Universal supercritical behavior for some skew-product maps

Hans Koch ††$\\!{}^{1}$ Department of Mathematics, The University of Texas at
Austin, Austin, TX 78712.

Abstract. We consider skew-product maps over circle rotations $x\mapsto
x+\alpha\;(\mathop{\eightrm mod}\nolimits\,1)$ with factors that take values
in ${\eightrm SL}(2,{\scriptstyle{{\tends R}}})$. In numerical experiments
with $\alpha$ the inverse golden mean, Fibonacci iterates of almost Mathieu
maps with rotation number $\scriptstyle 1/4$ and positive Lyapunov exponent
exhibit asymptotic scaling behavior. We prove the existence of such asymptotic
scaling for “periodic” rotation numbers and for large Lyapunov exponent. The
phenomenon is universal, in the sense that it holds for open sets of maps,
with the scaling limit being independent of the maps. The set of maps with a
given periodic rotation number is a real analytic codimension $1$ manifold in
a suitable space of maps.

1\. Introduction and main results

num xyz goto num count 0 1. Introduction and main results We consider the
asymptotic behavior of skew products

$\eqalign{A^{\ast q}_{\scriptscriptstyle\circ}(x)&=A^{\ast
q}\bigl{(}x-{\textstyle{q\over 2}}\alpha\bigr{)}\,,\cr A^{\ast
q}(x)&\,\mathrel{\mathop{=}^{\scriptscriptstyle\rm
def}}\,A(x+(q-1)\alpha)\cdots A(x+2\alpha)A(x+\alpha)A(x)\,,\cr}$ $None$

as $q\to\infty$ along certain subsequences, where $\alpha$ is an irrational
number and $A$ is a real analytic function from the circle ${{T}}={{\tends
R}}/{{\tends Z}}$ to the group ${\rm SL}(2,{{\tends R}})$. Products for
negative $q$ are defined by replacing the factors $A$ in the above equation by
$A(\,\hbox{\bf.}-\alpha)^{-1}$.

The typical growth of such products is described by the Lyapunov exponent. The
Lyapunov exponent of a pair $G=(\alpha,A)$ with $A:{{T}}\to{\rm SL}(2,{{\tends
R}})$ continuous is defined as

$L(G)=\lim_{q\to\infty}{1\over q}\log\bigl{\|}A^{\ast q}(x)\bigr{\|}\,.$
$None$

Assuming that $\alpha$ is irrational, this limit exists by ergodicity and is
a.e. constant in $x$.

The work presented in this paper was motivated in part by the observation in
[28] of systematic slower-than-typical growth of certain products, as will be
described below. We restrict to factors that are reversible in the following
sense.

Definition 1.1. The symmetric factor $A_{\scriptscriptstyle\circ}$ associated
with a pair $G=(\alpha,A)$ is defined by setting
$A_{\scriptscriptstyle\circ}(x)=A\bigl{(}x-\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}\bigr{)}$ for all $x$. We say that
$G$ is reversible if
$S^{-1}A_{\scriptscriptstyle\circ}(x)S=A_{\scriptscriptstyle\circ}(-x)^{\dagger}$
for all $x$, where

$S={\eighttt\eightrm\left[\matrix{0&1\cr 1&0\cr}\right]}\,,\qquad
M^{\dagger}={\eighttt\eightrm\left[\matrix{d&-b\cr-c&a\cr}\right]}\quad{\rm
if}\quad M={\eighttt\eightrm\left[\matrix{a&b\cr c&d\cr}\right]}\,.$ $None$

Notice that $M^{\dagger}M=\mathop{\rm det}\nolimits(M){\bf 1}$. We will refer
to $M^{\dagger}$ as the quasi-inverse of $M$.

To be more precise about the type of products being considered, let
$\alpha={\sqrt{5}-1\over 2}$ be the inverse golden mean and $p_{n}$ the $n$-th
Fibonacci number. Then the following holds.

Theorem 1.2. Let $A:{{T}}\to{\rm SL}(2,{{\tends R}})$ be real analytic and
reversible. Assume that $G=(\alpha,A)$ has a positive Lyapunov exponent. Then
there exist real numbers $M_{1},M_{2},M_{3},\ldots$ such that every
subsequence of

$n\mapsto M_{n}\,A^{\ast
p_{n}}_{\scriptscriptstyle\circ}\bigl{(}\alpha^{n}\hbox{\bf.}\bigr{)}$ $None$

has a subsequence that converges to a nonzero function $z\mapsto
b_{\diamond}(z){\tt B}$, where ${\tt B}$ is a constant $2\times 2$ matrix of
rank one. The scalar factor $b_{\diamond}$ is an even entire function, and
convergence is uniform on compact subsets of ${{\tends C}}$.

A more restrictive version of this theorem was given in [28]. The main objects
under investigation in [28] were parametrized factors $A$ at the boundary
between zero Lyapunov exponent (subcritical or critical factors) and positive
Lyapunov exponent (supercritical factors). In this context, products of
supercritical factors were computed numerically, and limit functions
$b_{\diamond}$ were found with surprisingly regular zeros. Our goal here is to
describe this type of limits.

Our main focus is on Schrödinger factors

$A(x)=\left[\matrix{\lambda v(t+x)-E&-1\cr 1&0\cr}\right]\,,$ $None$

and perturbations of such factors. Here, $\lambda$, $t$, and $E$ are real
parameters, and $v$ is a $1$-periodic continuous function on ${{\tends R}}$.
The choice $v(x)=-2\cos(2\pi x)$ defines the almost Mathieu (AM) family of
factors [15,22,24]. Schrödinger factors arise naturally in the study of
Schrödinger operators on $\ell^{2}({{\tends Z}})$ of the form ††$\\!{}^{2}$ We
adopt here the “physical” choice of signs, which makes the kinetic term
positive.

$({\cal H}_{\lambda}^{t}u)_{q}=\lambda
v(t+q\alpha)u_{q}-u_{q+1}-u_{q-1}\,,\qquad q\in{{\tends Z}}\,.$ $None$

To be more specific, $A^{\ast q}(0)$ is the transfer matrix that maps
$\bigl{[}{u_{0}\atop u_{-1}}\bigr{]}$ to $\bigl{[}{u_{q}\atop
u_{q-1}}\bigr{]}$, for a formal solution $u$ of the equation ${\cal
H}_{\lambda}^{t}u=Eu$. Many properties of ${\cal H}_{\lambda}^{t}$ are
reflected by properties of the corresponding products (1.1), and vice versa.
The Lyapunov exponent of the AM factors with $\lambda>0$ is known to be
$L(G)=\max\\{0,\log\lambda\\}$ for all energies in the spectrum [19].

The family of AM operators $t\mapsto{\cal H}_{\lambda}^{t}$ describe the
motion of an electron on ${{\tends Z}}^{2}$ under the influence of a magnetic
flux $2\pi\alpha$ per unit cell, after restricting to wave functions
$\phi(q,p)=e^{-2\pi ipt}u_{q}$. The full Hamiltonian for this system is known
as the Hofstadter Hamiltonian [1,6].

Another quantity associated with ${\cal H}_{\lambda}^{t}$ that is relevant in
the present context is the integrated density of states
$E\mapsto{\hbox{\magnineeufm k}}(E)$. Notice that ${\cal H}_{\lambda}^{t}$ is
self-adjoint and bounded. Denote by $P_{\scriptscriptstyle L}$ the canonical
inclusion map from $\ell^{2}\bigl{(}{{\tends Z}}\cap[-L,L]\bigr{)}$ into
$\ell^{2}({{\tends Z}})$. Given $E\in{{\tends R}}$, the integrated density of
states ${\hbox{\magnineeufm k}}_{\scriptscriptstyle L}({\cal
H}_{\lambda}^{t},E)$ is defined as the fraction of eigenvalues of
$P_{\scriptscriptstyle L}^{\ast}{\cal H}_{\lambda}^{t}P_{\scriptscriptstyle
L}$ that belong to $(-\infty,E]$. The integrated density of states for ${\cal
H}_{\lambda}^{t}$ can be obtained as the limit ${\hbox{\magnineeufm
k}}(E)=\lim_{L\to\infty}{\hbox{\magnineeufm k}}_{\scriptscriptstyle L}({\cal
H}_{\lambda}^{t},E)$. Clearly, k is an increasing function, taking the value
${\hbox{\magnineeufm k}}(E)=0$ for $E$ below the spectrum of ${\cal
H}_{\lambda}^{t}$ and the value ${\hbox{\magnineeufm k}}(E)=1$ for $E$ above
the spectrum. And it is constant on any spectral gaps (connected components of
the resolvent set). For the AM operator, it is known that there are no
isolated eigenvalues [21], so k is continuous. For $\lambda>1$, the spectrum
is pure point, with exponentially decreasing eigenfunctions [18]. (This holds
for arbitrary Diophantine $\alpha$, and for “nonresonant” values of $t$ that
include our choice $\scriptstyle\alpha$$\scriptstyle/$$\scriptstyle 2$ below.)

Among the intriguing features of the AM operator is the following resonance
phenomenon. ${\cal H}^{t}_{\lambda}$ has an infinite number of spectral gaps.
Each gap can be labeled canonically by an integer $k$, known as the Hall
conductance, and ${\hbox{\magnineeufm k}}(E)\equiv k\alpha\;(\mathop{\rm
mod}\nolimits 1)$ holds for all energies $E$ in the spectral gap labeled by
$k$. For details we refer to [7,8,11,14,23].

A related quantity for Schrödinger pairs $G=(\alpha,A)$ is the rotation number
$\mathop{\rm rot}\nolimits(G)$. In cases where $A$ is of the form (1.5), with
a potential $v$ that is continuous, we can define the rotation number by the
equation

$\mathop{\rm rot}\nolimits(G)=\lim_{q\to\infty}{\mathop{\rm
Rot}\nolimits_{\scriptscriptstyle N}(G)\over N}\,,\qquad\qquad\hbox{\rm(for
Schr\"{o}dinger $G$)}$ $None$

where $\mathop{\rm Rot}\nolimits_{\scriptscriptstyle N}(G)$ is half the number
of sign changes on $\\{1,2,\ldots,N\\}$ of a nonzero solution $u$ of the
equation ${\cal H}_{\lambda}^{t}u=Eu$, adding $\scriptstyle
1$$\scriptstyle/$$\scriptstyle 2$ if $u(N)=0$. Notice that $0\leq\mathop{\rm
rot}\nolimits(G)\leq\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt}$. The rotation number for more general pairs $G$, and its
properties, will be described in Section 3. Assuming that $\alpha$ is
irrational, the value of $\mathop{\rm rot}\nolimits(G)$ is independent of the
solution $u$. For an AM pair $G_{\scriptscriptstyle E}$ with spectral energy
$E$, it can be shown [11] that the rotation number is related to the
integrated density of states via ${\hbox{\magnineeufm k}}(E)=2\mathop{\rm
rot}\nolimits(G_{\scriptscriptstyle E})$. Roughly speaking, this relation
expresses the fact that the wave number (inverse wave length) of
eigenfunctions increases with the energy $E$.

Convention 1.  In what follows, we fix the phase $t$ in the definition of the
Schrödinger factor (1.5) and the Schrödinger operator (1.6) to the value
$t=\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}$, unless specified otherwise. This
makes the symmetric AM factors $A_{\scriptscriptstyle\circ}$ reversible in the
sense of Definition 1.1.

The above-mentioned gap-labeling property associates spectral gaps with
rotation numbers that belong to ${1\over 2}{{\tends Z}}[\alpha]$. We will now
describe a different phenomenon that is associated with some other rotation
numbers in ${{\tends Q}}[\alpha]$.

Definition 1.3. A number $\rho\in[0,\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt}]$ will be called positive periodic (as a rotation number) if
$\rho={w\over v}+{u\over v}\alpha$ for some integers $u,v,w$ with $v>2$ and
$\mathop{\rm gcd}\nolimits(u,v,w)=1$, and if $\left|{u\over v}-{w\over
v}\alpha\right|\leq{1\over 2}$.

We will call $\rho\in[-\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt},0]$ negative periodic if $\rho+\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}$ is positive periodic. Notice that,
for any given integer $v>2$, the (positive or negative) periodic rotation
numbers in ${1\over v}{{\tends Z}}[\alpha]$ constitute a discrete subset of
${{\tends R}}$.

In what follows, $\alpha$ denotes the inverse golden mean, unless specified
otherwise. Let $p_{n}$ be the $n$-th Fibonacci number, and define
$q_{n}=p_{n+1}$. Our main result in this paper is the following.

Theorem 1.4. Let $\rho$ be positive periodic. Then there exists a positive
integer $n$ and two nonzero even entire functions $b_{\diamond}$ and
$a_{\diamond}$, such that the following holds. Consider the family of AM maps
$G_{\lambda,E}$ parametrized by the coupling constant $\lambda>0$ and the
energy $E$. Then there exists a real analytic function $\epsilon$, defined on
an open neighborhood of zero, such that if $\delta=\lambda^{-1}$ belongs to
the domain of $\epsilon$, and if $E=\lambda\epsilon(\delta)$, then $G$ has
rotation number $\mathop{\rm rot}\nolimits(G)=\rho$. In this case, there exist
sequences of positive real numbers $k\mapsto M_{k}$ and $k\mapsto W_{k}$, such
that

$\eqalign{\lim_{k\to\infty}M_{k}\,A^{\ast
p_{3nk}}_{\scriptscriptstyle\circ}\bigl{(}\alpha^{3nk}x\bigr{)}&=b_{\diamond}(x){\tt
A}^{\dagger}\,,\cr\lim_{k\to\infty}W_{k}\,A^{\ast
q_{3nk}}_{\scriptscriptstyle\circ}\bigl{(}\alpha^{3nk}x\bigr{)}&=a_{\diamond}(x){\tt
A}\,,\cr}$ $None$

for some constant $2\times 2$ matrix ${\tt A}$ of rank one. Convergence in
(1.8) is uniform on compact subsets of ${{\tends C}}$. An analogous result
holds for any two-parameter family that is sufficiently close to the AM family
(in a sense described later).

Both $b_{\diamond}$ and $a_{\diamond}$ have infinitely many zeros, all simple
and on the real axis. The zeros can be determined rather explicitly, and this
is used to construct the functions $b_{\diamond}$ and $a_{\diamond}$. We note
that the function $\epsilon=\epsilon(\delta)$ and the matrix ${\tt A}$ in the
above theorem can depend on the family. But the scaling limits $b_{\diamond}$
and $a_{\diamond}$ are universal.

We expect that an analogous result holds for all $\lambda>1$. Numerically,
this is observed at energy $E=0$, which corresponds to $\rho=\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 4$}\hskip 1.0pt}$. For this particular value of
$\rho$, a proof seems feasible, since the emergence of the limiting zeros is
quite transparent in this case [28]. We also expect that Theorem 1.4
generalizes to arbitrary quadratic irrationals $\alpha$.

Remark 2.  For non-Schrödinger maps $G=(\alpha,A)$, the rotation number
$\mathop{\rm rot}\nolimits(G)$ in Theorem 1.4 has to be replaced by the
rotation number $\varrho(G)$ defined in Section 3. Using $\varrho(G)$ in place
of $\mathop{\rm rot}\nolimits(G)$, an analogous theorem holds for negative
periodic rotation numbers as well. This corresponds to replacing $A$ by $-A$.
For a Schrödinger factor $A$, we can get $-A$ via a conjugacy by
$\bigl{[}{1~{}\phantom{-}0\atop 0~{}-1}\bigr{]}$ and then replacing $v,E$ by
their negatives. The conjugacy is equivalent to replacing $u$ by
$q\mapsto(-1)^{q}u_{q}$ in the equation (1.6).

Theorem 1.4 is best understood in a dynamics context. The pairs $G=(\alpha,A)$
with $A(x)\in{\rm SL}(2,{{\tends R}})$ are naturally associated with skew-
product maps

$G(x,y)=\bigl{(}x+\alpha,A(x)y\bigr{)}\,,\qquad x\in X\,,\quad y\in Y\,,$
$None$

on $X\times Y$, where $X={{T}}$ and $Y={{\tends R}}^{2}$. The $q$-th iterate
of $G$ is $G^{q}=\bigl{(}q\alpha,A^{\ast q})$, with $A^{\ast q}$ as defined in
(1.1). So Theorem 1.4 can be viewed as describing the asymptotic behavior of
long orbits for skew-product maps. Due to the scaling involved in (1.8), we
will also need to consider the case $X={{\tends R}}$. The condition
$\mathop{\rm det}\nolimits(A)=1$ will be weakened as well, and $Y={{\tends
R}}$ will be used for limit cases.

Our proof of Theorem 1.4 involves the use of a renormalization transformation
${\cal R}$ that acts on pairs $P=(F,G)$ of skew-product maps $F=(1,B)$ and
$G=(\alpha,A)$. We consider families of pairs $P_{\delta,\epsilon}$ that admit
a scaling limit as $\delta\to 0$. For Schrödinger factors (1.5), we use
$\delta=\lambda^{-1}$, multiply $A$ by $\delta$, and set $E=\lambda\epsilon$.
If $\varrho(G_{\delta,\epsilon})=\rho$ with $\rho$ positive periodic, then the
asymptotic behavior (1.8) is governed by a fixed point $P_{\diamond}$ of (a
modified version of) the transformation ${\cal R}^{3n}$. The local stable
manifold of this transformation at $P_{\diamond}$ is of codimension $1$ and
agrees with the level set $\varrho(G)=\rho$. So the real analytic curve
$\delta\mapsto\epsilon(\delta)$ described in Theorem 1.4 characterizes the
intersection of the given family with this manifold.

Given the relation $2\mathop{\rm rot}\nolimits(G_{\scriptscriptstyle
E})={\hbox{\magnineeufm k}}(E)$ for the AM family, the above suggests that the
energies $E=\lambda\epsilon\bigl{(}\lambda^{-1}\bigr{)}$ are eigenvalues of
the operator ${\cal H}^{\alpha/2}_{\lambda}$. Analytic curves of eigenvalues
are a standard feature in real analytic families of compact self-adjoint
operators, even in the presence of level crossings [5]. This applies e.g. to
the operators $P_{\scriptscriptstyle L}^{\ast}{\cal
H}^{t}_{\lambda}P_{\scriptscriptstyle L}$ mentioned earlier. So it seems
possible that $E=\lambda\epsilon\bigl{(}\lambda^{-1}\bigr{)}$ is a limit as
$L\to\infty$ of such eigenvalue curves.

From a dynamics perspective, the zeros at $\pm\rho$ of the limit function
$a_{\diamond}$ seem associated with factors $A_{\scriptscriptstyle\circ}(x)$
in large products that map the expanding direction (from the product to the
right of this factor) to the contracting direction of the product to the left
of this factor. This suggest that $x$ is near $\pm\rho$, which is where
$A_{\scriptscriptstyle\circ}(x)$ can cover a wide range of rotation angles.
The location of such factors may be associated with peaks of an eigenvector.

We note that the Hofstadter model has a large number of symmetries [13,16,17]
and other important arithmetic features. It seems fair to say that their
interplay with analysis is rather poorly understood. This is our motivation
for focusing on the inverse golden mean. Periodicity may be regarded as
exceptional, but periodic orbits play an important role in dynamical systems,
since they heavily influence the motion nearby. The AM family can be regarded
as exceptional as well, but, as Theorem 1.4 shows, some of its “rigid”
behavior is shared by nearby families.

The remaining part of this paper is organized as follows. In Section 2 we
introduce the renormalization transformation ${\cal R}$ and prove a result
that implies Theorem 1.2. Section 3 is devoted to limits of Schrödinger
factors and the rotation number. In Section 4 we construct the periodic orbits
of ${\cal R}$ that are associated with positive periodic rotation numbers. A
proof of Theorem 1.4 is given in Section 5.

Remarks 3.

$\circ$ To leading order, the scaling constants in Theorem 1.4 are $M_{k}\sim
e^{-p_{3nk}L}$ and $W_{k}\sim e^{-q_{3nk}L}$, where $L$ is the Lyapunov
exponent of $G$. For the AM factors, we suspect that a choice
$M_{k}=Me^{-q_{3nk}L}$ and $W_{k}=We^{-p_{3nk}L}$ will work. But trying to
verify this would go beyond the scope of this paper.

$\circ$ If we allowed $v=1$ and $v=2$ in Definition 1.3, then
$\\{0,\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}\\}$ would count as positive periodic
as well. The associated energies for the AM operators correspond to spectral
gaps. We believe that Theorem 1.4 can be proved for these rotation numbers as
well, but they would require special treatment.

$\circ$ Numerically, critical fixed points (for the appropriate power of
${\cal R}^{3}$) have been found for rotation numbers in $\\{0,\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 8$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 6$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}-\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 4$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle 2$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 6$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle 3$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 8$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}\\}$. Existence proofs have been
given in [26,28] in the case of rotation numbers $\\{0,\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 4$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}\\}$.

$\circ$ Conjecture 1.1 in [28] on critical periods only covers rational
rotation numbers. A reasonable amendment would be to include all (positive or
negative) periodic rotation numbers.

$\circ$ Numerically, the unstable manifold associated with the critical fixed
point of ${\cal R}^{3}$ described in Theorem 2.3 of [28] leads into a
supercritical fixed point $P_{\diamond}$ of the type described here. This
concerns the case $\rho=\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
4$}\hskip 1.0pt}$, where the curve
$E=\lambda\epsilon\bigl{(}\lambda^{-1}\bigr{)}$ is trivial, namely $E=0$ by
symmetry. Still, a possible renormalization picture would be that something
similar occurs for any periodic rotation number $\rho$.

2\. Renormalization

num xyz goto num count 0 2. Renormalization We use renormalization as a
combinatorial tool for generating factors of the type that appear in (1.8).
From an arithmetic point of view, the renormalization (RG) transformation
${\cal R}$ defined below lifts the Gauss map for real numbers to pairs of maps
$(F,G)$. From an analysis point of view, ${\cal R}$ constitutes a dynamical
system on a space of such pairs.

Consider a pair $(F,G)$ of skew-product maps $F=(1,B)$ and $G=(\alpha,A)$ on
${{\tends R}}\times{{\tends R}}^{2}$, with factors $B,A:{{\tends R}}\to{\rm
SL}(2,{{\tends R}})$. If $G$ commutes with $F$, then $G$ can be viewed as a
map on a cylinder in ${{\tends R}}\times{{\tends R}}^{2}$ whose points are the
orbits of $F$. Assume that $\alpha<1$ is a positive irrational number, and
denote by $c$ the integer part of $\alpha^{-1}$. Then the basic renormalized
pair is defined by

${\cal
R}(F,G)=\bigl{(}\check{F},\check{G}\bigr{)}\,,\qquad\check{F}=\Lambda^{-1}G\Lambda\,,\quad\check{G}=\Lambda^{-1}FG^{-c}\Lambda\,,$
$None$

where $\Lambda(x,y)=(\alpha x,y)$. By construction, the first component of
$\check{F}$ is again $1$, while the first component of $\check{G}$ is
$\check{\alpha}=\alpha^{-1}-c$. Notice that $\alpha\mapsto\check{\alpha}$ is
the Gauss map that appears in the continued fraction expansion of $\alpha$.

In what follows, we restrict to the inverse golden mean, which satisfies
$\alpha^{-1}-1=\alpha$. So we can fix $c=1$ in (2.1). The transformation
${\cal R}$ extends readily to factors in ${\rm GL}(2,{{\tends R}})$. In this
case, $G^{-1}$ is replaced by the quasi-inverse $G^{\dagger}$ defined below.
We note that a map $G=(\alpha,A)$ is reversible in the sense of Definition 1.1
precisely if

${\cal S}^{-1}G{\cal
S}=G^{\dagger}\,\mathrel{\mathop{=}^{\scriptscriptstyle\rm
def}}\,\bigl{(}-\alpha,A(\,\hbox{\bf.}-\alpha)^{\dagger}\bigr{)}\,,\qquad{\cal
S}(x,y)=(-x,Sy)\,.$ $None$

A pair $P=(F,G)$ is said to be reversible if both $F$ and $G$ are reversible.

The conjugacy by $\Lambda$ only renormalizes the translational parts of our
skew-products. We will often have to renormalize the matrix part as well. When
considering periodic orbits of length $k$, it is best to do this only once per
period. So for ${\cal R}^{k}$ we use a scaling

$\Lambda_{k}={\cal L}_{k}\Lambda^{k}\,,\qquad{\cal L}_{k}(x,y)=(x,L_{k}y)\,,$
$None$

where $L_{k}$ a suitable matrix in $\pm{\rm SL}(2,{{\tends R}})$ that commutes
with $S$. This matrix can depend on the pair $P=(F,G)$. It will be specified
as needed, when the choice matters.

The definition (2.1) of ${\cal R}$ involves the basic composition operator
${\cal C}(F,G)=(G,FG^{\dagger})$. Due to the symmetry $A\mapsto-A$ mentioned
in Remark 2, we restrict to powers of ${\cal C}$ that are multiples of $3$.
Then it is convenient to replace ${\cal C}^{3}$ by the transformation ${\cal
C}_{3}$ defined by

${\cal
C}_{3}(F,G)=\bigl{(}GF^{\dagger}G\,\hbox{\bf,}\,G^{\dagger}FG^{\dagger}FG^{\dagger}\bigr{)}\,.$
$None$

The only difference between ${\cal C}^{3}$ and ${\cal C}_{3}$ is the order of
the factors. This is irrelevant for commuting pairs; but for non-commuting
pairs, which need to be included in our analysis, the transformation ${\cal
C}^{3}$ does not in general preserve reversibility, while ${\cal C}_{3}$ does.

Given any $n\geq 1$, we define the RG transformation ${\cal R}_{3n}$ by
setting

${\cal
R}_{3n}(F,G)=\bigl{(}\Lambda_{3n}^{-1}\hat{F}\Lambda_{3n}\,\hbox{\bf,}\,\Lambda_{3n}^{-1}\hat{F}\Lambda_{3n}\bigr{)}\,,\qquad\bigl{(}\hat{F},\hat{G}\bigr{)}={\cal
C}_{3}^{n}(F,G)\,.$ $None$

Consider first the case $n=1$. Let $\bigl{(}\tilde{F},\tilde{G}\bigr{)}={\cal
R}_{3}(F,G)$. A straightforward computation shows that the symmetric factor
$\tilde{B}_{\scriptscriptstyle\circ}$ for $\tilde{F}$ is

$\tilde{B}_{\scriptscriptstyle\circ}(x)=L_{3}^{-1}A_{\scriptscriptstyle\circ}\bigl{(}\alpha^{3}x-{\textstyle{1-\alpha\over
2}}\bigr{)}B_{\scriptscriptstyle\circ}\bigl{(}\alpha^{3}x\bigr{)}^{\dagger}A_{\scriptscriptstyle\circ}\bigl{(}\alpha^{3}x+{\textstyle{1-\alpha\over
2}}\bigr{)}L_{3}\,,$ $None$

and that the symmetric factor $\tilde{A}_{\scriptscriptstyle\circ}$ of
$\tilde{G}$ is

$\eqalign{\tilde{A}_{\scriptscriptstyle\circ}(x)&=L_{3}^{-1}A_{\scriptscriptstyle\circ}\bigl{(}\alpha^{3}x+(1-\alpha)\bigr{)}^{\dagger}B_{\scriptscriptstyle\circ}\bigl{(}\alpha^{3}x+{\textstyle{1-\alpha\over
2}}\bigr{)}A_{\scriptscriptstyle\circ}\bigl{(}\alpha^{3}x\bigr{)}^{\dagger}\times\cr&\quad\times
B_{\scriptscriptstyle\circ}\bigl{(}\alpha^{3}x-{\textstyle{1-\alpha\over
2}}\bigr{)}A_{\scriptscriptstyle\circ}\bigl{(}\alpha^{3}x-(1-\alpha)\bigr{)}^{\dagger}L_{3}\,.\cr}$
$None$

Notice that, if $B_{\scriptscriptstyle\circ}$ and
$A_{\scriptscriptstyle\circ}$ are analytic in a strip $|\mathop{\rm
Im}\nolimits x|<\delta$, then $\tilde{B}_{\scriptscriptstyle\circ}$ and
$\tilde{A}_{\scriptscriptstyle\circ}$ are analytic in the strip $|\mathop{\rm
Im}\nolimits x|<\alpha^{-3}\delta$. The following is equally straightforward
to verify.

Let ${r_{{}_{\hskip-1.0ptB}}}$ and ${r_{{}_{\hskip-1.0ptA}}}$ be positive real
numbers.

Proposition 2.1. Assume that
$\alpha^{3}{r_{{}_{\hskip-1.0ptA}}}+{\textstyle{1\over
2}}\alpha^{2}\leq{r_{{}_{\hskip-1.0ptB}}}\leq\alpha^{-3}{r_{{}_{\hskip-1.0ptA}}}-{\textstyle{1\over
2}}\alpha^{-1}$. Notice that ${r_{{}_{\hskip-1.0ptB}}}=\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}$ and
${r_{{}_{\hskip-1.0ptA}}}=\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt}$ satisfy this condition. If the domains of
$B_{\scriptscriptstyle\circ}$ and $A_{\scriptscriptstyle\circ}$ include the
(real or complex) disks $|x|\leq{r_{{}_{\hskip-1.0ptB}}}$ and
$|x|\leq{r_{{}_{\hskip-1.0ptB}}}$, respectively, then the domains of
$\tilde{B}_{\scriptscriptstyle\circ}$ and
$\tilde{A}_{\scriptscriptstyle\circ}$ include the same disks. Assume now that

${r_{{}_{\hskip-1.0ptB}}}>{\textstyle{\alpha\over
2}}\,,\qquad{r_{{}_{\hskip-1.0ptA}}}>{\textstyle{1\over
2}}\,,\qquad\alpha^{3}{r_{{}_{\hskip-1.0ptA}}}+{\textstyle{1\over
2}}\alpha^{2}<{r_{{}_{\hskip-1.0ptB}}}<\alpha^{-3}{r_{{}_{\hskip-1.0ptA}}}-{\textstyle{1\over
2}}\alpha^{-1}\,.$ $None$

Suppose that the domains of $B_{\scriptscriptstyle\circ}$ and
$A_{\scriptscriptstyle\circ}$ include the disks $|x|<{r_{{}_{\hskip-1.0ptB}}}$
and $|x|<{r_{{}_{\hskip-1.0ptB}}}$, respectively. Then the domains of the
symmetric factors associated with ${\cal R}_{3}^{n}(F,G)$ include the same
disks. Furthermore, these domains grow asymptotically like $\alpha^{-3n}$, as
$n\to\infty$. Moreover, the values of these factors on any given compact set
are determined (for sufficiently large $n$) by the values of
$B_{\scriptscriptstyle\circ}$ on the disk $|x|<{r_{{}_{\hskip-1.0ptB}}}$ and
the values of $A_{\scriptscriptstyle\circ}$ on the disk
$|x|<{r_{{}_{\hskip-1.0ptA}}}$.

This motivates the following choice of function spaces. Given $r>0$, denote by
${\cal F}_{r}$ the space of all real analytic functions $f$ on $(-r,r)$ that
have a finite norm

$\|f\|_{r}=\sum_{n=0}^{\infty}\bigl{|}f^{(n)}(0)\bigr{|}\,{r^{n}\over n!}\,.$
$None$

Notice that every function $f\in{\cal F}_{r}$ extends analytically to the
complex disk $|x|<r$. Furthermore, ${\cal F}_{r}$ is a Banach algebra under
pointwise multiplication of functions. This was crucial in [26,28] but is less
important here.

The space of matrix functions

$x\mapsto
A_{\scriptscriptstyle\circ}(x)=\left[\matrix{a_{\scriptscriptstyle\circ}(x)&b_{\scriptscriptstyle\circ}(x)\cr
c_{\scriptscriptstyle\circ}(x)&d_{\scriptscriptstyle\circ}(x)\cr}\right]$
$None$

with entries in ${\cal F}_{r}$ will be denoted by ${\cal F}_{r}^{4}$. The norm
of $A_{\scriptscriptstyle\circ}\in{\cal F}_{r}^{4}$ is defined as
$\|A_{\scriptscriptstyle\circ}\|_{r}=\|a_{\scriptscriptstyle\circ}\|_{r}+\|b_{\scriptscriptstyle\circ}\|_{r}+\|c_{\scriptscriptstyle\circ}\|_{r}+\|d_{\scriptscriptstyle\circ}\|_{r}$.
Given a pair $r=({r_{{}_{\hskip-1.0ptB}}},{r_{{}_{\hskip-1.0ptA}}})$ of
positive real numbers, we define ${\cal H}_{r}$ to be the vector space of all
pairs ${\cal P}=(B_{\scriptscriptstyle\circ},A_{\scriptscriptstyle\circ})$ in
${\cal F}_{r_{{}_{\hskip-1.0ptB}}}^{4}\times{\cal
F}_{r_{{}_{\hskip-1.0ptA}}}^{4}$, equipped with the norm $\|{\cal
P}\|_{r}=\|B_{\scriptscriptstyle\circ}\|_{r_{{}_{\hskip-1.0ptB}}}+\|A_{\scriptscriptstyle\circ}\|_{r_{{}_{\hskip-1.0ptA}}}$.

Given that every skew-product map that appears in our analysis has a pre-
determined first component, we will identify a skew-product map $G=(\alpha,A)$
with its symmetric factor $A_{\scriptscriptstyle\circ}$. Referring to (2.10),
we note that $G$ is reversible if and only if $a_{\scriptscriptstyle\circ}$
and $d_{\scriptscriptstyle\circ}$ are even, while
$b_{\scriptscriptstyle\circ}(x)=-c_{\scriptscriptstyle\circ}(-x)$. The
subspace of reversible pairs in ${\cal H}_{r}$ will be denoted by ${\cal
H}_{r}^{\prime}$.

Convention 4.  We always assume that domain parameters
$r=({r_{{}_{\hskip-1.0ptB}}},{r_{{}_{\hskip-1.0ptA}}})$ satisfy the domain
condition (2.8). And whenever $L_{k}:{\cal H}_{r}\to\pm{\rm SL}(2,{{\tends
R}})$ remains unspecified, it is assumed to be continuous.

Lemma 2.2. The transformation ${\cal R}_{3}:{\cal H}_{r}\to{\cal H}_{r}$ is
compact and preserves reversibility.

The reversibility-preserving property is a consequence of the fact that the
products $GF^{\dagger}G$ and $G^{\dagger}FG^{\dagger}FG^{\dagger}$ that appear
in (2.4) are palindromic. Compactness is a consequence of the fact that ${\cal
R}_{3}$ is analyticity-improving, in the sense that ${\cal R}_{3}$ maps
bounded sets in ${\cal H}_{r}$ to bounded sets in ${\cal H}_{r^{\prime}}$, for
some $r_{\scriptscriptstyle B}^{\prime}>{r_{{}_{\hskip-1.0ptB}}}$ and
$r_{\scriptscriptstyle A}^{\prime}>{r_{{}_{\hskip-1.0ptA}}}$. Now combine this
with the fact that the inclusion maps ${\cal F}_{r_{\scriptscriptstyle
B}^{\prime}}\to{\cal F}_{r_{{}_{\hskip-1.0ptB}}}$ and ${\cal
F}_{r_{\scriptscriptstyle A}^{\prime}}\to{\cal F}_{r_{{}_{\hskip-1.0ptA}}}$
are compact.

The same holds of course for the transformations ${\cal R}_{3n}$ with $n>1$.

When the exact growth of products is not needed, we also consider a modified
version of our RG transformation, defined by

${\hbox{\teneufm R}}_{3n}={\hbox{\teneufm N}}\circ{\cal R}_{3n}\,,$ $None$

where N is a suitable normalization. Unless specified otherwise, N consists in
normalizing
$B_{\scriptscriptstyle\circ}\mapsto\|B_{\scriptscriptstyle\circ}\|_{r_{{}_{\hskip-1.0ptB}}}^{-1}B_{\scriptscriptstyle\circ}$
and
$A_{\scriptscriptstyle\circ}\mapsto\|A_{\scriptscriptstyle\circ}\|_{r_{{}_{\hskip-1.0ptA}}}^{-1}A_{\scriptscriptstyle\circ}$.

The following implies Theorem 1.2.

Theorem 2.3. Pick $L_{3}=S$. Let $P^{\prime}=(F^{\prime},G^{\prime})$ be a
pair in ${\cal H}_{r}$, of the form $F^{\prime}=(1,{\bf 1})$ and
$G^{\prime}=(\alpha,A^{\prime})$. Assume hat $G^{\prime}$ has a positive
Lyapunov exponent and is reversible. Denote by $K$ the set of all accumulation
points of the sequence $n\mapsto{\hbox{\teneufm R}}_{3}^{n}(P^{\prime})$. Then
every pair $P=(F,G)$ in $K$ has the following property. $F=(1,B)$ and
$G=(\alpha,A)$, with
$B_{\scriptscriptstyle\circ}(x)=b_{\scriptscriptstyle\circ}(x){\tt
A}^{\dagger}$ and
$A_{\scriptscriptstyle\circ}(x)=a_{\scriptscriptstyle\circ}(x){\tt A}$ for all
$x$, where $b_{\scriptscriptstyle\circ}$ and $a_{\scriptscriptstyle\circ}$ are
nonzero even entire functions and ${\tt A}$ is a constant $2\times 2$ matrix
of rank $1$.

Proof. First, notice that
$SC_{\scriptscriptstyle\circ}(x)S=C_{\scriptscriptstyle\circ}(-x)^{\dagger}$
for any reversible map $H=(\gamma,C)$. So the conjugacy by $L_{3}=S$ in (2.6)
and (2.7) just replaces each factor $C(x)$ by $C(-x)^{\dagger}$. In
particular, the leftmost factor in (2.7) becomes
$A_{\scriptscriptstyle\circ}\bigl{(}-\alpha^{3}x-(1-\alpha)\bigr{)}$.

Since the transformation ${\hbox{\teneufm R}}_{3}$ is compact, the set $K$ is
compact and non-empty. And by Proposition 2.1, the symmetric factors
$B_{\scriptscriptstyle\circ}$ and $A_{\scriptscriptstyle\circ}$ of any pair
$P\in K$ are entire analytic. They have norm $1$ and thus cannot be zero.
Without the normalization N, the norm of ${\cal R}_{3}^{n}(P_{0})$ tends to
infinity, since $G$ has a positive Lyapunov exponent. So the factors
$B_{\scriptscriptstyle\circ}$ and $A_{\scriptscriptstyle\circ}$ have
determinant zero and rank $1$.

Consider the representation (2.10) for $A_{\scriptscriptstyle\circ}$. By
reversibility, the functions $a_{\scriptscriptstyle\circ}$ and
$d_{\scriptscriptstyle\circ}$ are even, while
$b_{\scriptscriptstyle\circ}(x)=-c_{\scriptscriptstyle\circ}(-x)$. Consider
first the case where one of
$a_{\scriptscriptstyle\circ},b_{\scriptscriptstyle\circ},c_{\scriptscriptstyle\circ},d_{\scriptscriptstyle\circ}$
is identically zero. Then
$a_{\scriptscriptstyle\circ}d_{\scriptscriptstyle\circ}=b_{\scriptscriptstyle\circ}c_{\scriptscriptstyle\circ}=0$,
implying that
$A_{\scriptscriptstyle\circ}=a_{\scriptscriptstyle\circ}\bigl{[}{1~{}0\atop
0~{}0}\bigr{]}$ or
$A_{\scriptscriptstyle\circ}=d_{\scriptscriptstyle\circ}\bigl{[}{0~{}0\atop
0~{}1}\bigr{]}$. Using now (2.6) and (2.7), we see that
$\tilde{P}={\hbox{\teneufm R}}_{3}(P)$ has symmetric factors with the
properties described in Theorem 2.3.

Assume now that none of
$a_{\scriptscriptstyle\circ},b_{\scriptscriptstyle\circ},c_{\scriptscriptstyle\circ},d_{\scriptscriptstyle\circ}$
is identically zero. Consider the ratio
$r_{0}=c_{\scriptscriptstyle\circ}/a_{\scriptscriptstyle\circ}$. This is a
meromorphic function describing the angle between the range of
$A_{\scriptscriptstyle\circ}$ and one of the coordinate axis. Denote by
$r_{n}$ the corresponding ratio for the pair $P_{n}={\hbox{\teneufm
R}}_{3}^{n}(P)$. By (2.7) we have

$r_{1}(x)=r_{0}\bigl{(}-\alpha^{3}x-(1-\alpha)\bigr{)}\,.$ $None$

Notice that the function $x\mapsto-\alpha^{3}x-(1-\alpha)$ has
$x_{0}=-\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}$ as a fixed point. Setting
$R_{n}(t)=r_{n}\bigl{(}x_{0}+t)$, the identity (2.12) becomes

$R_{n}(t)=R_{0}\bigl{(}(-\alpha)^{3n}t\bigr{)}\,,$ $None$

with $n=1$. The same holds for any $n>0$ by iteration. Unless $R$ has a pole
at the origin, we see that $R_{n}\to R_{0}(0)$ on some open neighborhood of
the origin. If $R$ has a pole at the origin, we repeat the above with
$r_{0}=a_{\scriptscriptstyle\circ}/c_{\scriptscriptstyle\circ}$ in place of
$r_{0}=c_{\scriptscriptstyle\circ}/a_{\scriptscriptstyle\circ}$. Using
compactness again, this shows that $K$ includes a pair $P$ with constant ratio
$r_{0}$ in some open neighborhood of $x_{0}$. In the case where
$r_{0}=a_{\scriptscriptstyle\circ}/c_{\scriptscriptstyle\circ}$, this implies
that $a_{\scriptscriptstyle\circ}=r_{0}c_{\scriptscriptstyle\circ}$ and
$b_{\scriptscriptstyle\circ}=r_{0}d_{\scriptscriptstyle\circ}$ on all of
${{\tends C}}$. If $r_{0}=0$, then we are in the case discussed at the
beginning. Otherwise, the reversibility property
$b_{\scriptscriptstyle\circ}=-c_{\scriptscriptstyle\circ}$ shows that
$a_{\scriptscriptstyle\circ},b_{\scriptscriptstyle\circ},c_{\scriptscriptstyle\circ},d_{\scriptscriptstyle\circ}$
are all constant multiples of $c_{\scriptscriptstyle\circ}$. A similar
argument applies in the case
$r_{0}=c_{\scriptscriptstyle\circ}/a_{\scriptscriptstyle\circ}$. In either
case, since $a_{\scriptscriptstyle\circ}$ and $d_{\scriptscriptstyle\circ}$
are even and not both identically zero, $K$ includes a pair $P$ that admits
the representation described in Theorem 2.3. QED

We are interested in describing the points $x$ where the limiting factors
$B_{\scriptscriptstyle\circ}$ and $A_{\scriptscriptstyle\circ}$ in Theorem 2.3
are zero. For a limit pair $P=(F,G)$ with $F=(1,B)$ and $G=(\alpha,A)$, the
equations (2.6) and (2.7) can be used to describe how zeros propagate under
renormalization. Let ${\cal B}$ be the set of zeros of
$B_{\scriptscriptstyle\circ}$ and ${\cal A}$ the set of zeros of
$A_{\scriptscriptstyle\circ}$. Let $\tilde{\cal B}$ and $\tilde{\cal A}$ be
the set of zeros of the symmetric factors for $\tilde{P}={\cal R}_{3}(P)$.
From (2.6) and (2.7) we see that

$\tilde{\cal B}=\alpha^{-3}\hat{\cal B}\,,\qquad\hat{\cal B}={\cal
B}\bigcup_{m=\pm 1}\Bigl{(}{\cal A}+{\textstyle{m\over
2}}\alpha^{2}\Bigr{)}\,,$ $None$

and

$\tilde{\cal A}=\alpha^{-3}\hat{\cal A}\,,\qquad\hat{\cal A}=\bigcup_{m=\pm
1}\Bigl{(}{\cal B}+{\textstyle{m\over 2}}\alpha^{2}\Bigr{)}\bigcup_{m=0,\pm
1}\Bigl{(}{\cal A}+m\alpha^{2}\Bigr{)}\,.$ $None$

The map $({\cal B},{\cal A})\mapsto\bigl{(}\tilde{\cal B},\tilde{\cal
A}\bigr{)}$ will be described in more detail in Section 4.

3\. Limit skew-products and the rotation number

num xyz goto num count 0 3. Limit skew-products and the rotation number In
order to describe the behavior of factors (1.5) with large values of
$\lambda$, we divide these matrices by $\lambda$ and consider the resulting
factors

$A(x)=\left[\matrix{v(t+x)-\epsilon&-\delta\cr\delta&0\cr}\right]\,,$ $None$

where $\delta=\lambda^{-1}$ and $\epsilon=\lambda^{-1}E$. Here, and in what
follows, we assume that $\lambda>1$. Orbits for the associated skew-product
map $G=(\alpha,A)$ on ${{T}}\times{{\tends R}}^{2}$ are in one-to-one
correspondence with solutions of the equation $H_{\delta}^{t}u=\epsilon u$,
where

$(H_{\delta}^{t}u)_{q}=v(t+q\alpha)u_{q}-\delta u_{q+1}-\delta
u_{q-1}\,,\qquad q\in{{\tends Z}}\,.$ $None$

Clearly, the operator $H_{\delta}^{t}$ on $\ell^{2}({{\tends Z}})$ has a well-
defined limit as $\delta\to 0$, given by setting $\delta=0$ in the above
equation. A sequence $u\in\ell^{2}({{\tends Z}})$ is an eigenvector of
$H_{0}^{t}$ with energy $\epsilon$ if and only if
$[v(t+q\alpha)-\epsilon]u_{q}=0$ for all $q$. So for every integer $q$, the
function $u=\delta_{q}$ supported at $q$ is an eigenvector of $H_{0}^{t}$ with
eigenvalue $v(t+q\alpha)$. Given that these eigenvectors span a dense subspace
of $\ell^{2}({{\tends Z}})$, the spectrum of $H_{0}^{t}$ is the closure of the
range of $v$, and it consists purely of eigenvalues and their accumulation
points.

The integrated density of states ${\hbox{\magnineeufm k}}(\epsilon)$ is the
fraction (in the usual limit sense) of integers $q$ for which
$v(t+q\alpha)\leq\epsilon$. By ergodicity, this is simply the measure of the
set of all points $x\in{{T}}$ for which $v(x)\leq\epsilon$.

Let us compute ${\hbox{\magnineeufm k}}(\epsilon)$ for the AM potential
$v(x)=-2\cos(2\pi x)$. For simplicity consider $t=\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}$. Then the symmetric limit factor
associated with (3.1) is

$A_{\scriptscriptstyle\circ}(x)=a_{\scriptscriptstyle\circ}(x){\eighttt\eightrm\left[\matrix{1&0\cr
0&0\cr}\right]}\,,\qquad a_{\scriptscriptstyle\circ}(x)=-\epsilon-2\cos(2\pi
x)\,.$ $None$

Given $\epsilon$ in the spectrum $[-2,2]$ of $H_{0}^{t}$, we define
$\rho=\rho(\epsilon)$ in $[0,\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt}]$ by the equation $\epsilon=-2\cos(2\pi\rho)$. Then

$a_{\scriptscriptstyle\circ}(x)=2\cos(2\pi\rho)-2\cos(2\pi
x)=4\sin(\pi(x-\rho))\sin(\pi(x+\rho))\,.$ $None$

If we represent ${{T}}$ as the interval $[-\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}]$ with the endpoints identified,
then $a_{\scriptscriptstyle\circ}$ is negative on $(-\rho,\rho)$ and positive
on the complement of $[-\rho,\rho]$. This shows that ${\hbox{\magnineeufm
k}}(\epsilon)=2\rho$.

In order to determine the rotation number of the skew-product map
$G_{\epsilon}=(\alpha,A)$ with symmetric factor (3.3), let us first
reformulate the definition of the number $\mathop{\rm
Rot}\nolimits_{\scriptscriptstyle N}(G)$ that appears in (1.7). It counts half
the number of times that the first component of the vector $A^{\ast k}(t)y$
changes sign as $k$ is increased from $1$ to $N$. The limit (1.7) is
independent of $t$ and of the initial condition $y\in{{\tends
R}}^{2}\setminus\\{0\\}$. We choose the same definition for degenerate skew-
products with $A(x)=a_{\scriptscriptstyle\circ}(x){\tt A}$, but we assume that
${\tt A}y\neq 0$.

Notice that $A^{\ast k}(t)y$ changes sign as $k$ is increased to $k+1$
precisely when $a_{\scriptscriptstyle\circ}(t+k\alpha)$ is negative. So by
ergodicity, the average number of sign changes is the measure of
$[-\rho,\rho]$. The limit (1.7) is half this number, which is $\rho$. So in
summary, we have

${\hbox{\magnineeufm k}}(\epsilon)=2\rho\,,\qquad\mathop{\rm
rot}\nolimits(G_{\epsilon})=\rho\,,\qquad\epsilon=-2\cos(2\pi\rho)\,.$ $None$

In particular, ${\hbox{\magnineeufm k}}(\epsilon)=2\mathop{\rm
rot}\nolimits(G_{\epsilon})$, which corresponds to the known relation for the
AM factors. Notice that $H_{0}^{t}+2\delta{\rm I}\geq H_{\delta}^{t}\geq
H_{0}^{t}-2\delta{\rm I}$, where $H\geq K$ means that $H-K$ is a positive
operator. Using the definition of ${\hbox{\magnineeufm k}}_{\scriptscriptstyle
L}$ given in the introduction, we have ${\hbox{\magnineeufm
k}}_{\scriptscriptstyle L}\bigl{(}H_{0}^{t}+2\delta{\rm
I},\epsilon\bigr{)}\leq{\hbox{\magnineeufm k}}_{\scriptscriptstyle
L}\bigl{(}H_{\delta}^{t},\epsilon\bigr{)}\leq{\hbox{\magnineeufm
k}}_{\scriptscriptstyle L}\bigl{(}H_{0}^{t}-2\delta{\rm I},\epsilon\bigr{)}$.
Taking $L\to\infty$ and then $\delta\to 0$, and using that
${\hbox{\magnineeufm k}}(\epsilon)=2\mathop{\rm rot}\nolimits(G_{\epsilon})$
for $0\leq\delta<1$, we obtain the following.

Proposition 3.1. Let $G_{\delta}=(\alpha,A)$, with $A$ given by (3.1) and
$v=-2\cos(2\pi\,\hbox{\bf.})$. Then $\mathop{\rm
rot}\nolimits(G_{\delta})\to\mathop{\rm rot}\nolimits(G_{0})$ as $\delta\to
0$, uniformly in $\epsilon$.

The same holds for real analytic potentials close to
$v=-2\cos(2\pi\,\hbox{\bf.})$.

Notice that a skew-product map $G=(\alpha,A)$ with
$A(x)=a(x)\bigl{[}{1~{}0\atop 0~{}0}\bigr{]}$ is for practical purposes a
skew-product map $g=(\alpha,a)$ on ${{T}}\times{{\tends R}}$, where $a(x)$
acts on ${{\tends R}}$ by multiplication. The rotation number of $g$,
determined via sign changes, is clearly the same as the rotation number of
$G$.

Recall our definition (1.7) of the rotation number $\mathop{\rm
rot}\nolimits(G)$ is restricted to Schrödinger factors as defined by (1.5).
The formula (1.7) can be applied to other pairs $G=(\alpha,A)$ as well. But it
does not yield the desired result, in general, not even modulo $\scriptstyle
1$$\scriptstyle/$$\scriptstyle 2$ .

For a more general definition of a rotation number for skew-product maps
$G=(\alpha,A)$ on ${{T}}\times{{\tends R}}^{2}$, assume that $A:{{T}}\to{\rm
SL}(2,{{\tends R}})$ is continuous. Consider the map $y\mapsto A(x)y$ on
${{\tends R}}\setminus\\{0\\}$. Writing $y$ as $\|y\|\bigl{[}{\cos t\atop\sin
t}\bigr{]}$ and $\eta=A(x)y$ as
$\|\eta\|\bigl{[}{\cos\tau\atop\sin\tau}\bigr{]}$, we have a continuous
function $\tau=\tau(x,t)$ from ${{T}}\times{{S}}$ to ${{S}}$, where
${{S}}={{\tends R}}/(2\pi{{\tends Z}})$. Each of the maps $t\mapsto\tau(x,t)$
on ${{S}}$ admits a continuous lift $t\mapsto t+g(x,t)$ to ${{\tends R}}$.
This lift is unique up to an additive constant $2\pi m_{x}$ with
$m_{x}\in{{\tends Z}}$. Assuming now that $A$ is homotopic to $x\mapsto{\bf
1}$, $g$ can be chosen continuous, with $m_{x}=m$ for all $x$. Using the map
${\cal G}$ defined by ${\cal G}(x,t)=(x+\alpha,t+g(x,t))$, one now defines

$\varrho(G)=\lim_{N\to\infty}{\Sigma_{\scriptscriptstyle N}(G)\over
N}\,,\qquad\Sigma_{\scriptscriptstyle
N}(G)\mathrel{\mathop{=}^{\scriptscriptstyle\rm def}}{1\over
2\pi}\sum_{n=1}^{N}g\bigl{(}{\cal G}^{n}(x,t)\bigr{)}\,.$ $None$

Here, $\alpha$ can be any irrational number. Using the unique ergodicity of
$x\mapsto x+\alpha$ on ${{T}}$, it is possible to prove [8,9] that the limit
(3.6) exists, is independent of $(x,t)$, and that convergence is uniform in
$(x,t)$. Due to the freedom of choosing $m\in{{\tends Z}}$, the rotation
number $\varrho(G)$ is unique only modulo $1$.

$\Sigma_{\scriptscriptstyle N}(G)$ measures the amount of rotation on the
interval $\\{1,2,\ldots,N\\}$, in units of $1$ per revolution. For Schrödinger
maps $G=(\alpha,A)$, we assume that the lift $g$ has been chosen in such a way
that $g(0,0)\in[0,2\pi)$. Then $\Sigma_{\scriptscriptstyle N}(G)$ agrees
within an error $\leq 1$ with half the number of sign changes $\mathop{\rm
Rot}\nolimits_{\scriptscriptstyle N}(G)$. This follows from the fact that the
second row of $A(x)$ is $[1\;0]$, implying that $A(x)$ maps the right (left)
half-plane to the upper (lower) half-plane. So every full revolution is
associated with a single sign change positive $\to$ negative and a single sign
change negative $\to$ positive; see also [10,12,20]. This implies in
particular that $\mathop{\rm rot}\nolimits(G)=\varrho(G)$ for Schrödinger
factors (1.5) with continuous potentials $v$. (Replacing $A$ by $-A$ in this
case yields $\varrho\mapsto\varrho-\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt}$ modulo $1$, while sign counting would yield $\mathop{\rm
rot}\nolimits\mapsto\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt}-\mathop{\rm rot}\nolimits$.)

4\. Periodic orbits

num xyz goto num count 0 4. Periodic orbits In this section we consider pairs
of skew-product maps $p=(f,g)$ with $f=(1,b)$ and $g=(\alpha,a)$, where $b$
and $a$ are even real analytic functions whose values $a(x)$ and $b(x)$ act on
${{\tends R}}$ by multiplication. The RG transformations ${\cal R}_{3n}$
extend naturally to such pairs, if we define $b^{\dagger}=b$ and
$a^{\dagger}=a$. Conjugacy has no effect for scalar factors, so
$\Lambda_{k}=\Lambda^{k}$ for all $k$. The condition that $a$ and $b$ be even
corresponds to reversibility.

Our goal here is to construct periodic orbits under renormalization and to
describe some of their properties.

Convention 5.  Throughout this section, the second component $A$ in a pair
$(\alpha,A)$ representing a skew-product map denotes the symmetric factor. So
the composition rule here is $(\beta,B)(\alpha,A)=(\alpha+\beta,C)$ with
$C(x)=B(x+\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt})A(x-\hbox{\raise
2.2pt\hbox{$\scriptstyle\beta$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt})$.

First we need to establish some properties of the map $({\cal B},{\cal
A})\mapsto\bigl{(}\tilde{\cal B},\tilde{\cal A}\bigr{)}$ defined by the
equations (2.14) and (2.15). A crucial ingredient in the proof below is the
well-known identity

$p_{k}-q_{k}\alpha=(-\alpha)^{k+1}\,,\qquad q_{k}=p_{k+1}\,,$ $None$

where $p_{k}$ denotes the $k$-th Fibonacci number.

Lemma 4.1. Let ${\cal B}_{0}=\\{\\}$. Given ${\cal A}_{0}\subset{{\tends R}}$,
define ${\cal B}_{n}=\tilde{\cal B}_{n-1}$ and ${\cal A}_{n}=\tilde{\cal
A}_{n-1}$ for $n=1,2,3,\ldots$ For every $r>0$ define ${\cal A}_{n}(r)={\cal
A}_{n}\cap[-r,r]$ and ${\cal B}_{n}(r)={\cal B}_{n}\cap[-r,r]$. Consider now
${\cal A}_{0}=\\{\rho\\}$ with $0\leq\rho\leq\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}$. Then ${\cal B}_{n}(\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt})$ and ${\cal A}_{n}(\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt})$ contain at most one point, for
each $n$. The sequence $k\mapsto{\cal A}_{k}(\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt})$ is periodic if and only if $\rho$
is positive periodic or belongs to $\\{0,\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}\\}$. Furthermore, some set ${\cal
A}_{n}(\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt})$ includes a point $\tilde{\rho}\in{1\over 2}{{\tends
Z}}[\alpha]$ if and only if $\tilde{\rho}-\rho\in{{\tends Z}}[\alpha]$.

Proof. Let $B_{0}=\\{\\}$ and $A_{0}=\\{\rho\\}$, considered as sets on the
circle ${{T}}={{\tends R}}/{{\tends Z}}$. Let $B_{n}=\hat{B}_{n-1}$ and
$A_{n}=\hat{A}_{n-1}$ for $n=1,2,3,\ldots$, using the map
$(B,A)\mapsto\bigl{(}\hat{B},\hat{A}\bigr{)}$ defined by (2.14) and (2.15).
Consider the functions $b_{0},a_{0}:{{T}}\to{{\tends R}}$ defined by
$b_{0}(x)=1$ and $a_{0}(x)=\sin(\pi(x-\rho))$. If we set $b_{n}=\hat{b}_{n-1}$
and $a_{n}=\hat{a}_{n-1}$ for $n=1,2,3,\ldots$, where

$\eqalign{\hat{b}(x)&=a\bigl{(}x-{\textstyle{1-\alpha\over
2}}\bigr{)}b(x)a\bigl{(}x+{\textstyle{1-\alpha\over
2}}\bigr{)}\,,\cr\hat{a}(x)&=a\bigl{(}x+(1-\alpha)\bigr{)}b\bigl{(}x+{\textstyle{1-\alpha\over
2}}\bigr{)}a\bigl{(}x\bigr{)}b\bigl{(}x-{\textstyle{1-\alpha\over
2}}\bigr{)}a\bigl{(}\alpha^{3}x-(1-\alpha)\bigr{)}\,,\cr}$ $None$

then the zeros of $b_{n}$ and $a_{n}$ are precisely the sets $B_{n}$ and
$A_{n}$, respectively. Notice that

$\eqalign{b_{n}(x)&=a_{0}\bigl{(}x+{\textstyle{q_{3n-1}\over
2}}\alpha\bigr{)}\cdots a_{0}\bigl{(}x+{\textstyle{1\over
2}}\alpha\bigr{)}a_{0}\bigl{(}x-{\textstyle{1\over 2}}\alpha\bigr{)}\cdots
a_{0}\bigl{(}x-{\textstyle{q_{3n-1}\over 2}}\alpha\bigr{)}\,,\cr
a_{n}(x)&=a_{0}\bigl{(}x+{\textstyle{q_{3n}-1\over 2}}\alpha\bigr{)}\cdots
a_{0}\bigl{(}x+\alpha\bigr{)}a_{0}(x)a_{0}\bigl{(}x-\alpha\bigr{)}\cdots
a_{0}\bigl{(}x-{\textstyle{q_{3n}-1\over 2}}\alpha\bigr{)}\,.\cr}$ $None$

So the zeros of $a_{n}$ constitute an orbit on ${{T}}$ under translation
$x\mapsto x+\alpha\;(\mathop{\rm mod}\nolimits 1)$. This orbit has length
$q_{3n}$ and its center point is at $\rho$. By the three-gap theorem [2,3,4],
the gaps between adjacent points in $A_{n}$ can only take the values
$\alpha^{3n-2}$, $\alpha^{3n-1}$, and $\alpha^{3n}$. In fact, the gap
$\alpha^{3n-2}$ cannot occur, since it gets closed in the last step of the
orbit. So the gaps in the scaled orbit ${\cal A}_{n}=\alpha^{-3n}A_{n}$ are
either $\alpha^{-1}$ or $1$. Thus, ${\cal A}_{n}(\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt})$ contains at most one point.
Similarly, the gaps in $B_{n}$ are no shorter $\alpha^{3n-1}$, so the scaled
orbit ${\cal B}_{n}=\alpha^{-3n}B_{n}$ has at most one point in
$[-\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt},\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt}]$.

Consider now the condition that ${\cal A}_{n}$ contains a given real number
$\tilde{\rho}$. This condition is equivalent to
$\tilde{a}_{n}(\tilde{\rho})=0$, where
$\tilde{a}_{n}(x)=a_{n}\bigl{(}\alpha^{3n}x\bigr{)}$. From (4.3) we see that
$\tilde{a}_{n}(\tilde{\rho})=0$ exactly when

$\bigl{[}(-\alpha)^{3n}-1\bigr{]}\tilde{\rho}-\nu_{2}+\nu_{1}\alpha=\rho-\tilde{\rho}\,,\qquad|\nu_{1}|\leq{q_{3n}-1\over
2}\,,$ $None$

for some $\nu\in{{\tends Z}}^{2}$. Using that
$(p_{3n-1}-1)-p_{3n}\alpha=(-\alpha)^{3n}-1$, this can be written as

$\bigl{[}(p_{3n-1}-1)-p_{3n}\alpha\bigr{]}\tilde{\rho}-\nu_{2}+\nu_{1}\alpha=\rho-\tilde{\rho}\,,\qquad|\nu_{1}|\leq{q_{3n}-1\over
2}\,.$ $None$

If $\tilde{\rho}=0$, then (4.5) can be satisfied from some $n$ if and only if
$\rho\in{{\tends Z}}[\alpha]$. And a straightforward computation shows that
$\rho=0$ yields ${\cal A}_{n}(\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt})=\\{0\\}$ for all $n$. Consider now an arbitrary
$\tilde{\rho}\in{1\over 2}{{\tends Z}}[\alpha]$. In this case, the equation
(4.5) can be written as

$\left[{p_{3n-1}-1\over 2}-{p_{3n}\over
2}\alpha\right]2\tilde{\rho}-\nu_{2}+\nu_{1}\alpha=\rho-\tilde{\rho}\,,\qquad|\nu_{1}|\leq{q_{3n}-1\over
2}\,.$ $None$

Notice that $p_{3n-1}-1$ and $p_{3n}$ are even. So we must have
$\rho\in\tilde{\rho}+{{\tends Z}}[\alpha]$. For $\rho=\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}$ and $\rho=\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}$, an explicit computation shows that
${\cal A}_{n}(\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt})=\\{\pm\rho\\}$ for all $n$. We note also that (4.6) has no
solution for $\tilde{\rho}=\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt}-\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}$.

Consider now the periodicity condition $\tilde{a}_{n}(\rho)=0$, which
corresponds to setting $\tilde{\rho}=\rho$ in (4.5). Clearly, we need
$\rho\in{{\tends Q}}[\alpha]$ for (4.5) to have a solution in this case. So
write $\rho={w\over v}+{u\over v}\alpha$ for some $v>0$. Without loss of
generality, we can restrict restrict to values of $n$ for which
$p_{3n-1}\equiv 1$ and $p_{3n}\equiv 0$ modulo $v$. This condition is
satisfied e.g. if $n$ is a multiple of the Pisano period $\ell(v)$. By
definition, the Pisano period $\ell(v)$ is the period of the Fibonacci
sequence $k\mapsto p_{k}$ modulo $v$. Then we have $\tilde{a}_{n}(\rho)=0$, if
and only if

$\left[{p_{3n-1}-1\over v}-{p_{3n}\over
v}\alpha\right](w+u\alpha)-\nu_{2}+\nu_{1}\alpha=0\,,\qquad|\nu_{1}|\leq{q_{3n}-1\over
2}\,,$ $None$

for some $\nu\in{{\tends Z}}^{2}$. Multiplying out the product
$[\ldots](w+u\alpha)$ in the above equation and using that
$\alpha^{2}=1-\alpha$, one finds that

$\nu_{1}={p_{3n}\over v}w-{q_{3n}-1\over v}u\,,\qquad\nu_{2}={p_{3n-1}-1\over
v}w-{p_{3n}\over v}u\,.$ $None$

And the condition $|\nu_{1}|\leq{q_{3n}-1\over 2}$ becomes

$\left|{p_{3n}\over q_{3n}-1}{w\over v}-{u\over v}\right|\leq{1\over 2}\,.$
$None$

Assume now that $\rho$ is neither $0$ nor $\scriptstyle
1$$\scriptstyle/$$\scriptstyle 2$ , since these values have been covered
already. Then equality in (4.9) can hold only for finitely many values of $n$.
Given that ${p_{3n}\over q_{3n}-1}\to\alpha$, we have $\tilde{a}_{n}(\rho)=0$
as $n\to\infty$ along multiples of $\ell(v)$ if and only if $\bigl{|}{w\over
v}\alpha-{u\over v}\bigr{|}<{1\over 2}$, meaning that $\rho$ is positive
periodic or equal to $\scriptstyle\alpha$$\scriptstyle/$$\scriptstyle 2$ .
This show that the sequence $k\mapsto{\cal A}_{k}(\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt})$ is periodic if and only if $\rho$
is positive periodic or belongs to $\\{0,\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}\\}$. QED

We note that, since points evolve independently, having $m>1$ points in ${\cal
A}_{0}$ results in ${\cal A}_{n}(\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt})$ and ${\cal B}_{n}(\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt})$ containing $m$ points or none. Furthermore, if ${\cal
A}_{0}$ is invariant under $t\mapsto-t$, then so are all the sets ${\cal
A}_{n}$ and ${\cal B}_{n}$.

Convention 6.  In the remaining part of this paper, $\rho$ is a fixed but
arbitrary positive periodic rotation number. Referring to the periodicity
statement in Lemma 4.1, the fundamental period of the sequence $k\mapsto{\cal
A}_{k}$, generated from ${\cal B}_{0}=\\{\\}$ and ${\cal A}_{0}=\\{\rho\\}$,
will from now on be denoted by $n$.

Recall from (2.11) that ${\hbox{\teneufm R}}_{3n}={\hbox{\teneufm
N}}\circ{\cal R}_{3n}$, where N is a suitable multiplicative normalization.
The following theorem deals only with (zeros of) specific functions, so we
“normalize” here simply by identifying functions that are constant multiples
of each other.

Theorem 4.2. ${\hbox{\teneufm R}}_{3n}$ has a fixed point
$p_{\diamond}=((1,b_{\diamond}),(\alpha,a_{\diamond})$ with the following
properties. The functions $b_{\diamond}$ and $a_{\diamond}$ are entire and
even, and as functions of $\sqrt{x}$, they are of order $\scriptstyle
1$$\scriptstyle/$$\scriptstyle 2$ . The zeros of $b_{\diamond}$ and
$a_{\diamond}$ are all real and simple. $a_{\diamond}$ has zeros at $\pm\rho$
and no other zeros in the interval $[-\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt},\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt}]$.

Following an approach used in [27], we construct the functions $b_{\diamond}$
and $a_{\diamond}$ by first determining their zeros. In what follows, we
identify nonzero functions $u$ and $v$ that are constant multiples of each
other; in symbols, $u\equiv v$. The set of zeros of a function $u$ will be
denoted by ${\cal Z}(u)$.

By reversibility, the factors of our skew-product maps $f=(1,b)$ and
$g=(\alpha,a)$ admit a factorization $b(x)\equiv{\hbox{\magnineeufm
b}}(x){\hbox{\magnineeufm b}}(-x)$ and $a(x)\equiv{\hbox{\magnineeufm
a}}(x){\hbox{\magnineeufm a}}(-x)$. This factorization is canonical (via
orbits) and propagates under renormalization. The functions b and a will be
referred to as the semi-factors of $f$ and $g$, respectively. Consider the
sequence of zeros $j\mapsto b_{j}$ of b and the sequence of zeros $j\mapsto
a_{j}$ of a. If the sums $\sum_{j}|b_{j}|^{-2}$ and $\sum_{j}|a_{j}|^{-2}$ are
finite, then

$b(x)\equiv\prod_{j}\biggl{(}1-{x^{2}\over b_{j}^{2}}\biggr{)}\,,\qquad
a(x)\equiv\prod_{j}\biggl{(}1-{x^{2}\over b_{j}^{2}}\biggr{)}\,,$ $None$

with uniform convergence on compact subsets of ${{\tends C}}$. In the cases
considered below, all zeros are simple, so we may work with the sets ${\cal
B}={\cal Z}({\hbox{\magnineeufm b}})$ and ${\cal A}={\cal
Z}({\hbox{\magnineeufm a}})$ instead of the sequences $j\mapsto b_{j}$ and
$j\mapsto a_{j}$, respectively.

If ${\cal P}=({\cal B},{\cal A})$ is a pair of subsets of ${{\tends R}}$,
define the pair $\tilde{\cal P}=\bigl{(}\tilde{\cal B},\tilde{\cal A}\bigr{)}$
by the equations (2.14) and (2.15). Denoting the map ${\cal
P}\mapsto\tilde{\cal P}$ by $R$, we define ${\cal P}_{t}=R^{tn}({\cal P})$ for
$t=0,1,2,\ldots$. To recall the connection of $R$ with the transformation
${\cal R}_{3}$, consider two reversible maps $f=(1,b)$ and $g=(\alpha,a)$. If
${\cal B}={\cal Z}({\hbox{\magnineeufm b}})$ and ${\cal A}={\cal
Z}({\hbox{\magnineeufm a}})$, then we have ${\cal B}_{t}={\cal
Z}\bigl{(}{\hbox{\magnineeufm b}}^{t}\bigr{)}$ and ${\cal A}_{t}={\cal
Z}\bigl{(}{\hbox{\magnineeufm a}}^{t}\bigr{)}$. Here ${\hbox{\magnineeufm
b}}^{t}$ and ${\hbox{\magnineeufm a}}^{t}$ are the semi-factors associated
with the pair $p^{t}={\cal R}_{3}^{tn}(p)$, where $p=(f,g)$.

Proof of Theorem 4.2.  In order to simplify the description, consider the
explicit pair $p=((1,1),(\alpha,a))$ with
$a(x)=4\sin(\pi(x-\rho))\sin(\pi(x+\rho))$. Its semi-factors are given by

${\hbox{\magnineeufm b}}(x)=1\,,\qquad{\hbox{\magnineeufm
a}}(x)\equiv\sin(\pi(x-\rho))\,.$ $None$

Notice that a is periodic with period $2$. But its zeros are periodic with
period $1$, that is, ${\cal B}=\\{\\}$ and ${\cal A}=\\{\rho\\}+{{\tends Z}}$.
Since the transformation $R$ involves a scaling by $\alpha^{-3}$, the sets
${\cal B}_{t}$ and ${\cal A}_{t}$ are periodic with period
$r_{t}=\alpha^{-3nt}$. So we can regard them as subsets of the circle
${{T}}_{t}={{\tends R}}/(r_{t}{{\tends Z}})$. Representing this circle by the
interval $[-r_{t}/2,r_{t}/2]$ whose endpoints are being identified, the zero
sets ${\cal B}_{t}^{\prime}\subset{{T}}_{t}$ and ${\cal
A}_{t}^{\prime}\subset{{T}}_{t}$ associated with $p^{t}$ are given by ${\cal
B}_{t}^{\prime}={\cal B}_{t}\cap{{T}}_{t}$ and ${\cal A}_{t}^{\prime}={\cal
A}_{t}\cap{{T}}_{t}$, respectively.

As described in the proof of Lemma 4.1, the set ${\cal A}_{t}^{\prime}$ is a
scaled orbit of length $q_{3nk}$ under $x\mapsto x+\alpha$, centered at
$x=\rho$. As a result, the gaps between two adjacent zeros in ${\cal
A}_{t}^{\prime}$ are either $\alpha^{-1}$ or $1$. Similarly, the gaps between
two adjacent zeros in ${\cal B}_{t}^{\prime}$ are no shorter than
$\alpha^{-1}$. Recall also that ${\cal A}_{0}^{\prime}\subset{\cal
A}_{1}^{\prime}$ by Lemma 4.1. Thus, given that ${\cal
B}_{0}^{\prime}\subset{\cal B}_{1}^{\prime}$, the inclusions ${\cal
B}_{t}^{\prime}\subset{\cal B}_{t+1}^{\prime}$ and ${\cal
A}_{t}^{\prime}\subset{\cal A}_{t+1}^{\prime}$ hold for all $t$.

Now we can define ${\cal B}_{\infty}=\lim_{t}{\cal B}_{t}$ and ${\cal
A}_{\infty}=\lim_{t}{\cal A}_{t}$, using either the $\liminf$ or $\limsup$,
with the same result. Due to the above-mentioned gap properties, the zeros
$b^{t}_{j}$ of ${\hbox{\magnineeufm b}}^{t}$ and the zeros $a^{t}_{j}$ of
${\hbox{\magnineeufm a}}^{t}$ satisfy $\sum_{j}(b^{t}_{j})^{-2}<\infty$ and
$\sum_{j}(a^{t}_{j})^{-2}<\infty$, respectively. This includes the case
$t=\infty$ as well. We note that all zeros are real. And they are simple: we
have $\rho\not\in{1\over 2}{{\tends Z}}[\alpha]$, so $-\rho$ is not on the
orbit of $\rho$ under translation $x\mapsto x+\alpha$ on ${{T}}$.

Clearly, ${\cal P}_{\infty}=({\cal B}_{\infty},{\cal A}_{\infty})$ is a fixed
point of $R^{3n}$. Now define $b_{\diamond}$ and $a_{\diamond}$ as products
(4.10), for the zeros $b_{j}=b^{\infty}_{j}$ and $a_{j}=a^{\infty}_{j}$,
respectively. Since we are dealing with a class of entire functions (even, and
of order $\scriptstyle 1$$\scriptstyle/$$\scriptstyle 2$ in the variable
$\sqrt{x}$ ) that are determined up to a constant factor by their zeros, it is
not hard to see that the pair $p_{\diamond}$ with symmetric factors
$b_{\diamond}$ and $a_{\diamond}$ is a fixed point of ${\cal R}_{3n}$, up to a
normalization by constant factors. QED

We note that $b^{t}\to b_{\diamond}$ and $a^{t}\to a_{\diamond}$, uniformly on
compact subsets of ${{\tends C}}$. This is a straightforward consequence of
our uniform gap-bounds, together with the property that ${\cal
B}_{t}^{\prime}\subset{\cal B}_{t+1}^{\prime}$ and ${\cal
A}_{t}^{\prime}\subset{\cal A}_{t+1}^{\prime}$ for all $t$. But we need not
prove this here, since it follows from Theorem 4.3 below.

For a more detailed description of the convergence argument we refer to [27],
where a similar analysis has been carried out (related to the critical case)
for meromorphic factors $a(x)={\hbox{\tenib a}}(x)/{\hbox{\tenib a}}(-x)$ and
quadratic irrational $\alpha$.

We note that the product (4.10) for the function $a_{\diamond}$ is in effect
an infinite product of sine functions, albeit rescaled. The classical product
of sines is the Sudler product $P_{m}(\alpha)=\prod_{j=1}^{m}|2\sin\pi
j\alpha|$. Perturbed versions of this product, for $\alpha$ the golden mean,
have been investigated recently in [25].

Let us return to the issue of normalization. As can be seen from (2.6) and
(2.7), we may replace $b_{\diamond}\mapsto-b_{\diamond}$ and/or
$a_{\diamond}\mapsto-a_{\diamond}$, in such a way that $b_{\diamond}(0)>0$ and
$a_{\diamond}(0)<0$. This corresponds to the signs of the limit AM pair, with
$b_{\scriptscriptstyle\circ}=1$ and $a_{\scriptscriptstyle\circ}$ given by
(3.3).

The symmetric factors of $\tilde{p}_{\diamond}={\cal R}_{3n}(p_{\diamond})$
are of the form $\tilde{b}_{\diamond}=b_{\diamond}e^{\tilde{v}_{\diamond}}$
and $\tilde{a}_{\diamond}=a_{\diamond}e^{\tilde{u}_{\diamond}}$ for some real
constants $\tilde{v}_{\diamond}$ and $\tilde{u}_{\diamond}$. Consider a pair
$p=((1,a),\alpha,b))$ with symmetric factors $b=b_{\diamond}e^{v}$ and
$a=a_{\diamond}e^{u}$. If $u$ and $v$ are constants, then the symmetric
factors for $\tilde{p}={\cal R}_{3n}(p)$ are
$\tilde{b}=\tilde{b}_{\diamond}e^{\tilde{v}}$ and
$\tilde{a}=\tilde{a}_{\diamond}e^{\tilde{u}}$, where

$\left[\matrix{\tilde{v}\cr\tilde{u}\cr}\right]=\left[\matrix{\tilde{v}_{\diamond}\cr\tilde{u}_{\diamond}\cr}\right]+U^{3n}\left[\matrix{v\cr
u\cr}\right]\,,\qquad U=\left[\matrix{0&1\cr 1&1\cr}\right]\,.$ $None$

Since $U$ is hyperbolic, it is clear that we can determine $v$ and $u$ in such
a way that $\tilde{v}=v$ and $\tilde{u}=u$. Thus, we will assume from now on
that $b_{\diamond}$ and $a_{\diamond}$ have been normalized in such a way that
$p_{\diamond}$ is a fixed point of ${\cal R}_{3n}$.

For the normalization N that enters the definition (2.11) of ${\hbox{\teneufm
R}}_{3n}$, we now choose $(b,a)\mapsto\bigl{(}Mb,a\bigr{)}$, with $M$ defined
by the condition $Mb(0)=b_{\diamond}(0)$. Clearly, $p_{\diamond}$ is a fixed
point of ${\hbox{\teneufm R}}_{3n}$.

Denote by $\kappa$ the positive zero of $b_{\diamond}$ that is closest to the
origin. Presumably $\kappa>\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}$, but we are not sure. In any case,
all zeros of $p_{\diamond}$ are determined (via orbits) from the two zeros of
$a_{\diamond}$ at $\pm\rho$. Under renormalization, zeros of $b_{\diamond}$ in
$[-\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}]$ could only generate zeros of
$b_{\diamond}$ outside $[-\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}]$ and/or zeros of $a$ outside
$[\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt},\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt}]$.

The following holds for some open neighborhood $Z$ of $\kappa$.

Theorem 4.3. Let $b$ be even and real analytic on $[-\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt},\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}]$, satisfying $b(0)>0$. Let $a$ be
even and real analytic on $[-\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt},\hbox{\raise 2.2pt\hbox{$\scriptstyle
1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower 2.2pt\hbox{$\scriptstyle
2$}\hskip 1.0pt}]$, satisfying $a(0)<0$. Assume that $b$ has no zero in
$(0,\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}]\setminus Z$. Assume that $a$ has a
simple zero at $\rho$ and no other zeros in $(0,\hbox{\raise
2.2pt\hbox{$\scriptstyle 1$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}]$. If $p=((1,b),(\alpha,a))$, then
${\hbox{\teneufm R}}_{3n}^{k}(p)\to p_{\diamond}$ as $k\to\infty$, uniformly
on compact subsets of ${{\tends C}}$.

Given any real numbers ${r_{{}_{\hskip-1.0ptA}}}$ and
${r_{{}_{\hskip-1.0ptB}}}$ satisfying (2.8), define ${\cal G}_{r}={\cal
F}_{r_{{}_{\hskip-1.0ptB}}}\times{\cal F}_{r_{{}_{\hskip-1.0ptA}}}$ and denote
by ${\cal G}_{r}^{\prime}$ the subspace of reversible pairs.

Proof of Theorem 4.3.  Consider first the sequence of pairs $p_{m}={\cal
R}_{3n}^{m}(p)$. For sufficiently large $m$, the zero sets of $b_{m}$ and
$a_{m}$ agree with the zero sets of $b_{\diamond}$ and $a_{\diamond}$,
respectively, on any given compact subset of ${{\tends C}}$. This includes the
absence of non-real zeros, since their imaginary parts expand by a factor
$\alpha^{-3}$ under ${\cal R}_{3}$, as can be seen from (2.14) and (2.15).

Now pick ${r_{{}_{\hskip-1.0ptA}}}$ and ${r_{{}_{\hskip-1.0ptB}}}$ satisfying
(2.8). If $m_{0}$ has been chosen sufficiently large, then for $m\geq m_{0}$,
the pair $p_{m}$ belongs to ${\cal G}_{r}^{\prime}$, and its zeros agree with
those of $p_{\diamond}$ on the disks $|x|\leq{r_{{}_{\hskip-1.0ptB}}}$ for $b$
and $|x|\leq{r_{{}_{\hskip-1.0ptA}}}$ for $a$. To simplify notation, replace
now $p$ by $p_{m_{0}}$.

Then we have $b=b_{\diamond}e^{v}$ and $a=a_{\diamond}e^{u}$ for a pair of
functions $(u,v)\in{\cal G}_{r}$. And the symmetric factors of
$\tilde{p}={\hbox{\teneufm R}}_{3n}(p)$ are again of the form
$\tilde{b}=b_{\diamond}e^{\tilde{v}}$ and
$\tilde{a}=a_{\diamond}e^{\tilde{u}}$, with $(\tilde{u},\tilde{v})\in{\cal
G}_{r}$. The map $L:(v,u)\mapsto(\tilde{v},\tilde{u})$ on ${\cal
G}_{r}^{\prime}$ is linear, compact, and rather simple: its eigenvectors are
pairs of polynomials of degree $2k$ with eigenvalues $\alpha^{3n(2k\pm 1)}$;
except at degree $0$, where our normalization condition N acts. Without N, the
constant functions transform via the matrix $U^{3n}$ in (4.12). Our
normalization condition N eliminates the expanding eigenvalue $\alpha^{-3n}$
and makes the constant terms contract as well. Thus, $u$ and $v$ tend to zero
under iteration of $L$, implying that ${\hbox{\teneufm R}}_{3n}^{k}(p)\to
p_{\diamond}$ in ${\cal G}_{r}^{\prime}$, as $k\to\infty$. QED

Remark 7.  Theorem 1.2 and Lemma 4.1 indicate that the periodic orbits
$p_{\diamond}$ associated with periodic rotation numbers $\rho$ are merely
periodic points in a supercritical attractor for ${\hbox{\teneufm R}}_{3}$. It
should be possible to determine non-periodic orbits associated with many other
values of $\rho$. In such an analysis, there would be no reason to restrict
$\alpha$ to the inverse golden mean or another quadratic irrational. But we
have not looked into these questions.

5\. Hyperbolicity and consequences

num xyz goto num count 0 5. Hyperbolicity and consequences Let $p_{\diamond}$
be the fixed point of ${\hbox{\teneufm R}}_{3n}$ that was described in the
last section.

Theorem 4.3 implies that the stable subspace of $D{\hbox{\teneufm
R}}_{3n}(p_{\diamond})$ has codimension one and includes all reversible pairs
$p=((1,b),(\alpha,a))$ that satisfy $a(\rho)=0$. Perturbations with
$a(\rho)\neq 0$ are unstable: (2.14) and (2.15) show that small perturbations
of zeros grow at a rate $\alpha^{-3}$ under ${\cal R}_{3}$. This implies the
following.

Corollary 5.1. The derivative of $D{\hbox{\teneufm R}}_{3n}(p_{\diamond})$ on
${\cal G}_{r}^{\prime}$ has a simple eigenvalue $\alpha^{-3n}$ and no other
spectrum outside the open unit disk. The stable subspace is characterized by
the condition $a_{\scriptscriptstyle\circ}(\rho)=0$.

Consider now the transformation ${\cal R}_{3}$ for skew-products with factors
in ${\rm GL}(2,{{\tends R}})$. Given the form of our matrices (3.1), we
associate with the fixed point $p_{\diamond}$ the pair
$P_{\diamond}=(F_{\diamond},G_{\diamond})$ with symmetric factors

$B_{\diamond}(x)=b_{\diamond}(x){\tt A}^{\dagger}\,,\qquad
A_{\diamond}(x)=a_{\diamond}(x){\tt A}\,,\qquad{\tt
A}={\eighttt\eightrm\left[\matrix{1&0\cr 0&0\cr}\right]}\,.$ $None$

Recall from (2.5) that the transformation ${\cal R}_{3n}$ includes a scaling
$\Lambda_{3n}(x,y)=\bigl{(}\alpha^{3n}x,L_{3n}y\bigr{)}$. Consider first the
choice $L_{3n}=S^{n}$. Then it is straightforward to check that $P_{\diamond}$
is a fixed point of ${\cal R}_{3n}$. If we restrict ${\cal R}_{3n}$ to pairs
$P=(F,G)$ with symmetric factors
$B_{\scriptscriptstyle\circ}(x)=b_{\scriptscriptstyle\circ}(x){\tt
A}^{\dagger}$ and
$A_{\scriptscriptstyle\circ}(x)=a_{\scriptscriptstyle\circ}(x){\tt A}$, where
${\tt A}$ is fixed as in (5.1), then $D{\cal R}_{3n}(P_{\diamond})$ is clearly
equivalent to $D{\cal R}_{3n}(p_{\diamond})$. But in the full space ${\cal
H}_{r}^{\prime}$, the derivative of $D{\cal R}_{3n}(P_{\diamond})$ has an
eigenvalue $1$ associated with conjugacies by constant matrices that commute
with $S$. This eigenvalue can be eliminated as follows.

Let $\bigl{(}\check{F},\check{G}\bigr{)}={\cal R}_{3n}(F,G)$, still with the
choice $L_{3n}=S^{n}$. As an extra normalization step, we include a conjugacy
$\check{A}_{\scriptscriptstyle\circ}\mapsto\tilde{A}_{\scriptscriptstyle\circ}=e^{-\sigma_{3n}S}\check{A}_{\scriptscriptstyle\circ}e^{\sigma_{3n}S}$
and
$\check{B}_{\scriptscriptstyle\circ}\mapsto\tilde{B}_{\scriptscriptstyle\circ}=e^{-\sigma_{3n}S}\check{B}_{\scriptscriptstyle\circ}e^{\sigma_{3n}S}$,
in order to normalize the off-diagonal terms
$\tilde{b}_{\scriptscriptstyle\circ}$ and
$\tilde{c}_{\scriptscriptstyle\circ}$ of the matrix
$\tilde{A}_{\scriptscriptstyle\circ}$. To be more precise, we choose
$\sigma_{3n}=\sigma_{3n}(P)$ in such way that

$\tilde{c}_{\scriptscriptstyle\circ}(x_{0})=0\,,\qquad
x_{0}=-{\textstyle{\alpha\over 2}}\,.$ $None$

This is possible if $\check{A}_{\scriptscriptstyle\circ}$ is sufficiently
close to $A_{\diamond}$, since $a_{\diamond}(x_{0})\neq 0$. Here we use that
$\rho\neq x_{0}$. Our reason for normalizing at $x_{0}$ is that this is the
point about which we have the scaling (2.13). This normalization step can now
be incorporated into the scaling $\Lambda_{3n}$ by choosing

$L_{3n}=S^{n}e^{\sigma_{3n}S}=S^{n}\bigl{[}\cosh(\sigma_{3n}){\bf
1}+\sinh(\sigma_{3n})S\bigr{]}\,,\qquad\sigma_{3n}=\sigma_{3n}(P)\,.$ $None$

In what follows, the transformation ${\cal R}_{3n}$ is defined with the above
choice of $L_{3n}$.

We also have to define a proper version of ${\hbox{\teneufm
R}}_{3n}={\hbox{\teneufm N}}\circ{\cal R}_{3n}$. This can be done the same way
as for scalar factors. Let $\tilde{P}={\hbox{\teneufm R}}_{3n}(P)$. If
$\tilde{B}_{\scriptscriptstyle\circ}$ denotes the symmetric factor of
$\tilde{F}$, then N consists of the scaling
$\tilde{B}_{\scriptscriptstyle\circ}\mapsto
M\tilde{B}_{\scriptscriptstyle\circ}$, where $M$ is determined by the
condition that

$M\mathop{\rm
tr}\nolimits\bigl{(}\tilde{B}_{\scriptscriptstyle\circ}(0)\bigr{)}=b_{\diamond}(0)\,.$
$None$

Clearly, $P_{\diamond}$ is a fixed point of ${\hbox{\teneufm R}}_{3n}$.

Consider now the derivative of ${\hbox{\teneufm R}}_{3n}(P_{\diamond})$,
applied to a pair $P=(F,G)$ with symmetric factors

$B_{\scriptscriptstyle\circ}={\eighttt\eightrm\left[\matrix{a_{\scriptscriptstyle
B}&b_{\scriptscriptstyle B}\cr c_{\scriptscriptstyle B}&d_{\scriptscriptstyle
B}\cr}\right]}\,,\qquad
A_{\scriptscriptstyle\circ}={\eighttt\eightrm\left[\matrix{a_{\scriptscriptstyle
A}&b_{\scriptscriptstyle A}\cr c_{\scriptscriptstyle A}&d_{\scriptscriptstyle
A}\cr}\right]}\,.$ $None$

Theorem 5.2. $D{\hbox{\teneufm R}}_{3n}(P_{\diamond})$ is hyperbolic, with a
simple eigenvalue $\alpha^{-3n}$ and no other spectrum outside the open unit
disk. The stable subspace of $D{\hbox{\teneufm R}}_{3n}(P_{\diamond})$ is
characterized by the condition $a_{\scriptscriptstyle A}(\rho)=0$ in the
representation (5.5).

Proof. Consider a representation analogous to (5.5) for the pair
$\tilde{P}=D{\hbox{\teneufm R}}_{3n}(P_{\diamond})P$, with a tilde over the
matrices and their elements.

The factors $\tilde{B}_{\scriptscriptstyle\circ}$ and
$\tilde{A}_{\scriptscriptstyle\circ}$ naturally decompose into sums of
products of matrices from
$\\{B_{\scriptscriptstyle\circ},A_{\scriptscriptstyle\circ},B_{\diamond},A_{\diamond}\\}$.
But only a few products are nonzero, due to the fact that each product can
only have one factor $B_{\scriptscriptstyle\circ}$ or
$A_{\scriptscriptstyle\circ}$. The other factors all are $B_{\diamond}$ or
$A_{\diamond}$, which are “sparse” in the sense that they only have one
nonzero matrix element.

In order to describe some nonzero terms, let us split
$B_{\scriptscriptstyle\circ}$ into a “1d” part that has $a_{\scriptscriptstyle
B}=b_{\scriptscriptstyle B}=c_{\scriptscriptstyle B}=0$, and into a “2d” part
that has $d_{\scriptscriptstyle B}=0$. Similarly, we split
$A_{\scriptscriptstyle\circ}$ into a 1d part that has $b_{\scriptscriptstyle
A}=c_{\scriptscriptstyle A}=d_{\scriptscriptstyle A}=0$, and into a 2d part
that has $a_{\scriptscriptstyle A}=0$. If we split
$\tilde{B}_{\scriptscriptstyle\circ}$ and
$\tilde{A}_{\scriptscriptstyle\circ}$ analogously, then $D{\hbox{\teneufm
R}}_{3n}(P_{\diamond})$ becomes a $2\times 2$ “matrix”. A useful feature of
this matrix is that it is upper triangular, meaning that the entry “$\hbox{\rm
1d}\mapsto\hbox{\rm 2d}$” is zero. So the eigenvalues of $D{\hbox{\teneufm
R}}_{3n}(P_{\diamond})P$ are those of the operators “$\hbox{\rm
1d}\mapsto\hbox{\rm 1d}$” and “$\hbox{\rm 2d}\mapsto\hbox{\rm 2d}$”. The
former is hyperbolic, with a single expanding direction, as described in
Corollary 5.1.

Consider now the “$\hbox{\rm 2d}\mapsto\hbox{\rm 2d}$” part. The first
observation is that $\tilde{a}_{\scriptscriptstyle B}$ and
$\tilde{d}_{\scriptscriptstyle A}$ are zero when $\sigma_{3n}=0$. So these
functions are determined by $c_{\scriptscriptstyle A}(x_{0})$ only. As a
result, $D{\hbox{\teneufm R}}_{3n}(P_{\diamond})$ has an an eigenvalue $0$
with eigenvectors whose only nonzero components are $a_{\scriptscriptstyle B}$
and $d_{\scriptscriptstyle A}$. Furthermore, the terms $a_{\scriptscriptstyle
B}$ and $d_{\scriptscriptstyle A}$ do not contribute to the off-diagonal
entries of $\tilde{B}_{\scriptscriptstyle\circ}$ or
$\tilde{A}_{\scriptscriptstyle\circ}$. So it suffices to consider the operator
“off-diagonal $\mapsto$ off-diagonal”. Here, only the rightmost or leftmost
factors contribute, and this is always a factor $A_{\scriptscriptstyle\circ}$
or $A_{\scriptscriptstyle\circ}^{\dagger}$. So $\tilde{c}_{\scriptscriptstyle
B}$ and $\tilde{b}_{\scriptscriptstyle B}$ are determined by
$c_{\scriptscriptstyle A}$ and $b_{\scriptscriptstyle A}$. And
$D{\hbox{\teneufm R}}_{3n}(P_{\diamond})$ has an an eigenvalue $0$ with
eigenvectors whose only nonzero components are $c_{\scriptscriptstyle B}$ and
$b_{\scriptscriptstyle B}$.

Consider now the case $\sigma_{3n}=0$, where the conjugacy by
$e^{\sigma_{3n}S}$ is trivial. Then $\tilde{c}_{\scriptscriptstyle A}$
satisfies

${\tilde{c}_{\scriptscriptstyle A}\over
a_{\diamond}}(x_{0}+t)={c_{\scriptscriptstyle A}\over
a_{\diamond}}\bigl{(}x_{0}-\alpha^{3n}t\bigr{)}\,,$ $None$

near $t=0$, where $x_{0}=-\hbox{\raise
2.2pt\hbox{$\scriptstyle\alpha$}\hskip-1.2pt{$\scriptstyle/$}\hskip-0.9pt\lower
2.2pt\hbox{$\scriptstyle 2$}\hskip 1.0pt}$. This corresponds of course to the
relation (2.13). So $D{\hbox{\teneufm R}}_{3n}(P_{\diamond})$ has eigenvalues
$(-\alpha)^{3nm}$ with eigenvectors whose only nonzero components are

$c_{\scriptscriptstyle A}(x)=a_{\diamond}(x)(x-x_{0})^{m}\,,\qquad
c_{\scriptscriptstyle B}(x)=-b_{\diamond}(x)(x-x_{0})^{m}\,,$ $None$

as well as $b_{\scriptscriptstyle A}(x)=-c_{\scriptscriptstyle A}(-x)$ and
$b_{\scriptscriptstyle B}(x)=-c_{\scriptscriptstyle B}(-x)$. These
eigenfunctions are contracted for $m>0$. But for $m=0$ we have an eigenvalue
$1$. This is due to the fact that we considered $\sigma_{3n}=0$. With
$\sigma_{3n}=\sigma_{3n}(P)$ as defined by the condition (5.2), the eigenvalue
becomes $0$.

This shows that the only unstable direction of $D{\hbox{\teneufm
R}}_{3n}(P_{\diamond})$ is the one inherited from $D{\hbox{\teneufm
R}}_{3n}(p_{\diamond})$, which is associated with a nonzero value of
$a_{\scriptscriptstyle A}(\rho)$. So the stable subspace $W^{s}$ of
$D{\hbox{\teneufm R}}_{3n}(P_{\diamond})$ consists of all pairs $P$ with the
property that $a_{\scriptscriptstyle A}(\rho)=0$. QED

Our proof of Theorem 1.4 below involves the pre-limit versions

$\mathop{\rm rot}\nolimits_{\scriptscriptstyle N}(G)={\mathop{\rm
Rot}\nolimits_{N}(G)\over N}\,,\qquad\varrho_{\scriptscriptstyle
N}(G)={\Sigma_{N}(G)\over N}\,,$ $None$

of the rotation numbers defined in (1.7) and (3.6), respectively. Let
$P=(F,G)$ with $F=(1,{\bf 1})$. Then the rotation numbers (5.8) for
$N=q_{3nk}$ can be computed via iterates $P_{k}={\hbox{\teneufm
R}}_{3n}^{k}(P)$, using that for even $k$,

$P_{k}=(F_{k},G_{k})\,,\qquad
G_{k}=\Lambda_{3nk}^{-1}\bigl{(}F^{\dagger}\bigr{)}^{p_{3nk}}G^{q_{3nk}}\Lambda_{3nk}\,.$
$None$

Here, $\Lambda_{3nk}(x,y)=\bigl{(}\alpha^{3nk}x,e^{\sigma_{3nk}S}y\bigr{)}$,
and $\sigma_{3nk}$ is the sum $\sigma_{3nk}(P)=\sum_{j<k}\sigma_{3n}(P_{j})$.

Assuming that $G=(\alpha,A)$ commutes with $F$, meaning that $A$ is periodic
with period $1$, the factors $F^{\dagger}$ in (5.9) contribute nothing to the
matrix part of $G_{k}$. We will be using (5.9) in the case where $P_{k}\to
P_{\diamond}$, with the convergence being asymptotically geometric. Then the
sequence of normalization constant $k\mapsto\sigma_{3nk}(P)$ converges, since
$\sigma_{3n}(P_{\diamond})=0$.

Proof of Theorem 1.4.  Consider first the claim that (1.8) holds with
$k\mapsto e^{-p_{3nk}L}$ and $k\mapsto e^{-q_{3nk}L}$ replaced by suitable
sequences $k\mapsto M_{k}$ and $k\mapsto M_{k}^{\prime}$, respectively.

Let $R_{d}$ be a small rectangle in ${\cal H}_{r}^{\prime}$, centered at
$P_{\diamond}$. To be more precise, let us call the stable (codimension $1$)
subspace $W^{s}$ of $D{\hbox{\teneufm R}}_{3n}(P_{\diamond})$ vertical and the
unstable (dimension $1$) subspace $W^{u}$ horizontal. Then $R_{d}$ is a
rectangle centered at $P_{\diamond}$, of width and height $2d$, whose
left/right sides are vertical and top/bottom sides are horizontal. Denote by
${\cal W}^{s}$ (${\cal W}^{u}$) the local (un)stable manifold of
${\hbox{\teneufm R}}_{3n}$ at $P_{\diamond}$. It is tangent to
$P_{\diamond}+W^{s}$ ($P_{\diamond}+W^{u}$) at $P_{\diamond}$. If $d>0$ is
chosen sufficiently small, then ${\cal W}^{s}$ (${\cal W}^{u}$) leaves the
rectangle through the top/bottom (left/right) sides within ${\cal
O}\bigl{(}d^{2}\bigr{)}$ from their centers.

Consider first the AM factor, multiplied by $\delta=\lambda^{-1}$, with energy
$E=\lambda\epsilon$. So we have a pair $P=(F,G)$ with $F=(1,{\bf 1})$ and
$G=(\alpha,A)$, and the symmetric factor $A_{\scriptscriptstyle\circ}$ is

$A_{\scriptscriptstyle\circ}(x)=\left[\matrix{a_{\scriptscriptstyle\circ}(x)&-\delta\cr\delta&0\cr}\right]\,,\qquad
a_{\scriptscriptstyle\circ}(x)=-\epsilon-2\cos(2\pi x)\,.$ $None$

To indicate the dependence on $\delta$ and $\epsilon$, we will use the
notation $P^{\delta}$ or $P^{\delta}(\epsilon)$.

Consider first the case $\delta=0$. Let $\epsilon(0)=-2\cos(2\pi\rho)$. If we
fix $\epsilon=\epsilon(0)$, then the sequence $k\mapsto{\hbox{\teneufm
R}}_{3n}^{k}\bigl{(}P^{0}\bigr{)}$ converges to $P_{\diamond}$ by Theorem 4.3.
If we consider the family $\epsilon\mapsto P^{0}(\epsilon)$, with $\epsilon$
close to $\epsilon(0)$, then for sufficiently large $k$, the family
$\epsilon\mapsto{\hbox{\teneufm R}}_{3n}^{k}(P^{0}(\epsilon))$ enters and
leaves $R_{d}$ through the vertical sides and it is transversal to the stable
manifold ${\cal W}^{s}$. This follows from the way the zeros of
$a_{\scriptscriptstyle\circ}$ transform under renormalization; and we assume
that $d>0$ has been chosen sufficiently small.

The same holds for the family $\epsilon\mapsto P^{\delta}(\epsilon)$, for
$\delta\neq 0$ sufficiently close to $0$. So if $d>0$ has been chosen
sufficiently small, then the renormalized family
$\epsilon\mapsto{\hbox{\teneufm R}}_{3n}^{k}(P^{\delta}(\epsilon))$ crosses
${\cal W}^{s}$ transversally at some value $\epsilon=\epsilon(\delta)$.
Clearly, the curve $\delta\mapsto\epsilon(\delta)$ is real analytic near the
origin, and it does not depend on $k$.

What remains to be shown is that $G^{\delta}(\epsilon(\delta))$ has rotation
number $\rho$. To this end, let $P^{\delta}=P^{\delta}(\epsilon(\delta))$ and
$P_{k}^{\delta}={\hbox{\teneufm R}}_{3n}^{k}\bigl{(}P^{\delta}\bigr{)}$. To
simplify notation, let us restrict to $\delta>0$.

We note that, if $k$ is sufficiently large, then the symmetric factor
$A^{\delta}_{k,{\scriptscriptstyle\circ}}(0)$ of $G^{\delta}_{k}$ is close to
$A_{\diamond}(0)=a_{\diamond}(0)\bigl{[}{1~{}0\atop 0~{}0}\bigr{]}$. Thus,
given that $a_{\diamond}(0)\neq 0$, the first component of
$A^{\delta}_{k,{\scriptscriptstyle\circ}}(0)\bigl{[}{1\atop 0}\bigr{]}$ stays
bounded away from $0$ and thus does not change sign, as $\delta$ is varied.
This holds uniformly in $k$ and $\delta$, if $k\geq k_{0}$ and
$\delta\leq\delta_{0}$ for some $k_{0}>0$ and some $\delta_{0}>0$.

We need a slight variation of this property. Recall from (5.9) that
$A^{\delta}_{k,{\scriptscriptstyle\circ}}(0)$ is the product
$\bigl{(}A^{\delta}\bigr{)}^{\ast q_{3nk}}_{\scriptscriptstyle\circ}(0)$,
conjugated by a matrix $e^{\sigma_{3nk}S}$ with
$\sigma_{3nk}=\sigma_{3nk}\bigl{(}P^{\delta}\bigr{)}$. Here, and in what
follows, we assume that $k$ is even. Notice that
$\sigma_{3nk}\bigl{(}P^{\delta}\bigr{)}\to 0$ as $\delta\to 0$, uniformly in
$k$. Thus, by decreasing the value $\delta_{0}>0$, if necessary, the absence
of sign changes described above still holds if
$A^{\delta}_{k,{\scriptscriptstyle\circ}}(0)$ is replaced by
$\bigl{(}A^{\delta}\bigr{)}^{\ast q_{3nk}}_{\scriptscriptstyle\circ}(0)$.

Now choose a fixed but arbitrary positive $\delta\leq\delta_{0}$. Then the
above implies that

$\mathop{\rm rot}\nolimits_{N_{k}}\bigl{(}G^{\delta}\bigr{)}=\mathop{\rm
rot}\nolimits_{N_{k}}\bigl{(}G^{0}\bigr{)}\,,\qquad N_{k}=q_{3nk}\,,\quad
k\geq k_{0}\,.$ $None$

Here, and in what follows, we use as starting point $x={1-N\over 2}\alpha$ in
the definition of $\mathop{\rm Rot}\nolimits_{\scriptscriptstyle N}(G)$. This
corresponds to the argument of the rightmost factor in our symmetric products
$A^{\ast{\scriptscriptstyle N}}_{\scriptscriptstyle\circ}(0)$. Due to the
uniform convergence in (3.6), we still have $\mathop{\rm
Rot}\nolimits_{\scriptscriptstyle N}(G)\to\mathop{\rm rot}\nolimits(G)$ as
$N\to\infty$.

Let now $\varepsilon>0$. If $k\geq k_{0}$ is sufficiently large, then
$\bigl{|}\mathop{\rm
rot}\nolimits_{N_{k}}\bigl{(}G^{\delta}\bigr{)}-\mathop{\rm
rot}\nolimits\bigl{(}G^{\delta}\bigr{)}\bigr{|}<\varepsilon/2$ and
$\bigl{|}\mathop{\rm
rot}\nolimits_{N_{k}}\bigl{(}G^{0}\bigr{)}-\rho\bigr{|}<\varepsilon/2$.
Combining this with (5.11), we have

$\eqalign{\bigr{|}\mathop{\rm
rot}\nolimits\bigl{(}G^{\delta}\bigr{)}-\rho\bigl{|}&\leq\bigl{|}\mathop{\rm
rot}\nolimits\bigl{(}G^{\delta}\bigr{)}-\mathop{\rm
rot}\nolimits_{N_{k}}\bigl{(}G^{\delta}\bigr{)}\bigr{|}\cr&\quad+\bigl{|}\mathop{\rm
rot}\nolimits_{N_{k}}\bigl{(}G^{\delta}\bigr{)}-\mathop{\rm
rot}\nolimits_{N_{k}}\bigl{(}G^{0}\bigr{)}\bigr{|}+\bigl{|}\mathop{\rm
rot}\nolimits_{N_{k}}\bigl{(}G^{0}\bigr{)}-\rho\bigr{|}<\varepsilon\,.\cr}$
$None$

Since $\varepsilon>0$ was arbitrary, we conclude that $\mathop{\rm
rot}\nolimits\bigl{(}G^{\delta}\bigr{)}=\rho$.

Given a positive value $\delta_{1}\leq\delta_{0}$, the above can be extended
to more general real analytic families of maps. We have to assume that, after
a suitable rescaling and possible reparametrization, the resulting family of
factors $A^{\delta,\epsilon}_{\circ}$ is sufficiently close in ${\cal
F}_{r}^{4}$ to the rescaled AM family (5.10), uniformly in $\delta$ (close to
$\delta_{1}$) and $\epsilon$ (close to $\epsilon(\delta_{1})$). Then we can
repeat the above argument for this family (using $\Sigma_{\scriptscriptstyle
N}$ in place of $\mathop{\rm Rot}\nolimits_{\scriptscriptstyle N}$) and get
the same conclusion. The main point here is that transversality to ${\cal
W}^{s}$ is stable under small perturbations. QED

References num xyz goto num count 0 References

[1] P.G. Harper, Single band motion of conduction electrons in a uniform
magnetic field, Proc. Phys. Soc. Lond. A 68, 874–892 (1955).

[2] V.T. Sós, On the distribution mod $1$ of the sequence $n\alpha$, Ann.
Univ. Sci. Budapest, Eötvös Sect. Math. 1, 127–134 (1958).

[3] J. Surányi, Über die Anordnung der Vielfachen einer reelen Zahl mod $1$,
Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 1, 107–111 (1958).

[4] S. Świerczkowski, On successive settings of an arc on the circumference of
a circle, Fundamenta Mathematicae 46, 187–189 (1959).

[5] T. Kato, Perturbation theory for linear pperators, Springer Verlag, 1976.

[6] D.R. Hofstadter, Energy levels and wave functions of Bloch electrons in
rational and irrational magnetic fields, Phys. Rev. B 14, 2239–2249 (1976).

[7] D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Quantized Hall
conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49,
405–408 (1982).

[8] R. Johnson, J. Moser, The rotation number for almost periodic potentials,
Commun. Math. Phys. 84, 403–438 (1982).

[9] M.R. Herman, Une méthode pour minorer les exposants de Lyapounov et
quelques exemples montrant le caractère local d’un théorème d’Arnold et de
Moser sur le tore de dimension 2, Comment. Math. Helvetici 58, 453–502 (1983).

[10] F. Delyon, B. Souillard, The rotation number for finite difference
operators and its properties, Commun. Math. Phys. 89, 415–426 (1983).

[11] J. Avron, B. Simon, Almost periodic Schrödinger operators. II. The
integrated density of states, Duke Math. J. 50, 369–391 (1983).

[12] R.A. Johnson, A review of recent work on almost periodic differential and
difference operator, Acta Appl. Math. 1, 241–261 (1983)

[13] P.B. Wiegmann, A.V. Zabrodin, Quantum group and magnetic translations.
Bethe ansatz solution for the Harper’s equation, Modern Phys. Lett. B 8,
311–318 (1994).

[14] J. Bellissard, Gap labelling theorems for Schrödinger operators, in From
Number Theory to Physics, 538–630, Springer, Berlin, 1992.

[15] Y. Last, Almost everything about the almost Mathieu operator. I, In: XIth
International Congress of Mathematical Physics (Paris, 1994), pp. 366–372,
Cambridge MA: Internat. Press, 1995.

[16] L.D. Faddeev, R.M. Kashaev, Generalized Bethe ansatz equations for
Hofstadter problem, Commun. Math. Phys. 169, 181–191 (1995).

[17] Y. Hatsugai, M. Kohmoto, Y.-S. Wu, Quantum group, Bethe ansatz equations,
and Bloch wave functions in magnetic fields, Phys. Rev. B 53, 9697–9712
(1996).

[18] S. Jitomirskaya, Metal-insulator transition for the almost Mathieu
operator, Ann. of Math. 150, 1159–1175 (1999).

[19] J. Bourgain S. Jitomirskaya, Continuity of the Lyapunov exponent for
quasiperiodic operators with analytic potential, J. Stat. Phys. 108, 1203–1218
(2002).

[20] J. Puig, Cantor spectrum for the almost Mathieu operator, Commun. Math.
Phys. 244, 297–309 (2004).

[21] A. Avila, R. Krikorian, Reducibility or nonuniform hyperbolicity for
quasiperiodic Schrödinger cocycles, Ann. Math. 164, 911–940 (2006).

[22] D. Damanik, The spectrum of the almost Mathieu operator, Lecture series
in the CRC 701 (2008).

[23] M. Goldstein and W. Schlag, Fine properties of the integrated density of
states and a quantitative separation property of the Dirichlet eigenvalues,
Geom. Funct. Anal. 18, 755–869 (2008).

[24] S. Jitomirskaya, C.A. Marx, Analytic quasi-periodic cocycles with
singularities and the Lyapunov exponent of extended Harper’s model, Comm.
Math. Phys. 316, 237–267 (2012).

[25] C. Aistleitner, N. Technau, A. Zafeiropoulos, On the order of magnitude
of Sudler products, Preprint 2020, arXiv:2002.06602.

[26] H. Koch, Golden mean renormalization for the almost Mathieu operator and
related skew products, J. Math. Phys. 62, 1–12 (2021).

[27] H. Koch, On trigonometric skew-products over irrational circle-rotations,
Discrete Contin. Dynam. Systems A. 41, 5455–5471 (2021).

[28] H. Koch, Asymptotic scaling and universality for skew products with
factors in ${\ninerm SL}(2,{\scriptstyle{{\tends R}}})$, To appear in Erg.
Theor. Dyn. Syst.