I$ has measure

$\nu(I_{N})\geqslant\frac{1}{2r}.$ (5.7)

We claim that for $\bm{\mathrm{P}}$-almost every $\omega\in\Sigma$, there is
$\widetilde{\varepsilon}_{\rho}\to 0$ as $\rho\to 0^{+}$ (depending on
$\omega$) such that for every $1\leqslant n\leqslant N$ we have

$|I_{n}|\leqslant 2^{\lambda(N-n)+\widetilde{\varepsilon}_{\rho}N}.$

Thus, applying Theorem 5.4 (let $(G_{t},t_{N,n},T,X)$ be
$(\mathcal{O}(\cdot,t),|I_{n}|,F,\Sigma\times\SS^{1})$) we obtain that

$\frac{1}{N}\sum_{n=0}^{N-1}\mathcal{O}(\sigma^{-n}\omega,|I_{n}|)\to 0$

for $m$-almost every $(\omega,x)\in\Sigma\times\SS^{1}$. Combining (5.3),
(5.4), (5.7) and (5.6) we obtain the desired convergence (5.2).

To show the claim, recall that $\omega$ is regular for the random walk. Then
there exists positive numbers $\varepsilon_{n}^{\prime}\to 0$ such that

$\forall n\in{\mathbb{N}},\quad(f_{\omega}^{-n})^{\prime}(x)\leqslant
2^{(-\lambda+\varepsilon_{n}^{\prime})n}\quad\text{and}\quad\sum_{k=0}^{n-1}(f_{\omega}^{-k})^{\prime}(x)\leqslant
2^{(-\lambda+\varepsilon_{n}^{\prime})n}.$

Writing
$M=\max\left\\{\sup_{f\in\mathcal{S}^{-1}}\|f^{\prime}\|,\sup_{f\in\mathcal{S}^{-1}}\|\log
f^{\prime}\|_{\mathrm{Lip}},2\right\\}$, by Proposition 3.8, we have
$\varkappa(f_{\omega}^{-n},I_{0})\leqslant 1$ whenever

$\rho\leqslant\frac{1}{2M}2^{(\lambda-\varepsilon_{n}^{\prime})n}.$

For such $n,$ we have

$|I_{n}|\leqslant 2(f_{\omega}^{n})^{\prime}(x)|I_{0}|\leqslant
2^{(-\lambda+\varepsilon_{n}^{\prime})n+2}\rho.$

Recall (5.6). So we can write

$\rho=2^{(\lambda+\varepsilon_{\rho})N}$

with some $\varepsilon_{\rho}\to 0$ as $\rho\to 0^{+}$. The claim can be
verified easily, with the choice

$\widetilde{\varepsilon}_{\rho}=\varepsilon_{\rho}+\sup_{1\leqslant n\leqslant
N}\varepsilon_{n}^{\prime}\frac{n}{N}+\frac{2+\log M}{N}.\qed$

##  6 Dimension Formulas

In the previous section, we established the exact dimensionality of ergodic
stationary measures and the dimension is expressed as the ratio between the
Furstenberg entropy and the Lyapunov exponent. However, computing the
Furstenberg entropy can be challenging in practice. In this section, we will
demonstrate that the random walk entropy can be used as a substitute for the
Furstenberg entropy under a suitable separation condition, namely the local
discreteness of the group generated by the support of the measure.

Let us explain how the local discreteness helps in this setting. Firstly, we
use the language of entropy to interpret the dimension. Let $\alpha$ be the
exact dimension of an ergodic $\mu$-stationary measure $\nu.$ Then $\alpha$
also equals to the entropy dimension of $\nu,$ i.e.,

$\alpha=\lim_{n\to\infty}\frac{1}{n}H(\nu,\mathcal{D}_{n}).$

Recall that $\Pi(\omega)$ plays the role of the Furstenberg boundary.
Proposition 4.28 gives rise to some $\underline{x}\in\SS_{r}$ with which we
can interpret $\nu$ as the weak-* limit

$\nu=\lim_{n\to\infty}\mu^{*n}*u_{\underline{x}}.$

Combining the convergences above, we can expect that

$\frac{1}{n}H(\mu^{*n}*u_{\underline{x}},\mathcal{D}_{-\lambda
n})\to-\lambda\alpha,$

where $\lambda<0$ is the corresponding Lyapunov exponent. It relates the
action of $n$-step iterations of $\mu$ and the dimension $\alpha.$

If there were a gap between $h_{\mathrm{RW}}(\mu)$ and $-\lambda\alpha,$ then
there would be exponentially many elements in
$\operatorname{\mathrm{supp}}\mu^{*n}$ mapping a point in $\underline{x}$ into
the same atom of $\mathcal{D}_{-\lambda n}.$ Combining the derivative and
distortion controls for good words in $\operatorname{\mathrm{supp}}\mu^{*n},$
the restrictions of these elements to a fixed interval would be close to each
other. This allows us to construct elements of the form $g^{-1}f$ that are
arbitrarily ($C^{1}$-)close to the identity on an interval. This contradicts
the local discreteness assumption.

###  6.1 Dimension formula on the circle (Proof of Theorem 2.6)

We will prove Theorem 2.6 in this subsection. Let $\mu$ be a probability
measure on $\mathrm{Diff}_{+}^{2}(\SS^{1})$ whose support is finite and does
not preserve any Borel probability measure on $\SS^{1}$. Let $\nu$ to be an
ergodic $\mu$-stationary measure. We know that $\nu$ is exact dimensional, and
denote its dimension by $\alpha$.

We use the notation of Theorem 4.12 and without loss of generality, assume
that $\nu=\nu_{0}^{+}$. Recall $\underline{\nu}_{0}^{+}$ is the probability
measure on $\SS_{r}$ which is the image measure of $\bm{\mathrm{P}}$ by
$\Pi(\,\bm{\cdot}\,,0)$. We abbreviate $\underline{\nu}_{0}^{+}$ to
$\underline{\nu}$ in this section.

######  Lemma 6.1.

For every $\underline{x}\in\operatorname{\mathrm{supp}}\underline{\nu},$ we
have

$\lim_{n\to+\infty}\frac{1}{n}H(\mu^{*n}*u_{\underline{x}},\mathcal{D}_{-\lambda
n})=-\alpha\lambda.$

Recall that $u_{\underline{x}}$ denotes the uniform probability measure on
$\underline{x}\in\SS_{r}$.

###### Proof.

By Proposition 4.28, for every $\varepsilon>0$, for every $n$ large enough, we
have

$\bm{\mathrm{P}}(\Sigma^{\prime})\geqslant 1-\varepsilon$

where

$\Sigma^{\prime}=\left\\{\omega\in\Sigma:\underline{d}(f_{\sigma^{-n}\omega}^{n}\underline{x},\Pi(\omega,0))<2^{(\lambda+\varepsilon)n}\right\\}.$

From this we can construct a Borel probability measure $\theta_{n}$ on
$\SS^{1}\times\SS^{1}$ whose marginals distributions are
$\mu^{*n}*u_{\underline{x}}$ and $\nu$ and moreover

$\theta_{n}\left\\{(x,y):d(x,y)\leqslant
2^{(\lambda+\varepsilon)n}\right\\}\geqslant 1-\varepsilon.$ (6.1)

Indeed, write $\underline{x}=\\{x_{1},\dotsc,x_{r}\\}$ and for each
$\omega\in\Sigma$, label elements of $\Pi(\omega,0)$ as
$y_{1}(\omega),\dotsc,y_{r}(\omega)$ (in a measurable way). Define

$\theta_{n}\coloneqq\int_{\Sigma}\frac{1}{r}\sum_{i=1}^{r}\delta_{(f_{\sigma^{-n}\omega}^{n}x_{i},y_{i}(\omega))}\,\mathrm{d}\bm{\mathrm{P}}(\omega),$

so that $\theta$ is a probability measure on $\SS^{1}\times\SS^{1}$. Its
projection to the first coordinate is $\mu^{*n}*u_{\underline{x}}$ and to the
second coordinate is $\nu$. At this stage, we still have the freedom to choose
the labelling $\Pi(\omega,0)=\\{y_{1}(\omega),\dotsc,y_{r}(\omega)\\}$. For
$\omega\in\Sigma^{\prime}$, there is unique labelling such that for every
$i=1,\dotsc,r$, $d(f_{\sigma^{-n}\omega}^{n}x_{i},y_{i}(\omega))\leqslant
2^{(\lambda+\varepsilon)n}$, because $\Pi(\omega,0)$ is $c$-separated for some
$c=c(\mu)>0$ by Lemma 4.21 and
$d(f_{\sigma^{-n}\omega}^{n}\underline{x},\Pi(\omega,0))<2^{(\lambda+\varepsilon)n}<c/2$.
This shows (6.1).

We have the following elementary fact on entropies.

######  Lemma 6.2.

Let $\theta$ be a Borel probability measure on $\SS^{1}\times\SS^{1}$ such
that

$\theta\left\\{(x,y):d(x,y)\leqslant 2^{-(1-\varepsilon)n}\right\\}\geqslant
1-\varepsilon$

for some $n\in{\mathbb{N}}$ and $\varepsilon>0.$ Let $\eta$ and $\zeta$ be the
marginal distributions of $\theta$ on the two coordinates. Then

$|H(\eta,\mathcal{D}_{n})-H(\zeta,\mathcal{D}_{n})|\ll\varepsilon n+1.$

Applying this lemma, we obtain

$|H(\mu^{*n}*u_{\underline{x}},\mathcal{D}_{-\lambda
n})-H(\nu,\mathcal{D}_{-\lambda n})|\ll\varepsilon(-\lambda)n+1.$

Since $\varepsilon>0$ is arbitrarily small and
$\lim_{n\to+\infty}\frac{1}{n}H(\nu,\mathcal{D}_{n})\to\alpha$, the desired
convergence follows. ∎

######  Lemma 6.3.

Assume that $-\lambda\alpha<h_{\mathrm{RW}}(\mu).$ Then there are constants
$c,C>0$, a closed interval $J\subset\SS^{1}$ centered at a point in
$\operatorname{\mathrm{supp}}\nu$ and infinitely many integers
$n\in{\mathbb{N}}$ with a sequence of positive numbers $\varepsilon_{n}\to 0$
and subsets $T_{n}\subset T_{\mu},$ satisfying

  1. (1)

$\\#T_{n}\geqslant 2^{cn},$

  2. (2)

for every $f\in T_{n},$
$f^{\prime}|_{J}\in[2^{(\lambda-\varepsilon_{n})n},2^{(\lambda+\varepsilon_{n})n}]$
and $\widetilde{\varkappa}(f,J)\leqslant C,$

  3. (3)

$f(J)$, $f\in T_{n}$, falls in a common interval of length at most
$2^{(\lambda+\varepsilon_{n})n}$.

###### Proof.

By the assumption, we can take $c>0$ such that $-\lambda\alpha+10\ell
c<h_{\mathrm{RW}}(\mu),$ where
$\ell=\max\left\\{\\#\operatorname{\mathrm{supp}}\mu,-\lambda\right\\}.$

By concavity (3.2), for all $n\in{\mathbb{N}}$,

$\frac{1}{r}\sum_{x\in\underline{x}}H(\mu^{*n}*\delta_{x},\mathcal{D}_{-\lambda
n})\leqslant H(\mu^{*n}*u_{\underline{x}},\mathcal{D}_{-\lambda n}).$

Thus, by Lemma 6.1, there is $x\in\underline{x}$ such that

$\liminf_{n\to+\infty}\frac{1}{n}H(\mu^{*n}*\delta_{x},\mathcal{D}_{-\lambda
n})\leqslant-\lambda\alpha.$ (6.2)

Recall Lemma 4.20. Let $\varepsilon^{\prime}>0$ be small so that

$\forall
i\in[d],\quad\nu_{i}^{-}(B(x,3\varepsilon^{\prime}))\leqslant\frac{c}{dr}.$

Let

$J=[x-\varepsilon^{\prime},x+\varepsilon^{\prime}].$

Let $\Sigma_{\varepsilon}\subset\Sigma$, $C=C(\varepsilon)$ and
$U(\omega)=U(\omega,\varepsilon)$ be given by Proposition 4.24 applied to
$\varepsilon=\min\left\\{c,\varepsilon^{\prime}\right\\}$. By the uniform
convergence and a uniform distortion bound in $\Sigma_{\varepsilon},$ there
exists positive constants $\varepsilon_{n}\to 0$ such that for every $y\in
U(\omega)\cap W^{s}(\omega,0),$

$|\frac{1}{n}\log(f_{\omega}^{n})^{\prime}(y)-\lambda|\leqslant\varepsilon_{n}.$

Since $\underline{x}\in\operatorname{\mathrm{supp}}\underline{\nu}$, for
$\bm{\mathrm{P}}$-almost every $\omega\in\Sigma$, we have
$\underline{x}\subset W^{s}(\omega,0)$. In particular, $J\cap
W^{s}(\omega,0)\neq\varnothing$. Thus, for $\omega\in\Sigma_{\varepsilon},$ if
$J\not\subset U(\omega),$ then the $2\varepsilon^{\prime}$-neighborhood of $J$
must contain a point in $\Xi(\omega).$ Thus,

$\bm{\mathrm{P}}\left\\{\omega\in\Sigma_{\varepsilon}:J\not\subset
U(\omega)\right\\}\leqslant\bm{\mathrm{P}}\left\\{\omega:\Xi(\omega)\cap
B(x,3\varepsilon^{\prime})\neq\varnothing\right\\}\leqslant
r\sum_{j\in[d]}\nu_{j}^{-}(B(x,3\varepsilon^{\prime}))\leqslant c.$

If follow that the set

$\Sigma^{\prime}=\left\\{\omega\in\Sigma_{\varepsilon}:J\subset U(\omega)\cap
W^{s}(\omega,0)\right\\},$

satisfies $\bm{\mathrm{P}}(\Sigma^{\prime})\geqslant 1-2c.$

For every $n\in{\mathbb{N}},$ define
$\Sigma_{n}^{\prime}\coloneqq\left\\{f_{\omega}^{n}:\omega\in\Sigma^{\prime}\right\\},$
then $\mu^{*n}(\Sigma_{n}^{\prime})\geqslant 1-2c.$ Denote
$\tau_{n}=1-\mu^{*n}(\Sigma_{n}^{\prime})\leqslant 2c$ and let

$\mu_{n}=\frac{1}{1-\tau_{n}}\mu^{*n}|_{\Sigma_{n}^{\prime}}.$

By the concavity and almost convexity of the entropy (3.2), we have

$|H(\mu_{n})-H(\mu^{*n})|\leqslant
1+\tau_{n}(\\#\operatorname{\mathrm{supp}}\mu)n\leqslant 1+2\ell cn$

and

$|H(\mu_{n}*\delta_{x},\mathcal{D}_{-\lambda
n})-H(\mu^{*n}*\delta_{x},\mathcal{D}_{-\lambda n})|\leqslant
1+\tau_{n}(-\lambda)n\leqslant 1+2\ell cn.$

Remembering $\frac{1}{n}H(\mu^{*n})\to h_{\mathrm{RW}}(\mu)$, the choice of
$c$ and (6.2), we have

$H(\mu_{n})\geqslant H(\mu_{n}*\delta_{x},\mathcal{D}_{-\lambda n})+cn.$

for infinitely many $n$.

Define $e_{x}:\mathrm{Diff}^{2}_{+}(\SS^{1})\to\SS^{1}$ to be the map
$f\mapsto f(x),$ then $e_{x}^{-1}\mathcal{D}_{-\lambda n}$ is a finite
partition of $\mathrm{Diff}^{2}_{+}(\SS^{1}).$ We have

$H(\mu_{n}|e_{x}^{-1}\mathcal{D}_{-\lambda
n})=H(\mu_{n})-H(\mu_{n},e_{x}^{-1}\mathcal{D}_{-\lambda
n})=H(\mu_{n})-H(\mu_{n}*\delta_{x},\mathcal{D}_{-\lambda n})\geqslant cn.$

This implies the existence of $I\in\mathcal{D}_{-\lambda n}$ such that the set
$T_{n}=\left\\{f\in\Sigma_{n}^{\prime}:f(x)\in I\right\\}$ has cardinality
$\\#T_{n}\geqslant 2^{cn}.$ By the definition of $\Sigma_{n}^{\prime}$, for
any $f\in T_{n}$,

  1. (1)

$\widetilde{\varkappa}(f,J)\leqslant C,$ and

  2. (2)

$\forall y\in J$, $\log
f^{\prime}(y)\in[(\lambda-\varepsilon_{n})n,(\lambda+\varepsilon_{n})n]$.

Moreover, since $f(x)\in I$, we conclude that $f(J)$ is contained in the
$2^{(\lambda+\varepsilon_{n})n}$-neighborhood of $I$. After replacing
$\varepsilon_{n}$ by $\varepsilon_{n}+2/n,$ we obtain that $f(J)$ falls in a
common interval of length at most $2^{(\lambda+\varepsilon_{n})n}$ for $f\in
T_{n}.$ ∎

######  Lemma 6.4.

Let $c,C,J,\varepsilon_{n},T_{n}$ be as in Lemma 6.3. Let
$J^{\prime}=\frac{1}{2}J$ be the closed interval of the same center as $J$ and
but of half the length. Then there exists $g_{n}\neq f_{n}\in T_{n}$ such that
$f_{n}(J^{\prime})\subset g_{n}(J)$ and the sequence $g_{n}^{-1}f_{n}$
converges to the identity on $J^{\prime}$ in the $C^{1}$-topology.

###### Proof.

Define $k=k(n)$ for large $n$ to be the greatest integer such that

$4k+1<\frac{cn}{\lceil 2\varepsilon_{n}n+\sqrt{n}\rceil+\log(\lceil
2\varepsilon_{n}n^{2}\rceil+1)}.$

Since $\varepsilon_{n}\to 0$, $k(n)\to+\infty$ as $n\to+\infty$. Let
$y_{0},y_{1},\cdots,y_{4k}\in J$ be evenly spaced points on the circle such
that $J=[y_{0},y_{4k}]$. Then $d(y_{j},y_{j+1})\leqslant 1/(4k).$ For every
$f\in T_{n},$ each $f(y_{i})$ takes value in a fixed interval of length
$2^{(\lambda+\varepsilon_{n})n}$ and $\log f^{\prime}(x_{i})$ takes value in