3.1 Some basic facts and identities

Let $P=(F,G)$ be a pair of skew-product maps $F=(1,B)$ and $G=(\alpha ,A)$ whose factors B and A take values in ${\textrm {GL}}(2,\mathbb {R})$ and have positive determinants. Then the renormalized pair $\tilde P=\mathfrak {R}_3(P)$ is given by

(3.1) $$ \begin{align} \tilde P=(\tilde F,\tilde G),\quad \tilde F=\Lambda_3^{-1}\hat F\Lambda_3,\quad \tilde G=\Lambda_3^{-1}\hat G\Lambda_3, \end{align} $$

where

(3.2) $$ \begin{align} \hat F=GF^\dagger G,\quad \hat G=G^\dagger FG^\dagger FG^\dagger. \end{align} $$

Here, $F^\dagger $ and $G^\dagger $ denote the quasi-inverses of F and G, respectively, as defined in (2.8). The first component of $\hat F$ is $2\alpha -1=\alpha ^3$ . So after scaling by $\alpha ^3$ , the first component of $\tilde F$ is again $1$ . Similarly, the first component of $\hat G$ is $2-3\alpha =\alpha ^4$ . So after scaling by $\alpha ^3$ , the first component of $\tilde G$ is again $\alpha $ . The symmetric factor $\hat B$ of $\hat F$ is given by

(3.3) $$ \begin{align} \textstyle \hat B_\circ(x)=A_\circ\bigg(\dfrac{\alpha-1}{ 2}+x\bigg)B_\circ(x)^\dagger A_\circ\bigg(\dfrac{1-\alpha}{ 2}+x\bigg), \end{align} $$

and for the symmetric factor $\hat A$ of $\hat G$ we obtain

(3.4) $$ \begin{align} \textstyle \hat A_\circ(x)=A_\circ((1-\alpha)+x)^\dagger B_\circ\bigg(\dfrac{1-\alpha}{ 2}+x\bigg)A_\circ(x)^\dagger B_\circ\bigg(\dfrac{\alpha-1}{ 2}+x\bigg) A_\circ((\alpha-1)+x)^\dagger. \end{align} $$

The symmetric factors associated with $\tilde P=\mathfrak {R}_3(P)$ are now obtained via scaling:

(3.5) $$ \begin{align} \tilde B_\circ(x)=L_3^{-1}\hat B_\circ(\alpha^3 x)L_3,\quad \tilde A_\circ(x)=L_3^{-1}\hat A_\circ(\alpha^3 x)L_3. \end{align} $$

Notice that such a relation is obvious for the regular factors. But it holds for the symmetric factors as well, as a short computation shows. Our choice for the y-scaling matrices $L_3$ will be described in §3.2.

Next, let us consider some consequences of (anti-)reversibility. To this end, and for reference later on, define

(3.6) $$ \begin{align} J=\begin{bmatrix} {0}&{1}\\ {-1}&{0} \end{bmatrix},\quad S=\begin{bmatrix} {1}&{0}\\ {0}&{-1} \end{bmatrix},\quad M=2^{-1/2} \begin{bmatrix} {1}&{1}\\ {1}&{-1} \end{bmatrix}. \end{align} $$

With the exception of §4, (anti-)reversibility in this paper is defined with respect to the reflection $\Sigma =iJ$ . Notice that conjugacy by $iJ$ keeps real matrices real.

Assume now that $F=(1,B)$ is reversible and $G=(\alpha ,A)$ anti-reversible, both with respect to $\Sigma =iJ$ . A short computation shows that this condition is equivalent to

(3.7) $$ \begin{align} B_\circ(x)^\top=B_\circ(-x),\quad A_\circ(x)^\top=-A_\circ(-x). \end{align} $$

Here, $C^\top $ denotes the transpose of a matrix C.

What makes skew-products over irrational rotations difficult to deal with is that products $A^{\ast q}(x)$ with large q can vary vastly in size, as a function of x. If the Lyapunov exponent $L=L(G)$ is positive, then $A^{\ast q}(x)$ grows asymptotically like $e^{qL}$ for typical values of x. Particularly large factors can be obtained via the identity

(3.8) $$ \begin{align} (-1)^m A_\circ^{\ast 2m}(iy) =U(iy)^\ast U(iy), \end{align} $$

where $U(x)=A_\circ ((m-\tfrac {1}{2})\alpha +x) \cdots A_\circ ({\alpha }/{2}+x)$ , and where $U^\ast =\overline {U}^\top $ denotes the adjoint of U. So, in particular, $(-1)^m A_\circ ^{\ast 2m}(iy)$ is a positive matrix for $y\in \mathbb {R}$ . This fact will be used in several of our proofs.

On the other hand, products with many factors can be of order $1$ in size. Consider the case where $\mathop {\textrm {det}}(A)=1$ . Using that $A_\circ (-x)=-J^{-1}A_\circ (x)^{-1}J$ , we have

(3.9) $$ \begin{align} (-1)^m A^{\ast(2m+1)}_\circ(x) =V(x)A_\circ(x)J^{-1}V(-x)^{-1}J, \end{align} $$

where $V(x)=A_\circ (m\alpha +x)\cdots A_\circ (\alpha +x)$ . If $A_\circ (0)=-J$ , then this implies that

(3.10) $$ \begin{align} (-1)^m A_\circ^{\ast(2m+1)}(0)=-J. \end{align} $$

This applies, for example, to the AM map with $\xi ={\alpha }/{2}-\tfrac {1}{4}$ and $E=0$ . And it is independent of the value of $\unicode{x3bb} $ . So for $\unicode{x3bb}>1$ , sub-products that appear in $A^q$ that are of the form (3.10) are much smaller than sub-products of the form (3.8) for $y=0$ . This is the mechanism that produces the zeros described in Theorem 2.5.

In the remainder of this paper, we consider the AM maps with respect to the basis defined by the column vectors of the matrix M given in (3.6). In this representation, the symmetric factor of the anti-reversible AM map is given by

(3.11) $$ \begin{align} A_\circ= \begin{bmatrix} {t_\circ}&{t_\circ+1}\\ {t_\circ-1}&{t_\circ} \end{bmatrix},\quad t_\circ(x)=\unicode{x3bb}\sin(2\pi x). \end{align} $$

3.2 Scaling and normalization

As mentioned earlier, we choose $L_3$ to be a reflection matrix. In the coordinates considered here,

(3.12) $$ \begin{align} L_3=L(\vartheta)\mathrel{\mathop=^{\textrm{def}}} \begin{bmatrix} {\cos(\vartheta+\pi/4)}&{-\sin(\vartheta+\pi/4)}\\ {-\sin(\vartheta+\pi/4)}&{-\cos(\vartheta+\pi/4)} \end{bmatrix}\!. \end{align} $$

Notice that $L(\vartheta )=L(0)e^{-\vartheta J}$ , where $e^{-\vartheta J}$ is a rotation by $\vartheta $ . Notice also that $L_3^2=\textbf {1}$ . So $L_3$ drops out in a fixed point equation for $\mathfrak {R}_3^2$ . But if $P=(F,G)$ is a fixed point of $\mathfrak {R}_3^2$ , then a conjugacy by any rotation yields another fixed point.

The goal is to get uniqueness by choosing $\vartheta =\vartheta (P)$ in such a way that $\tilde P=\mathfrak {R}_3(P)$ satisfies a suitable normalization condition. With a y-scaling $L_3$ of the form (3.12), the symmetric factor of $\tilde F=\Lambda _3^{-1}\hat F\Lambda _3$ is given by

(3.13) $$ \begin{align} \tilde B_\circ(0) =e^{\vartheta J} \begin{bmatrix} {a}&{u}\\ {v}&{d} \end{bmatrix} e^{-\vartheta J},\quad \begin{bmatrix} {a}&{u}\\ {v}&{d} \end{bmatrix} \mathrel{\mathop=^{\textrm{def}}} -L(0)\hat B_\circ(0)L(0), \end{align} $$

where $\hat B_\circ $ is as described in (3.3). The negative sign on the right-hand side of this equation is due to the step $(B,A)\mapsto (-B,-A)$ mentioned in Remark 1. As a normalization condition, we impose that the two entries on the main diagonal of $\tilde B_\circ (0)$ agree. A straightforward computation shows that this determines $\mathfrak {c}=\cos (2\vartheta )$ and $\mathfrak {s}=\sin (2\vartheta )$ as follows:

(3.14) $$ \begin{align} \mathfrak{s}=\frac{a-d}{ q},\quad\mathfrak{c}=-\frac{u+v}{ q},\quad q=\sqrt{(u+v)^2+(a-d)^2}. \end{align} $$

As tedious as such computations may be, explicit expressions like (3.14) are needed in a computer-assisted proof that is by nature highly constructive. In order to get explicit expressions for the derivative $D\mathfrak {R}_3(P)\dot P$ , it is convenient to consider a family of pairs P that depend differentiably on a parameter. Then the quantities $a,u,v,d$ defined by (3.13) depend differentiably on the parameter as well. Using the ‘dot notation’ for derivatives with respect to the parameter, the derivatives of $\mathfrak {s}$ and $\mathfrak {c}$ are given by

(3.15) $$ \begin{align} \dot{\mathfrak{s}}=\frac{\mathfrak{c}}{ q} [\mathfrak{c}(\dot a-\dot d)+\mathfrak{s}(\dot u+\dot v)],\quad \dot{\mathfrak{c}}=-\frac{\mathfrak{s}}{ q} [\mathfrak{c}(\dot a-\dot d)+\mathfrak{s}(\dot u+\dot v)]. \end{align} $$

At this point we have defined the ‘basic’ version of our RG transformation $\mathfrak {R}_3$ for skew-product pairs $P=(F,G)$ . We are not assuming that $F=(1,B)$ and $G=(\alpha ,A)$ commute, nor that A and B have determinant $1$ . But the transformation does not behave as desired for non-commuting pairs or for factors that have determinants not equal to $1$ . Denote this basic version by $\mathcal {R}_3$ . Our extended version of $\mathfrak {R}_3$ is defined as

(3.16) $$ \begin{align} \mathfrak{R}_3=\mathfrak{N}\circ\mathcal{R}_3\circ\mathfrak{C}, \end{align} $$

where $\mathfrak {C}$ is a ‘commutator correction’ that will be defined later, and where $\mathfrak {N}$ performs a renormalization of determinants.

To normalize determinants, we simply choose

(3.17) $$ \begin{align} \mathfrak{N}((\gamma,C))=(\gamma,\mathcal{N}(C)),\quad \mathcal{N}(C)=[\mathop{\textrm{det}}(C)]^{-1/2}C. \end{align} $$

If the determinant of C is close to $1$ , then (3.17) is well defined and $\mathcal {N}(C)$ has determinant  $1$ . We note that, if $H=(\gamma ,C)$ is (anti-)reversible, then $\mathop {\textrm {det}}(C)$ is an even function, so $\mathfrak {N}(H)$ is still (anti-)reversible. For a pair $P=(F,G)$ we define $\mathfrak {N}$ componentwise.

For estimates of derivatives $D\mathfrak {R}_3(P)\dot P$ , we use that the derivative of $\mathcal {N}$ at $C=[{a~u\atop v~d}]$ is given by

(3.18) $$ \begin{align} D\mathcal{N}(C)\dot C= \mathop{\textrm{det}}(C)^{-1/2}\dot C -\tfrac{1}{2}\mathop{\textrm{det}}(C)^{-3/2}[a\dot d+d\dot a-u\dot v-v\dot u]C. \end{align} $$

3.3 Commutators

The linearization of the basic transformation $\mathcal {R}_3$ at the fixed point $P_\ast $ can have non-contracting directions that are associated with non-commuting perturbations of $P_\ast $ . Formally, one can see that $D\mathcal {R}_3(P_\ast )$ must have an eigenvalue $-1$ . And numerically, another eigenvalue is the number $\mathcal {V}=8.3524100320\ldots $ that appears in Theorem 2.1. The goal is to eliminate these two eigenvalues. This will be done in the next section.

First, we need some generalities. Consider the commutator $\Theta =FG(GF)^{-1}$ for a pair $P=(F,G)$ . A straightforward computation shows that the commutator for the renormalized pair $\tilde P=(\tilde F,\tilde G)$ is given by

(3.19) $$ \begin{align} \tilde{\Theta}=(G\Lambda_3)^{-1}\Theta^{-1}(G\Lambda_3). \end{align} $$

If we write $\Theta =(0,C)$ and $\tilde {\Theta }=(0,\tilde C)$ , then

(3.20) $$ \begin{align} \tilde C(x)=L(\vartheta)^{-1} A(\alpha^3x)^{-1}C(\alpha^3x+\alpha)^{-1} A(\alpha^3x)L(\vartheta). \end{align} $$

Consider the change of variables $x=({1+\alpha })/{ 2}+z$ and define

(3.21) $$ \begin{align} \mathcal{C}(P,z)=\textstyle C\bigg(\dfrac{1+\alpha}{ 2}+z\bigg),\quad \mathcal{A}(P,z)=A_0\bigg(\dfrac{1}{2}+\alpha^3z\bigg)L(\vartheta_P). \end{align} $$

Then equation (3.20) becomes

(3.22) $$ \begin{align} \mathcal{C}(\tilde P,z) =\mathcal{A}(P,z)^{-1}\mathcal{C}(P,\alpha^3 z)^{-1}\mathcal{A}(P,z). \end{align} $$

From this equation one can see that the eigenvalues of $\Theta \mapsto \tilde {\Theta }$ at $\Theta ={\textrm {I}}$ are determined by the behavior of $\mathcal {C}(P,z)$ near $z=0$ .

Thus, consider $\mathcal {C}(P)=\mathcal {C}(P,0)=C(({1+\alpha })/{ 2})$ . An explicit computation shows that

(3.23) $$ \begin{align} \textstyle \mathcal{C}(P) =XY^{-1},\quad X=B_0\bigg(\dfrac{\alpha}{ 2}\bigg) A_0\bigg(-\dfrac{1}{ 2}\bigg),\quad Y=A_0\bigg(\dfrac{1}{ 2}\bigg)B_0\bigg(-\dfrac{\alpha}{ 2}\bigg). \end{align} $$

Assume now that G is anti-reversible with respect to $\Sigma =iJ$ . Then $JXJ=Y^{-1}$ . So X, Y, and $XY^{-1}$ are of the form

(3.24) $$ \begin{align} X=\begin{bmatrix} {a}&{b}\\ {c}&{d} \end{bmatrix},\quad Y=-\begin{bmatrix} {a}&{c}\\ {b}&{d} \end{bmatrix},\quad XY^{-1}=\begin{bmatrix} {-1+b(b-c)}&{-a(b-c)}\\ {d(b-c)}&{-1-c(b-c)} \end{bmatrix}, \end{align} $$

with $ad-bc=1$ . If $XY^{-1}$ is the identity matrix, then $X=Y=J$ . So in our applications, the matrix elements a and d are close to zero.

3.4 Commutator corrections

The commutator correction map $\mathfrak {C}$ is the first step in our RG transformation (3.16). The goal is for $P'=\mathfrak {C}(P)$ to satisfy $\mathcal {C}(P')=\textbf {1}$ . We define $\mathfrak {C}$ as a composition of three maps. To simplify notation, each of these maps will be denoted by $P\mapsto P'$ .

Step 1. Here we replace the symmetric factor $A_\circ $ of G by

(3.25) $$ \begin{align} A_\circ'=RA_\circ R,\quad R=\begin{bmatrix} {\rho}&{r}\\ {r}&{\rho} \end{bmatrix},\quad \rho^2=1+r^2, \end{align} $$

while keeping $B_\circ '=B_\circ $ . The goal is to choose r in such a way that $\mathop {\textrm {tr}}(X')=0$ . Notice that R is reversible, in the sense that $J^{-1}RJ=R^{-1}$ . Since G is anti-reversible, this guarantees that the map $G'$ is anti-reversible as well. Write

(3.26) $$ \begin{align} A_\circ\bigg(-\frac{1}{2}\bigg) =\begin{bmatrix} {t_A+s_A}&{u_A}\\ {v_A}&{t_A-s_A} \end{bmatrix} ,\quad B_\circ\bigg(\frac{\alpha}{2}\bigg) =\begin{bmatrix} {t_B+s_B}&{u_B}\\ {v_B}&{t_B-s_B} \end{bmatrix}, \end{align} $$

and define

(3.27) $$ \begin{align} \begin{aligned} \varepsilon&=\mathop{\textrm{tr}}(X_\circ),\\ \tau&=4t_A t_B+(u_A+v_A)(u_B+v_B),\\ \sigma&=(u_A+v_A)2t_B+2t_A(u_B+v_B). \end{aligned} \end{align} $$

A tedious but trivial computation shows that $\mathop {\textrm {tr}}(X')=0$ if we choose

(3.28) $$ \begin{align} r=\frac{-\varepsilon}{ \sqrt{ \tfrac12 (\sigma^2-2\varepsilon\tau) +\tfrac12 \sqrt{(\sigma^2-2\varepsilon\tau)^2-4(\tau^2-\sigma^2)\varepsilon^2}}}. \end{align} $$

And for the derivative with respect to a parameter, we obtain

(3.29) $$ \begin{align} \dot r=-\frac{\dot{\varepsilon}+r(\rho\dot{\sigma}+r\dot{\tau})}{\varphi},\quad \varphi=\sigma\rho+r(2\tau+r\sigma\rho^{-1}). \end{align} $$

Step 2. Assume now that $\mathop {\textrm {tr}}(X)=0$ . The second correction $P\mapsto P'$ is defined via a transformation

(3.30) $$ \begin{align} A_\circ'=\mathcal{K} A_\circ\mathcal{K},\quad B_\circ'=\mathcal{K}^{-1} B_\circ\mathcal{K}^{-1},\quad \mathcal{K}= \begin{bmatrix}{\kappa^{1/2}}&{0}\\ {0}&{\kappa^{-1/2}} \end{bmatrix}, \end{align} $$

and the goal is to have

(3.31) $$ \begin{align} X'=\begin{bmatrix} {s}&{w}\\ {-w}&{-s} \end{bmatrix} \quad\textrm{if } X=\begin{bmatrix} {s}&{b}\\ {c}&{-s} \end{bmatrix}. \end{align} $$

Recall that $X\approx J$ in our applications, so that $s\approx 0$ , $b\approx 1$ , and $c\approx -1$ . Clearly $X'=\mathcal {K}^{-1}X\mathcal {K}$ is of the desired form if we choose

(3.32) $$ \begin{align} \kappa=\sqrt{-b/c}. \end{align} $$

Then $w=\sqrt {-bc}$ . The derivative with respect to a parameter is trivial, so we will not give it here.

Step 3. Assume now that $X=[{\phantom {-} s~\phantom {-} w\atop -w~-s}]$ . The third correction $P\mapsto P'$ is of the form

(3.33) $$ \begin{align} A_\circ'=RA_\circ R,\quad B_\circ'=R^{-1}B_\circ R^{-1}, \quad R=\begin{bmatrix} {\rho}&{r}\\ {r}&{\rho} \end{bmatrix}, \end{align} $$

with $\rho ^2=1+r^2$ . The goal is to determine r in such a way that $X'=R^{-1}XR$ is equal to J. An explicit computation shows that this is achieved with

(3.34) $$ \begin{align} r=\frac{-s}{\sqrt{2(w^2-s^2)+2w\sqrt{w^2-s^2}}}. \end{align} $$

For the derivative with respect to a parameter, we find that

(3.35) $$ \begin{align} \dot r=-\frac{\tfrac12 \dot s+r(\dot w\rho+\dot sr)}{\psi},\quad \psi=w\rho+r(2s+w\rho^{-1}r). \end{align} $$

3.5 The fixed point problem

Consider now the transformation $\mathfrak {R}_3$ defined by (3.16). Our first goal is to prove that $\mathfrak {R}_3$ has a fixed point $P_\ast $ that has potentially the properties described in Theorem 2.2. As is common in many computer-assisted proofs, we associate with the given transformation $\mathfrak {R}_3$ a quasi-Newton map $\mathfrak {M}$ that we hope to be a contraction near some approximate fixed point $\bar P$ . Picking an approximate inverse ${\textrm {I}}-M$ of ${\textrm {I}}-D\mathfrak {R}_3(\bar P)$ , we define

(3.36) $$ \begin{align} \mathfrak{M}(p)=\mathfrak{R}_3(\bar P+({\textrm{I}}-M)p)-\bar P+Mp. \end{align} $$

Here, the sum of map-pairs is defined componentwise, and $c_1(\gamma ,C_1)+c_2(\gamma ,C_2)$ is defined as $(\gamma ,c_1C_1+c_2C_2)$ . Notice that, if p is a fixed point of $\mathfrak {M}$ , then $P=\bar P+ ({\textrm {I}}-M)p$ is a fixed point of $\mathfrak {R}_3$ .

The following function spaces have already been used in [Reference Koch36]. Given $\rho>0$ , denote by $\mathcal {G}_\rho $ the space of all real analytic functions g on $(-\rho ,\rho )$ that have a finite norm

(3.37) $$ \begin{align} \|g\|_\rho=\sum_{n=0}^\infty|g_n|\rho^n,\quad g(x)=\sum_{n=0}^\infty g_n x^n. \end{align} $$

Notice that every function $g\in \mathcal {G}_\rho $ extends analytically to the complex disk $|x|<\rho $ . Furthermore, $\mathcal {G}_\rho $ is a Banach algebra under the pointwise product of functions.

The space of matrix functions

(3.38) $$ \begin{align} C_\circ= \begin{bmatrix} {t_\circ+s_\circ}&{u_\circ}\\ {v_\circ}&{t_\circ-s_\circ} \end{bmatrix} , \end{align} $$

with $t_\circ $ , $u_\circ $ , $v_\circ $ , and $s_\circ $ belonging to $\mathcal {G}_\rho $ , will be denoted by $\mathcal {G}_\rho ^4$ . The norm of $C_\circ \in \mathcal {G}_\rho ^4$ is defined as $\|C_\circ \|_\rho =\|t_\circ \|_\rho +\|u_\circ \|_\rho +\|v_\circ \|_\rho +\|s_\circ \|_\rho $ .

Given a pair $\rho =(\rho _{\text {F}},\rho _{\text {G}})$ of positive real numbers, we define $\mathcal {F}_\rho $ to be the vector space of all pairs $\mathcal {P}=(B_\circ ,A_\circ )$ in $\mathcal {G}_{\rho _{\text {F}}}^4\times \mathcal {G}_{\rho _{\text {G}}}^4$ , equipped with the norm $\|\mathcal {P}\|_\rho =\|B_\circ \|_{\rho _{\text {F}}}+\|A_\circ \|_{\rho _{\text {G}}}$ . The subspace of pairs $\mathcal {P}\in \mathcal {F}_\rho $ that satisfy the (anti-)reversibility conditions (3.7) will be denoted by $\mathcal {F}_\rho ^{\hskip 2.3pt r}$ .

For simplicity, and when no confusion can arise, we will identify a skew-product map $H=(\gamma ,C)$ with its symmetric factor $C_\circ $ . Referring to the representation (3.38), we note that H is reversible with respect to $iJ$ if and only if the functions $t_\circ $ and $s_\circ $ are even, while $v_\circ (-x)=u_\circ (x)$ . Or $C_\circ $ is anti-reversible precisely if $t_\circ $ and $s_\circ $ are odd, while $v_\circ (x)=-u_\circ (-x)$ .

In our applications, we always choose $\varrho _{\mathrm F}\le \varrho _{\mathrm G}$ . Under these conditions, (3.3) and (3.4) show that $\mathfrak {R}_3$ is well defined on $\mathcal {F}_\rho $ if

(3.39) $$ \begin{align} \tfrac{1}{2}<\rho_{\text{G}}\le \rho_{\text{F}}<\alpha^{-3}\rho_{\text{G}}-\tfrac{1}{2}\alpha^{-1}. \end{align} $$

To be more precise, the conditions needed in the normalization step $\mathfrak {N}$ and for the commutator correction $\mathfrak {C}$ (all of which represent ad hoc choices) also require some mild non-degeneracy properties. We note that the domain conditions for the transformation $\mathfrak {R}$ are more restrictive than the conditions (3.39) for $\mathfrak {R}_3$ . But both are satisfied with comfortable margins in the case $\rho _{\kern0.7pt\textrm{F}}=2$ and $\rho _{\kern0.7pt\textrm{G}}={11}/{8}$ considered below.

For reference later on, we note that the transformation $\mathfrak {R}_3$ is compact, due to the analyticity-improving property of $\mathfrak {R}_3$ . To be more precise, the transformation $(B_\circ ,A_\circ )\mapsto (\tilde B_\circ ,\tilde A_\circ )$ defined by equations (3.3), (3.4), and (3.5) maps bounded sets in $\mathcal {F}_\rho $ to bounded sets in $\mathcal {F}_{\rho '}$ , for some choice of $\rho _{\kern0.7pt\textrm{F}}'>\rho _{\kern0.7pt\textrm{F}}$ and $\rho _{\kern0.7pt\textrm{G}}'>\rho _{\kern0.7pt\textrm{G}}$ . And the inclusion map from $\mathcal {F}_{\rho '}$ into $\mathcal {F}_\rho $ is compact.

Lemma 3.1. Let $\rho =(2,{11}/{8})$ . Then there exist a pair $\bar P$ in $\mathcal {F}_\rho ^{\hskip2.3pt r}$ , a bounded linear operator M on $\mathcal {F}_\rho ^{\hskip 2.3pt r}$ , and positive constants $\varepsilon ,K,\delta $ satisfying $\varepsilon +K\delta <\delta $ , such that the transformation $\mathfrak {M}$ defined by (3.36) is analytic in $B_\delta $ and satisfies

(3.40) $$ \begin{align} \|\mathfrak{M}(0)\|_\rho\le\varepsilon,\quad \|D\mathfrak{M}(p)\|_\rho\le K,\quad p\in B_\delta, \end{align} $$

where $B_\delta $ denotes the open ball of radius $\delta $ in $\mathcal {F}_\rho ^{\hskip 2.3pt r}$ , centered at the origin. Every pair $p\in B_\delta $ has the following properties. The matrix components of $P=\bar P+({\textrm {I}}-M)p$ are non-constant and satisfy the bound $\|P-\bar P\|_\rho <10^{-450}$ . Furthermore, the angle $\vartheta =\vartheta (P)$ satisfies $\sin (2\vartheta )=-0.01760801\ldots .$

Our proof of this lemma is computer-assisted, as described in §8. These estimates will be used in §5.1 to give a proof of Theorem 2.2.

4.1 Scaling and normalization

We work in a basis where the AM factor $A_\circ $ for $\xi ={\alpha }/{2}$ and $E=0$ takes the form (3.11), with the sine replaced by a cosine. The corresponding AM map G is reversible with respect to $\Sigma =S$ , with S as defined in (3.6). So throughout this section, we restrict to pairs that are reversible with respect to $\Sigma =S$ . Referring to (3.38), reversibility of $H=(\gamma ,C)$ is equivalent to the functions $t_\circ ,u_\circ ,v_\circ $ being even and $s_\circ $ odd.

The matrix $L_3$ that enters the definition $\Lambda _3(x,y)=(\alpha ^3x,L_3 y)$ of the scaling used for $\mathfrak {R}^3$ and $\mathfrak {R}_3$ is taken to be of the form

(4.1) $$ \begin{align} L_3=Se^{\sigma_3 S}= \begin{bmatrix}{e^{\sigma_3}}&{0}\\ {0}&{-e^{-\sigma_3}} \end{bmatrix}, \end{align} $$

with $\sigma _3=\sigma _3(P)$ depending on the pair P being renormalized. Notice that $L_3$ commutes with S, so conjugacy by $\Lambda_3$ preserves reversibility.

Instead of $\mathfrak {R}^6$ , we first consider the transformation

(4.2) $$ \begin{align} \mathfrak{R}_6=\mathfrak{R}_3^2\circ\mathfrak{C},\quad \mathfrak{R}_3=\mathfrak{N}\circ\mathcal{R}_3, \end{align} $$

where $\mathcal {R}_3$ is the ‘basic’ RG transformation $P\mapsto \tilde P$ defined by equations (3.1) and (3.2). The transformation $\mathfrak {N}$ renormalizes determinants, as described after (3.17). The transformation $\mathfrak {C}$ is a commutator correction that will be described below.

With a y-scaling $L_3$ of the form (4.1), the symmetric factor of $\tilde F=\Lambda _3^{-1}\hat F\Lambda _3$ is given by

(4.3) $$ \begin{align} \tilde B_\circ(0)=\Lambda_3^{-1}\hat B_\circ(0)\Lambda_3 =\begin{bmatrix} {a}&{-e^{-2\sigma_3}u}\\ {-e^{2\sigma_3}v}&{d} \end{bmatrix},\quad \begin{bmatrix} {a}&{u}\\ {v}&{d} \end{bmatrix} \mathrel{\mathop=^{\textrm{def}}}\hat B_\circ(0), \end{align} $$

where $\hat B_\circ $ is as described in (3.3). We determine $\sigma _3=\sigma _3(P)$ is such a way that the off-diagonal elements of $\tilde B_\circ (0)$ are equal in modulus. In other words, $e^{-2\sigma _3}|u|=e^{2\sigma _3}|v|$ . Unless $uv=0$ , which does not occur in the cases considered, this trivially determines the scaling exponent $\sigma _3(P)$ .

4.2 Commutator correction

Unlike in the anti-reversible case, the largest eigenvalue of $\mathfrak {R}_3^2(P_\star )$ in the non-commuting direction appears to be $1$ . Our goal here is to eliminate this eigenvalue. One reason is that an eigenvalue of $1$ makes a quasi-Newton map ill-defined. Another reason is that the correction will be needed to prove that the pair $P_\star $ is in fact commuting.

The commutator for $P=(F,G)$ at $x=({1+\alpha })/{ 2}$ is again given by equation (3.23). By reversibility, we have $SXS=Y^{-1}$ . So X, Y, and $XY^{-1}$ are of the form

(4.4) $$ \begin{align} X=\begin{bmatrix} {a}&{b}\\ {c}&{d} \end{bmatrix},\quad Y=\begin{bmatrix} {d}&{b}\\ {c}&{a} \end{bmatrix},\quad XY^{-1}=\textbf{1}+(a-d)\begin{bmatrix} {a}&{-b}\\ {c}&{-d} \end{bmatrix}, \end{align} $$

with $ad-bc=1$ . In particular, $\mathop {\textrm {tr}}(XY^{-1})=1+(a-d)^2$ . So reversibility implies that the trace of the commutator does not change to first order. This motivates what follows. Consider a commutator correction $\mathfrak {C}:P\mapsto P'$ of the form

(4.5) $$ \begin{align} A_\circ'=RA_\circ R,\quad B_\circ'=R^{-1}B_0R^{-1},\quad R=\begin{bmatrix} {\rho}&{r}\\ {r}&{\rho} \end{bmatrix}, \end{align} $$

with $\rho ^2=1+r^2$ . Then $\mathfrak {C}(P')=X'{Y'}^{-1}$ , with $X'=R^{-1}XR$ and $Y'=RYR^{-1}$ . Ideally, we can find r in such away that $X'=Y'$ , or equivalently, that

(4.6) $$ \begin{align} Y=R^{-2}XR^2. \end{align} $$

As it turns out, this can be achieved not only to first order, but exactly, by choosing

(4.7) $$ \begin{align} r=\frac{q}{ 2}\bigg(\frac{2}{(1-q^2)^{1/2}+1-q^2}\bigg)^{1/2}, \quad q=-\frac{a-d}{ c-b}. \end{align} $$

This completes the definition (4.2) of the transformation $\mathfrak {R}_6$ . For estimates of the derivative $D\mathfrak {R}_6(P)\dot P$ we use that

(4.8) $$ \begin{align} \dot r=\frac{\dot q}{ 2}\cdot\frac{\rho}{ 1-q^2},\quad \dot q=\frac{(\dot a-\dot d)-q(\dot c-\dot b)}{ c-b}. \end{align} $$

4.3 Proof of Theorem 2.1

In this subsection we give a proof of Theorem 2.1 based on estimates that have been verified with the aid of a computer. We start by solving the fixed point equation for $\mathfrak {R}_6$ . To this end, we use again a quasi-Newton map of the type (3.36), namely

(4.9) $$ \begin{align} \mathfrak{M}(p)=\mathfrak{R}_6(\bar P+({\textrm{I}}-M)p)-\bar P+Mp, \end{align} $$

where $\bar P$ is an approximate fixed point of $\mathfrak {R}_6$ , and where ${\textrm {I}}-M$ is an approximation for the inverse of ${\textrm {I}}-D\mathfrak {R}_3(\bar P)$ . The relevant functions spaces are the spaces $\mathcal {G}_\rho $ and $\mathcal {F}_\rho $ defined in §3.5. But $\mathcal {F}_\rho ^{\hskip 2.3pt r}$ now denotes the subspace of $\mathcal {F}_\rho $ of pairs that are reversible with respect to S.

Lemma 4.1. Let $\rho =(3,2)$ . Then there exist a pair $\bar P$ in $\mathcal {F}_\rho ^{\hskip2.3pt r}$ , a bounded linear operator M on $\mathcal {F}_\rho ^{\hskip 2.3pt r}$ , and positive constants $\varepsilon ,K,\delta $ satisfying $\varepsilon +K\delta <\delta $ , such that the transformation $\mathfrak {M}$ defined by (4.9) is analytic in $B_\delta $ and satisfies

(4.10) $$ \begin{align} \|\mathfrak{M}(0)\|_\rho\le\varepsilon,\quad \|D\mathfrak{M}(p)\|_\rho\le K,\quad p\in B_\delta, \end{align} $$

where $B_\delta $ denotes the open ball of radius $\delta $ in $\mathcal {F}_\rho ^{\hskip 2.3pt r}$ , centered at the origin. Every pair $p\in B_\delta $ has the following properties. The matrix components of $P=\bar P+({\textrm {I}}-M)p$ are non-constant and satisfy the bound $\|P-\bar P\|_\rho <10^{-439}$ . Furthermore, the six-step scaling factor $\mathcal {V}=e^{\sigma _6(P)}$ satisfies the bound described in Theorem 2.1.

Our proof of this lemma is computer-assisted, as described in §8.

By the contraction mapping theorem, Lemma 4.1 guarantees the existence of a fixed point $p_\star \in B_\delta $ for $\mathfrak {M}$ and thus a fixed point $P_\star =\bar P+({\textrm {I}}-M)p_\star $ for $\mathfrak {R}_6$ . The symmetric factors for $F_\star $ and $G_\star $ are analytic in the disks $|x|<\rho _F$ and $|x|<\rho _G$ , respectively. A trivial computation, using the expressions (3.3) and (3.4) for the symmetric factors of $\hat F=GF^\dagger G$ and $\hat G=G^\dagger FG^\dagger FG^\dagger $ , respectively, shows that radii of the domains of analyticity increase with each iteration of $\mathfrak {R}_3$ by a factor larger than $1$ . The factor approaches $\alpha ^{-3}$ as the number of iterations increases. This shows that $B_\star $ and $A_\star $ extend to entire functions.

What remains to be proved is that the components $F_\star $ and $G_\star $ of the pair $P_\star $ commute. To this end, consider the commutator factor $\mathcal {C}(P,z)$ defined by (3.21). It admits a representation (3.22), with

(4.11) $$ \begin{align} \mathcal{A}(P,z)=A_0\big(\tfrac{1}{2}+\alpha^3z\big)Se^{\sigma_3(P)S}. \end{align} $$

Let $P_3=\mathfrak {R}_3(P)$ and $P_6=\mathfrak {R}_3(P_3)$ . Applying the identity (3.22) twice, we obtain

(4.12) $$ \begin{align} \mathcal{C}(P_6,z)=\mathcal{A}_2(P,z)^{-1} \mathcal{C}(P,\alpha^6 z)\mathcal{A}_2(P,z), \end{align} $$

where

(4.13) $$ \begin{align} \mathcal{A}_2(P,z)=\mathcal{A}(P,\alpha^3z)\mathcal{A}(P_3,z). \end{align} $$

Let $\mathcal {C}(P)=\mathcal {C}(P,0)$ .

Consider now the pair $P=\mathfrak {C}(P_\star )$ . Then $\mathcal {C}(P)$ is the identity matrix. This follows from our definition of the commutator correction $\mathfrak {C}$ . Given that $P_\star $ is a fixed point of $\mathfrak {R}_6=\mathfrak {R}_3^2\circ \mathfrak {C}$ and thus $P_6=P_\star $ , we see from (4.12) that $\mathcal {C}(P_\star )$ is the identity matrix as well. This implies, in particular, that $P=P_\star $ , so that $P_\star $ is a fixed point of $\mathfrak {R}_3^2$ .

In the case $P=P_\star $ , equation (4.12) is a linear fixed point equation for the function $z\mapsto \mathcal {C}(P_\star ,z)$ . We already know that $\mathcal {C}(P_\star ,z)$ is the identity matrix for $z=0$ . Whether or not the same holds for $z\ne 0$ depends on the eigenvalues of the matrix $\mathcal {A}_2(P_\star )=\mathcal {A}_2(P_\star ,0)$ .

Our proof of this lemma is computer-assisted, as described in §8. It also verifies that the origin $z=0$ belongs to the domain of analyticity of the function that appears in (4.12). But this could easily be checked by hand as well.

We note that $\nu $ appears to satisfy the equation $\nu +\nu ^{-1}=2\alpha ^{-1}$ . If this is the case, and if the scaling factor $\mathcal {V}$ in Theorem 2.1 satisfies $\mathcal {V}+\mathcal {V}^{-1}=2\alpha ^{-3}$ , then $\nu ^2=\mathcal {V}$ .

In some open neighborhood of the origin in $\mathbb {C}$ , we have either $\mathfrak {C}(P_\star ,z)=\textbf {1}$ for all z, or else

(4.14) $$ \begin{align} \mathcal{C}(P_\star,z)=\textbf{1}+z^n[\mathcal{C}_n+\mathcal{O}(1)] \end{align} $$

for some non-zero matrix $\mathcal {C}_n$ and some integer $n\ge 1$ . Substituting this expression for $\mathcal {C}(P_\star ,z)$ into (4.12) yields the identity

(4.15) $$ \begin{align} \mathcal{C}_n=\alpha^{6n}\mathcal{A}_2(P_\star)^{-1}\mathcal{C}_n\mathcal{A}_2(P_\star). \end{align} $$

The eigenvalues of $\mathcal {C}_n\mapsto \alpha ^{6n}\mathcal {A}_2(P_\star )^{-1}\mathcal {C}_n\mathcal { A}_2(P_\star )$ are $\alpha ^{6n}$ and $\alpha ^{6n}\nu ^{\pm 2}$ . They are all less than $1$ , so equation (4.15) cannot have a solution $\mathcal {C}_n\ne 0$ . This shows that the commutator of $F_\star $ and $G_\star $ is constant and equal to the identity in some open neighborhood of $x=({1+\alpha })/{ 2}$ . Given that $B_\star $ and $A_\star $ are entire analytic, this implies that $F_\star $ and $G_\star $ commute.

At this point, the proof of Theorem 2.1 is reduced to the task of verifying the bounds in Lemmas 4.1 and 4.2.

5.1 Proof of Theorem 2.2

Our goal here is to prove Theorem 2.2, with the exception of the inequality $\mu _2\ge \alpha ^{-3}$ , based on estimates that can be (and have been) verified with the aid of a computer. The inequality $\mu _2\ge \alpha ^{-3}$ will be proved in §6.

By the contraction mapping theorem, Lemma 3.1 guarantees the existence of a fixed point $p_\ast \in B_\delta $ for $\mathfrak {M}$ and thus a fixed point $P_\ast =\bar P+({\textrm {I}}-M)p_\ast $ for $\mathfrak {R}_3$ . For the same reasons as in the reversible case, the factors $B_\ast $ and $A_\ast $ associated with $P_\ast $ extend to entire functions.

In order to prove hyperbolicity and related properties, we consider the transformation $\mathfrak {T}$ defined by

(5.1) $$ \begin{align} \mathfrak{T}(p)=\mathcal{L}^{-1}[\mathfrak{R}_3^2(P_\ast+\mathcal{L} p)-P_\ast], \end{align} $$

where $\mathcal {L}$ is a suitable linear isomorphism of $\mathcal {F}_\rho ^{\hskip 2.3pt r}$ . Clearly $p_\ast =0$ is a fixed point of $\mathfrak {T}$ . We expect the derivative $D\mathfrak {T}(0)$ to have an eigenvalue $\alpha ^{-6}$ and no other spectrum outside the open unit disk. Thus, we consider a decomposition $\mathcal {F}_\rho ^{\hskip 2.3pt r}=\mathcal {U}\oplus \mathcal {W}$ , where $\mathcal {U}$ is a convenient one-dimensional subspace of $\mathcal {F}_\rho ^{\hskip 2.3pt r}$ . We will refer to $\mathcal {U}$ and $\mathcal {W}$ as the vertical and horizontal subspaces, respectively. Now the isomorphism $\mathcal {L}$ is chosen in such a way that the expected expanding direction of $D\mathfrak {T}(0)$ is roughly vertical. Writing an element $q\in \mathcal {F}_\rho ^{\hskip 2.3pt r}$ as $q=[{u\atop w}]$ , with $u\in \mathcal {U}$ and $w\in \mathcal {W}$ , we obtain a representation

(5.2) $$ \begin{align} D\mathfrak{T}(p)q= \begin{bmatrix} {M_{uu}(p)}&{M_{uw}(p)}\\ {M_{wu}(p)}&{M_{ww}(p)} \end{bmatrix} \begin{bmatrix}{u}\\ {w} \end{bmatrix},\quad q=\begin{bmatrix} {u}\\{w} \end{bmatrix}. \end{align} $$

By choosing $\mathcal {L}$ properly, the operators $M_{uw}(p):\mathcal {W}\to \mathcal {U}$ and $M_{wu}(p):\mathcal {U}\to \mathcal {W}$ can be made small for all p near $p_\ast =0$ . And $M_{uu}(p)$ should be close to $\alpha ^{-6}\simeq 18$ . In order to simplify notation, we identify $\mathcal {U}$ with $\mathbb {R}$ by choosing a unit vector $u_0\in \mathcal {U}$ and identifying the vector $tu_0$ with the coefficient t.

Specific estimates are obtained in terms of an enclosure

(5.3) $$ \begin{align} N_{uu}^{-}\le M_{uu}(p)\le N_{uu}^{+} \end{align} $$

and upper bounds

(5.4) $$ \begin{align} \|M_{uw}(p)\|\le N_{uw},\quad \|M_{wu}(p)\|\le N_{wu},\quad \|M_{ww}(p)\|\le N_{ww} \end{align} $$

that hold for all pairs p in a suitable cylinder $C_1$ . Here, and in what follows, $\|.\|$ denotes the norm in $\mathcal {F}_\rho $ . To be more precise, we determine two cylinders $C_0$ and $C_1$ such that

(5.5) $$ \begin{align} C_0\subset C_1,\quad C_j=[-h_j,h_j]\times\{w\in\mathcal{W}: \|w\|<r\}, \end{align} $$

with $0<r<h_0<h_1$ . Notice that both cylinders are centered at $p_\ast =0$ . The goal is to show that $\mathfrak {T}$ maps $C_0$ into $C_1$ and $C_1\setminus C_0$ into the complement of $C_0$ . To this end, it suffices to prove that

(5.6) $$ \begin{align} \begin{aligned} N_{uu}^{+}h_0+N_{uw}r&<h_1,\quad N_{wu}h_0+N_{ww}r<r,\\ N_{uu}^{-}h_0-N_{uw}r&>h_0. \end{aligned} \end{align} $$

Here, we have used that $\mathfrak {T}(p+q)-\mathfrak {T}(p)=\int _0^1 D\mathfrak {T}(p+sq)q\,ds$ whenever p and $p+q$ both belong to $C_1$ .

Lemma 5.1. There exists a linear isomorphism $\mathcal {L}$ of $\mathcal {F}_\rho ^{\hskip 2.3pt r}$ , as well as positive real numbers $r<h_0<h_1,N_{uu}^{-}<N_{uu}^{+},N_{uw},N_{wu},N_{ww}$ that satisfy (5.6), such that the derivative of the transformation $\mathfrak {T}$ defined by (5.1) satisfies the bounds (5.3) and (5.4) for every $p\in C_1$ . So $\mathfrak {T}$ maps $C_0$ into $C_1$ and $C_1\setminus C_0$ to the complement of $C_0$ , with room to spare for taking interiors and/or closures. Let

(5.7) $$ \begin{align} a_{\pm}=N_{uu}^{\pm}\pm N_{uw},\quad b=N_{wu}+N_{ww},\quad c=N_{uu}^{-}-N_{uw}-N_{wu}-N_{ww}. \end{align} $$

Then $a_{+}<19$ , $b<\tfrac {1}{4}$ , and $c>17$ .

Our proof of this lemma is computer-assisted, as described in §8.

One of the consequences of the ‘uniform hyperbolicity’ described in Lemma 5.1 is the following corollary.

Proof. Let $p\in C_0$ . Since the orbit of p can exit $C_0$ only via the set $C_1\setminus C_0$ , it suffices to consider the case where $p_n=\mathfrak {R}^n(p)$ belongs to $C_0$ for all $n\ge 0$ .

Write $p_n=[{u_n\atop w_n}]$ with $u_n\in \mathcal {U}$ and $w_n\in \mathcal {W}$ . From (5.3) and (5.4) we see that

(5.8) $$ \begin{align} |u_{n+1}|\ge N_{uu}^{-}|u_n|-N^{uw}\|w_n\|,\quad \|w_{n+1}\|\le N_{wu}|u_n|+N_{ww}\|w_n\|, \end{align} $$

for all $n\ge 0$ . Assume for contradiction that $|u_m|-\|w_m\|>0$ for some $m\ge 0$ . Then

(5.9) $$ \begin{align} |u_{m+1}|-\|w_{m+1}\| \ge N_{uu}^{-}|u_m|-N_{uw}\|w_m\|-N_{wu}|u_m|-N_{ww}\|w_m\|>c\|w_m\|, \end{align} $$

with $c>0$ as defined in (5.7). So we have $|u_n|-\|w_n\|>0$ for all $n\ge m$ . Combining this with the first inequality in (5.8), we find that

(5.10) $$ \begin{align} |u_{n+1}|>a|u_n|, \end{align} $$

for all $n\ge m$ , with $a=a_{-}$ as defined in (5.7). Given that $a>1$ , this leads to a contradiction. So we must have $|u_n|\le \|w_n\|$ for all $n\ge 0$ . By the second inequality in (5.8), this implies that

(5.11) $$ \begin{align} \|w_{n+1}\|\le N_{wu}|u_n|+N_{ww}\|w_n\|\le b\|w_n\|, \end{align} $$

for all $n\ge 0$ , with b as defined in (5.7). Given that $b<1$ , we find that $w_n\to 0$ as $n\to \infty $ . But $|u_n|\le \|w_n\|$ for all $n\ge 0$ , so $u_n\to \infty $ as well. Thus, $p_n\to 0$ as claimed.

The bounds $a_{-}>17$ and $b<\tfrac {1}{4}$ from Lemma 5.1 yield information about the spectrum of $D\mathfrak {T}(0)$ , using the theorem below, for example. We note that the operator $D\mathfrak {T}(0)$ is compact, for the reasons described before Lemma 3.1.

Here, in the real setting, a non-real number $\xi +i\eta $ is said to be an eigenvalue of A if there exist non-zero vectors x and y such that $Ax=\xi x-\eta y$ and $Ay=\xi y+\eta x$ .

As a consequence of Lemma 5.1 and Theorem 5.3 we have the following corollary.

The existence and real analyticity of the local unstable manifold near $p_\ast =0$ follows from standard theorems on invariant manifolds. Its extension is obtained by iterating $\mathfrak {T}$ and using Corollary 5.2.

Clearly Corollary 5.4 translates trivially to an analogous result for the transformation $\mathfrak {R}_3^2$ . In fact, an analogous result holds for $\mathfrak {R}_3$ as well, since $P_\ast $ is a fixed point of $\mathfrak {R}_3$ . Our reason for considering the second iterate of $\mathfrak {R}_3$ in this section is that it was easier to find a good isomorphism $\mathcal {L}$ in this case.

5.2 Proof of Theorem 2.3, Part I

Our goal here is to prove Theorem 2.3, based on estimates that can be (and have been) verified with the aid of a computer. A transversality condition that is needed, and the claim that $s_\ast =0$ , will be proved in §6.

Here we consider the unstable manifold of $\mathfrak {R}_3$ at $P_\ast $ to be a curve in the cylinder $C_0'=P_\ast +\mathcal {L} C_0$ rather than a graph. A possible parametrization of this curve is given by $\texttt{P}_\ast (t)=P_\ast +\mathcal {L}(tu_0+\texttt{P}_\ast (t))$ , with t ranging in $[-h_0,h_0]$ . Here $u_0\in \mathcal {U}$ is the unit vector mentioned earlier. The projection of $\texttt{P}_\ast $ onto $\mathcal {L}\mathcal {U}$ is a strictly increasing function; and as t increases from $-h_0$ to $h_0$ , the curve $\texttt{P}$ connects the bottom of $C_0'$ to the top.

Given a real number $c>0$ , consider the extension of $\mathfrak {R}_3$ to one-parameter families of pairs $s\mapsto \texttt{P}(s)$ , defined by the equation

(5.12) $$ \begin{align} \mathfrak{F}_c(\texttt{P})(s)=\mathfrak{R}_3(\texttt{P}(cs)). \end{align} $$

Our (computer-assisted) proof of this lemma implements the transformation $\mathfrak {F}_{\alpha ^3}$ on a space of curves $s\mapsto \texttt{P}(s)$ in $\mathcal {F}_\rho $ that are analytic in a disk $|s|<\delta $ of radius $\delta =2^{-96}$ . In this space, we determine bounds on the curve $\texttt{P}_0=\mathfrak {F}_{\alpha ^3}^{2m}(\texttt{P})$ for $m=66$ that imply the claims of Theorem 2.3 via strict inequalities. For further details we refer to §8.

Some immediate consequences of Lemma 5.5 are as follows. There exist an increasing sequence $n\mapsto s_n^{-}$ and a decreasing sequence $n\mapsto s_n^{+}$ , with $s_n^{-}<s_n^{+}$ for all n, such that the curve $\texttt{P}_n=\mathfrak {F}_1^{2n}(\texttt{P}_0)$ enters the bottom of the cylinder $C_0'$ at the parameter value $s_n^{-}$ and leaves the top of the cylinder for the first time at a parameter value $s_n^{+}$ .

Pick a parameter value $s_\infty $ that belongs to $[s_n^{-},s_n^{+}]$ for every n. Then the orbit $n\mapsto \mathfrak {R}_3^{2n}(\texttt{P}_0(s_\infty ))$ converges to $P_\ast $ by Corollary 5.2. If n is sufficiently large, then the pair $\texttt{P}_n(s_\infty )$ is close enough to $P_\ast $ for perturbative arguments to apply. In particular, if $\texttt{P}_n$ intersects the local stable manifold transversally, then we can use the graph transform [Reference Hirsch, Pugh and Shub29] to characterize convergence.

The graph transform $\mathfrak {F}$ associated with $\mathfrak {R}_3^2$ takes the form $\mathfrak {F}(\texttt{P})=\mathfrak {R}_3^2\circ \texttt{P}\circ R$ . Here, $R=R(\texttt{P})$ is a real analytic function defined near $s_\infty $ . Its dependence on $\texttt{P}$ is real analytic and can be chosen in such a way that $\mathfrak {F}$ has an attracting fixed point. By construction, this fixed point is the (canonically parametrized) local unstable manifold of $\mathfrak {R}_3$ at $P_\ast $ . If $\texttt{P}$ is any curve in the domain of $\mathfrak {F}$ , then the sequence $k\mapsto \mathfrak {F}^k(\texttt{P})$ converges to the fixed point of $\mathfrak {F}$ , and $k\mapsto R(\mathfrak {F}^k(\texttt{P}))$ converges to the function $s\mapsto \mu _2^{-1}s$ . In fact, R can be chosen affine, at the expense of possibly weakening the rate of convergence.

Assume now that $s_\infty $ must have the value $0$ , and that the curves $\texttt{P}_n$ , for n sufficiently large, are transversal to the local stable manifold of $\mathfrak {R}_3$ at $P_\ast $ . These properties will be proved in §6.

In this case, the reparametrization function R can be chosen linear. Since convergence of $k\mapsto R(\mathfrak {F}^k(\texttt{P}_n))$ to the function $s\mapsto \mu _2^{-2}s$ is exponential, we can in fact choose the fixed reparametrization $s\mapsto \mu _2^{-2}s$ at each step. This show that if $\texttt{P}$ is the AM family with parameter $\unicode{x3bb} =e^s$ , then the sequence $n\mapsto \mathfrak {F}_{\mu _2^{-2}}^n(\texttt{P}_0)$ converges to $\texttt{P}_\ast $ , modulo a one-time linear reparametrization. Here we assume that the local unstable manifold $\texttt{P}_\ast $ has been parametrized in such a way that it is a fixed point of $\mathfrak {F}_{\mu _2^{-2}}$ .

The same holds if $\texttt{P}$ is replaced by $\mathfrak {F}_{\alpha ^3}(\texttt{P})$ . So the above arguments can be repeated for the graph transform associated with $\mathfrak {R}_3$ . We note that the pairs $\texttt{P}_\ast (t)$ are limits of renormalized AM pairs, so they are commuting.

Since transversality to the local stable manifold is stable under small perturbations, we can repeat the same arguments for a sufficiently small perturbation of the AM curve. The only difference is that a one-time affine reparametrization is needed.

6.1 The critical coupling

Let $G=(\alpha ,A)$ be the anti-reversible AM map with coupling constant $\unicode{x3bb} =e^s$ . Let $\unicode{x3bb} _\ast $ be a value of the parameter $\unicode{x3bb} $ for which the pair $P=(F,G)$ with $F=(1,\textbf {1})$ gets attracted to $P_\ast $ under the iteration of $\mathfrak {R}_3^2$ . The goal here is to show that $\unicode{x3bb} _\ast =1$ .

First, we claim that $\unicode{x3bb} _\ast \le 1$ . To see why, consider $\unicode{x3bb}>1$ . Then the Lyapunov exponent $L(G)=\log \unicode{x3bb} $ is positive. Thus, by Proposition 6.4 in §6.2, the sequence of functions $n\mapsto \|A^{\ast q_n}(\alpha ^n.)\|$ cannot stay bounded on $[-1,1]$ as $n\to \infty $ . So we cannot have $\mathfrak {R}^n(P)\to P_\ast $ as $n\to \infty $ along multiples of $3$ .

Our next goal is to exclude the possibility $\unicode{x3bb} _\ast <1$ . The following result is Theorem 3.4 in [Reference Avila4]. It applies to the AM map $G=(\alpha ,A)$ , for any irrational $\alpha $ whose continued fraction denominators $q_n$ satisfy $\lim _n q_n^{-1}\log q_{n+1}=0$ .

Theorem 6.1. [Reference Avila4]

Let $0<\unicode{x3bb} <1$ and assume that E belongs to the spectrum of $H_\unicode{x3bb} ^\alpha $ . There exist constants $a,b,c>0$ such that for all $q>0$ ,

(6.1) $$ \begin{align} \|A^{\ast q}(z)\|\le bq^a,\quad |\mathrm{Im}\, z|\le c. \end{align} $$

An analogous result for Diophantine $\alpha $ was probably proved earlier. Corollary 4.5 in [Reference Avila and Jitomirskaya7] comes close, but it considers only the real domain.

In what follows, if $G=(\alpha ,A)$ is an arbitrary skew-product map, we will write $A^{\alpha \ast q}$ instead of $A^{\ast q}$ for the product (1.1), in order to emphasize the dependence on $\alpha $ .

Let G be an anti-reversible AM map for energy zero and $\alpha $ the inverse golden mean. Consider the corresponding pair $P=(F,G)$ with $F=(1,\textbf {1})$ , and its RG iterates $P_n=\mathfrak {R}_3^{2n}$ . We choose here even powers of $\mathfrak {R}_3$ , so that the reflections $L(0)$ that are part of the scaling $L_3=L(0)e^{-\vartheta J}$ cancel. And for the fixed point $P_\ast $ , the rotations cancel as well, since $L(0)e^{-\vartheta J}L(0)=e^{-\vartheta J}$ . Thus, in order to simplify notation, consider $\mathfrak {R}_3$ with ${L_3=\textbf {1}}$ . Then we can perform an initial rotation $A\mapsto e^{\vartheta J}Ae^{-\vartheta J}$ in such a way that $P_n\to P_\ast $ with $L_3=\textbf {1}$ fixed. Then $P_n=(F_n,G_n)$ with $F_n=(1,B_n)$ and $G_n=(\alpha ,A_n)$ , where

(6.2) $$ \begin{align} A_n(x)=A^{\alpha\ast q_{6n}}(\alpha^{6n}x),\quad B_n(x)=A^{\alpha\ast q_{6n-1}}(\alpha^{6n}x). \end{align} $$

As described earlier, the factors $A_\ast $ and $B_\ast $ of the fixed point $P_\ast $ are entire, due to the analyticity-improving property of $\mathfrak {R}_3$ . For the same reason, we have convergence $A_n\to A_\ast $ and $B_n\to B_\ast $ , uniformly on compact subsets of $\mathbb {C}$ .

Now let u and v be fixed but arbitrary non-negative integers, not both zero. Then

(6.3) $$ \begin{align} A_n^{\alpha\ast u}(.+v)B_n^{1\ast v} \to A_\ast^{\alpha\ast u}(.+v)B_\ast^{1\ast v}, \end{align} $$

uniformly on compact subsets of $\mathbb {C}$ .

Proof. Let M be some fixed $2\times 2$ matrix and define

(6.4) $$ \begin{align} f_n=\mathop{\textrm{tr}}(M A_n^{\alpha\ast u}(.+v)B_n^{1\ast v}),\quad f_\ast=\mathop{\textrm{tr}}(M A_\ast^{\alpha\ast u}(.+v)B_\ast^{1\ast v}). \end{align} $$

By Theorem 6.1 we have a bound

(6.5) $$ \begin{align} |f_n(z)|\le 2b(uq_{6n-1}+vq_{6n})^a \le C(\alpha u+v)^a\alpha^{-6na},\quad |\textrm{Im}\, z|\le c\alpha^{-6n}, \end{align} $$

for some fixed constant $C>0$ . Now restrict to a disk $|z|\le r$ of radius $r>0$ . If n is sufficiently large, then

(6.6) $$ \begin{align} \partial_z^k f_n(z)=\frac{k!}{ 2\pi i}\int_{\Gamma_n} \frac{f_n(\zeta)}{(\zeta-z)^{k+1}}\,d\zeta, \end{align} $$

where $\Gamma _n$ is the path along the two circles in $\mathbb {C}/(\alpha ^{-6n}\mathbb {Z})$ at $\mathop {\textrm {Im}}\zeta =\pm c\alpha ^{-6n}$ . Using the bound (6.5) and the fact that $|\Gamma _n|=2\alpha ^{-6n}$ , we have

(6.7) $$ \begin{align} \begin{aligned} |\partial_z^k f_n(z)| &\le\frac{k!}{ 2\pi}2\alpha^{-6n}(c\alpha^{-6n}-r)^{-k-1} C(\alpha u+v)^a\alpha^{-6na}\\[3pt] &\le C_k(\alpha u+v)^a\alpha^{6n(k-a)}, \end{aligned} \end{align} $$

for some constant $C_k>0$ . Thus, if $k>a$ , then the derivative $\partial _z^k f_\ast (z)=\lim _n\partial _z^k f_n(z)$ vanishes on the disk $|z|<r$ .

This shows that $f_\ast $ is a polynomial of degree $\lfloor a\rfloor $ or less. Since M was arbitrary, we conclude that $x\mapsto A_\infty ^{\alpha \ast u}(x+v)B_\infty ^{1\ast v}(x)$ is a polynomial of degree $\lfloor a\rfloor $ or less.

6.2 A Lyapunov exponent for pairs

Let $\alpha $ be an irrational number between $0$ and $1$ . To a pair $P=(F,G)$ , with $F=(1,B)$ and $G=(\alpha ,A)$ , we associate a renormalized pair as in (2.1) by setting

(6.8) $$ \begin{align} \mathfrak{R}(P)=(\Lambda^{-1}G\Lambda,\Lambda^{-1}FG^{-c}\Lambda). \end{align} $$

For simplicity, we restrict here to the trivial scaling $\Lambda =\Lambda (P)$ , given by $\Lambda (x,y)=(\alpha x,y)$ . Consider now the iterates $P_n=\mathfrak {R}^n(P)$ . The components of $P_n$ are of the form $F_n=(1,B_n)$ and $G_n=(\alpha _n,A_n)$ . After choosing a suitable norm for the factors $A_n$ , a Lyapunov-type exponent for pairs can be defined by setting

(6.9) $$ \begin{align} \ell(P)=\limsup_{n\to\infty}q_n^{-1}\log\kern-2pt\|A_n\|, \end{align} $$

where $q_n$ is the nth continued fraction denominator for $\alpha $ .

Consider first the case where $F=(1,\textbf {1})$ . Then the functions $A_n$ are scaled versions of $A^{q_n}$ . To be more precise, $A_n(x)=A^{\ast q_n}(\bar \alpha _n x)$ , where $\bar \alpha _n=\alpha _0\alpha _1\ldots \alpha _{n-1}$ . Taking the sup norm in (6.9) on a domain $|x|<\varepsilon $ , we have $\ell (P)=\ell _\varepsilon (G)$ , where

(6.10) $$ \begin{align} \ell_\epsilon(G)=\limsup_{n\to\infty} q_n^{-1}\log\sup_{|x|\le\epsilon}\|A^{\ast q_n}(\bar\alpha_n x)\|. \end{align} $$

Here, and in what follows, we assume that A is a continuous $1$ -periodic function on $\mathbb {R}$ , taking values in ${\textrm {SL}}(2,\mathbb {R})$ .

Proof. We may assume that $L(G)>0$ . Then, by Oseledets’s theorem [Reference Oseledets47], we have

(6.11) $$ \begin{align} \lim_q\frac{1}{ q}\log\frac{\|A^{\ast q}(x_0)v_0\|}{\|v_0\|}=L(G), \end{align} $$

for almost every $x_0\in \mathbb {R}$ , and for all vectors $v_0$ outside some one-dimensional subspace of $\mathbb {R}^2$ that can depend on $x_0$ . Now fix such an $x_0$ and $v_0$ .

Using the three-gap theorem [Reference Sós53, Reference Surányi55, Reference Świerczkowski56], we can find an integer $k>0$ and sequences $n\mapsto t_n$ and $n\mapsto s_n$ of positive integers, such that

(6.12) $$ \begin{align} q_n\le t_n\le q_{n+k},\quad |x_0+t_n\alpha-s_n|\le\epsilon\bar\alpha_n. \end{align} $$

Setting $x_n=x_0+t_n\alpha -s_n$ and $v_n=A^{\ast t_n}(x_0)v_0$ , we have

(6.13) $$ \begin{align} \frac{1}{ q_n}\log\frac{\|A^{\ast q_n}(x_n)v_n\|}{\|v_n\|} &=\bigg(1+\frac{t_n}{ q_n}\bigg)\frac{1}{ q_n+t_n}\log\frac{\|A^{\ast(q_n+t_n)}(x_0)v_0\|}{\|v_0\|}\nonumber\\[3pt] &\quad-\frac{t_n}{ q_n}\;\frac{1}{ t_n}\log\frac{\|A^{\ast t_n}(x_0)v_0\|}{\|v_0\|}. \end{align} $$

Notice that $1\le t_n/q_n\le C$ for some fixed constant C. Thus, the right-hand side of (6.13) converges to $L(G)$ as $n\to \infty $ . This yields a lower bound $\ell _\epsilon (G)\ge L(G)$ . The upper bound $\ell _\epsilon (G)\le L(G)$ follows from Furman’s theorem [Reference Furman21].

In what follows, we assume that $\alpha $ is the inverse golden mean. Then $c=1$ in equation (6.8), and $\alpha _n=\alpha $ for all n. In addition, we restrict to pairs in the set $\mathcal {F}_\rho \kern-2pt'$ defined below, where $\rho =(\rho _{\kern0.7pt\textrm{F}},\rho _{\kern0.7pt\textrm{G}})$ is assumed to satisfy

(6.14) $$ \begin{align} \tfrac{1}{2}\alpha^{-2}<{\rho_{_{\textrm{G}}}},\quad \tfrac{1}{2}\alpha+\alpha{\rho_{_{\textrm{G}}}}\le{\rho_{_{\textrm{F}}}}\le\alpha^{-1}{\rho_{_{\textrm{G}}}}. \end{align} $$

Condition (6.14) guarantees that $\mathfrak {R}$ defines a dynamical system on $\mathcal {F}_\rho \kern-2pt'$ . This condition is satisfied, for example, for the values $\rho =(2,{11}/{8})$ and $\rho =(3,2)$ that are used in Lemma 3.1 and Lemma 4.1, respectively.

Using that $q_n^{-1}=(1+\alpha ^2)\alpha ^n+\mathcal {O}(\alpha ^{3n})$ , we can rewrite (6.9) as

(6.15) $$ \begin{align} \ell(P)=\limsup_{n\to\infty}L_n(P), \end{align} $$

where

(6.16) $$ \begin{align} L_n(P)=(1+\alpha^2)\alpha^n\log\kern-2pt\|A_{n,\circ}\|_{\rho_{_{\textrm{G}}}},\quad A_{n,\circ}(x)=A^{\ast q_n}_\circ(\alpha^n x). \end{align} $$

Here, we have used that scaling and ‘symmetrizing’ commute, as mentioned after (3.5).

Our main reason for considering a Lyapunov exponent based on the functionals $L_n$ is that it transforms conveniently under renormalization:

(6.17) $$ \begin{align} L_n(\mathfrak{R}(P))=\alpha^{-1}L_{n-1}(P),\quad n=1,2,\ldots. \end{align} $$

So we have $\ell (\mathfrak {R}(P))=\alpha ^{-1}\ell (P)$ . Clearly, $\ell $ vanishes on any periodic orbit of $\mathfrak {R}$ .

Proof. Let $r=\rho _G$ . Consider the sup norm in (6.10) for small $\epsilon>0$ . Since the evaluation map $g\mapsto g(x)$ is continuous on $\mathcal {G}_r$ for $|x|\le r$ , this norm is bounded from above by $C\|A_{n,\circ }\|_r$ for some fixed constant $C>0$ . So we have $\liminf _{n\to \infty }L_n(P)\ge L(G)$ .

Next, consider the AM map $G_\delta $ with factor $A_\delta (x)=A(x+i\delta )$ . Using the (well-known) fact that $\log\kern-2pt \|A^{\ast q}(x+i\delta )\|$ is a convex function of $\delta $ , we have

(6.18) $$ \begin{align} L(P)\le \limsup_{n\to\infty}q_n^{-1} \sup_{|x|\le\epsilon}\log\kern-2pt\|A^{\ast q_n}_\circ(\alpha^n x\pm i\delta)\| \le\ell_\epsilon(G_\delta)=L(G_\delta), \end{align} $$

for $\epsilon>0$ sufficiently large (depending only on r) and every $\delta>0$ . The ‘ $\pm $ ’ in equation (6.18) includes a maximum over the two signs. But by anti-reversibility, the supremum over $|x|\le \epsilon $ does not depend on the sign.

Now we can use the fact [Reference Avila5] that $L(G_\delta )=\max \{0,\log \unicode{x3bb} +2\pi \delta \,\}$ . Thus, taking $\delta \to 0$ in (6.18) yields $\limsup _{n\to \infty }L_n(P)\le L(G)$ . This proves the claim in Proposition 6.6.

For more general pairs in $\mathcal {F}_\rho \kern-2pt'$ , the $\limsup $ in (6.15) may not be a limit. An easy way to cure this problem is to define a slightly different Lyapunov-type exponent as follows:

(6.19) $$ \begin{align} L(P)=\limsup_{n\to\infty}\frac{1}{ 1+\alpha^2} [L_n(P)+\alpha^2 L_{n-1}(P)]. \end{align} $$

Proof. For $n\ge 0$ define

(6.20) $$ \begin{align} b_n=b_n(\rho)=\log\kern-2pt\|B_{n,\circ}\|_{\rho_{_{\textrm{F}}}},\quad a_n=a_n(\rho)=\log\kern-2pt\|A_{n,\circ}\|_{\rho_{_{\textrm{G}}}}. \end{align} $$

Assume now that $n\ge 1$ . Then the domain conditions (6.14) guarantee that

(6.21) $$ \begin{align} b_n\le a_{n-1},\quad a_n\le b_{n-1}+a_{n-1}. \end{align} $$

Using that $1+\alpha =\alpha ^{-1}$ , this yields the bound

(6.22) $$ \begin{align} L_n(P)+\alpha^2L_{n-1}(P) &=(1+\alpha^2) [\alpha^n a_n+\alpha^{n+1}a_{n-1}]\nonumber\\[3pt] &\le(1+\alpha^2) [\alpha^n b_{n-1}+\alpha^n a_{n-1}+\alpha^{n+1}a_{n-1}]\nonumber\\[3pt] &\le(1+\alpha^2) [\alpha^n a_{n-2}+\alpha^n(1+\alpha)a_{n-1}]\nonumber\\[3pt] &=\alpha^2L_{n-2}(P)+L_{n-1}(P), \end{align} $$

for $n\ge 2$ . This shows that the sequence $n\mapsto L_n(P)+\alpha ^2L_{n-1}(P)$ is decreasing and thus has a limit. Since each $L_k$ is continuous, the limit is upper semicontinuous.

Next consider another domain parameter $\varrho $ satisfying (6.14). Let us write $L_n(\varrho )$ instead of $L_n(P)$ when the domain parameter is $\varrho $ , and $L_n(\rho )$ when the domain parameter is $\rho $ . Using that $\mathfrak {R}$ is analyticity-improving, there exist $k>0$ and a constant $c_k>0$ such that

(6.23) $$ \begin{align} \|B_{n+k}\|_{\varrho_{_{\textrm{F}}}}\le e^{c_k}\|B_n\|_{\rho_{_{\textrm{F}}}}^{q_{n-2}}\|A_n\|_{\rho_{_{\textrm{G}}}}^{q_{n-1}},\quad \|A_{n+k}\|_{\varrho_{_{\textrm{F}}}}\le e^{c_k}\|B_n\|_{\rho_{_{\textrm{F}}}}^{q_{n-1}}\|A_n\|_{\rho_{_{\textrm{G}}}}^{q_n}, \end{align} $$

for every $n\ge 0$ . Taking logarithms and using the identity

(6.24) $$ \begin{align} \begin{bmatrix} {q_{k-2}}&{q_{k-1}}\\ {q_{k-1}}&{q_k} \end{bmatrix} \begin{bmatrix}{\alpha}\\ {1} \end{bmatrix} = \begin{bmatrix} {0}&{1}\\ {1}&{1} \end{bmatrix}^k \begin{bmatrix} {\alpha}\\ {1} \end{bmatrix} =\alpha^{-k} \begin{bmatrix} {\alpha}\\ {1} \end{bmatrix}, \end{align} $$

we find that

(6.25) $$ \begin{align} \alpha^2L_{n+k-1}(\varrho)+L_{n+k}(\varrho) \le C_k\alpha^n+\alpha^2L_{n-1}(\rho)+L_n(\rho), \end{align} $$

with $C_k=(1+\alpha ^2)^2\alpha ^kc_k$ . An analogous inequality holds if $\varrho $ and $\rho $ are exchanged. This shows that $L(P)$ is independent of the choice of $\rho $ .

Using upper semicontinuity and (6.17), together with the fact that the pair $P_\ast $ commutes, we immediately obtain the following corollary.

6.3 Transversal intersection

At this point we can complete the proof of Theorem 2.3. We already know from §6.1 that $s_\ast =1$ . What remains to be proved is that some RG iterate of the AM family intersects the local stable manifold $\mathcal {W}^s$ of $\mathfrak {R}_3$ transversally, and that $\mu _2\ge \alpha ^{-3}$ .

To this end, consider the splitting $\mathcal {F}_\rho =W^u\oplus W^s$ , where $W^u$ is the unstable subspace for the operator $D\mathfrak {R}_3(P_\ast )$ and $W^s$ the stable subspace. Then $P_\ast +W^s$ is tangent to $\mathcal {W}^s$ at $P_\ast $ . And $\mathcal {W}^s$ is the graph of a real analytic function from $W^s$ to $P_\ast +W^u$ . To be more precise, this holds locally, near $P_\ast $ . So we restrict our analysis to a suitable open ball B in $\mathcal {F}_\rho $ that is centered at $P_\ast $ . Using that $\mathcal {W}^s$ is a graph, we can define the height of a pair $P\in B$ relative to $\mathcal {W}^s$ in the direction of $W^u$ .

Consider the AM family $s\mapsto \texttt{P}(s)$ associated with the factor (3.11) for $\unicode{x3bb} =e^s$ . Define $\texttt{P}_n=\mathfrak {F}_1^n(\texttt{P})$ , where $\mathfrak {F}_1$ denotes the pointwise version of $\mathfrak {R}_3$ defined by (5.12). Notice that $\mathfrak {R}_3$ may be replaced by $\mathfrak {R}^3$ , since the AM pairs commute. Recall that Lyapunov exponent of $\texttt{P}(s)$ is $\max \{0,s\}$ . So by (2.7), the Lyapunov exponent of $\texttt{P}_n(s)$ is $\alpha ^{-3n}s$ . And recall from §6.1 that $0$ is the unique value of s for which $\texttt{P}(s)$ is attracted to $P_\ast $ under the iteration of $\mathfrak {R}_3$ . If n is sufficiently large, so that the pair $\texttt{P}_n(0)$ lies on $\mathcal {W}^s$ in B, then the pairs $\texttt{P}_n(s)$ for $s>0$ cannot lie on $\mathcal {W}^s$ . Here, we have Corollary 6.8. In what follows, k denotes some (large) value of n for which this holds.