Monotonic cocycles

Abstract

We develop a “local theory” of multidimensional quasiperiodic ${\mathrm {SL}}(2,{\mathbb R})$ cocycles which are not homotopic to a constant. It describes a $C^1$-open neighborhood of cocycles of rotations and applies irrespective of arithmetic conditions on the frequency, being much more robust than the local theory of ${\mathrm {SL}}(2,{\mathbb R})$ cocycles homotopic to a constant. Our analysis is centered around the notion of monotonicity with respect to some dynamical variable. For such monotonic cocycles, we obtain a sharp rigidity result, minimality of the projective action, typical nonuniform hyperbolicity, and a surprising result of smoothness of the Lyapunov exponent (while no better than Hölder can be obtained in the case of cocycles homotopic to a constant, and only under arithmetic restrictions). Our work is based on complexification ideas, extended “à la Lyubich” to the smooth setting (through the use of asymptotically holomorphic extensions). We also develop a counterpart of this theory centered around the notion of monotonicity with respect to a parameter variable, which applies to the analysis of ${\mathrm {SL}}(2,{\mathbb R})$ cocycles over more general dynamical systems and generalizes key aspects of Kotani Theory. We conclude with a more detailed discussion of one-dimensional monotonic cocycles, for which results about rigidity and typical nonuniform hyperbolicity can be globalized using a new result about convergence of renormalization.

Appendices

Appendix A: Conformal barycenter

Let ${\mathcal M}$ be the set of probability measures on ${\mathbb D}$, and for $\mu \in {\mathcal M}$, let $\Phi (\mu )=\int _{\mathbb D}\frac{1}{1-|z|^2} d\mu (z)$. For $w \in {\mathbb D}$, let $\Phi _w(\mu )=\Phi (\mu ')$ where $\mu '$ is the pushforward of $\mu$ by some Moebius transformation of ${\mathbb D}$ taking $w$ to $0$. Notice that if $\Phi (\mu )<\infty$ then $\Phi _w(\mu )<\infty$ for every $w$. For every $1 \le K<\infty$, let ${\mathcal M}_K=\{\mu \in {\mathcal M}, \Phi (\mu ) \le K\}$, and let ${\mathcal M}_\infty =\cup {\mathcal M}_K$. Notice that ${\mathcal M}_K$ is compact in the weak-* topology for every $K<\infty$.

The next proposition can be proved using the conformal barycenter of Douady–Earle [19]. The construction is sufficiently simple for us to give the details here.

Proposition 5.1

There exists a Borelian function ${\mathcal B}:{\mathcal M}\rightarrow {\mathbb D}$, equivariant with respect to Möebius transformations of ${\mathbb D}$ and such that $\Phi (\delta _{{\mathcal B}(\mu )}) \le \Phi (\mu )$.

Proof

Following an idea of Yoccoz, let us define a pairing ${\mathbb D}\times {\mathbb D}\rightarrow {\mathbb D}$ by setting $z * w$ as the midpoint of the hyperbolic geodesic passing through $z$ and $w$ if $z \ne w$, and $z * z=z$. This pairing is continuous and equivariant, and we have

$$\begin{aligned} u_s(z,w) \equiv \Phi _s \left( \frac{\delta _z+\delta _w}{2} \right) -\Phi _s(\delta _{z*w})\ge 0, \end{aligned}$$

(5.1)

with equality if and only if $z=w$. Notice that

$$\begin{aligned} u_s(z,s)=(2 \Phi _s(\delta _{z*s})-1) (\Phi _s(\delta _{z*s})-1) \ge \Phi _s(\delta _{z*s})-1. \end{aligned}$$

(5.2)

Extend the pairing $*$ to ${\mathcal M}\times {\mathcal M}\rightarrow {\mathcal M}$ linearly. Thus

$$\begin{aligned} \mu * \nu =\int _{{\mathbb D}\times {\mathbb D}} \delta _{z * w} d\mu (z) d\nu (w). \end{aligned}$$

(5.3)

If $\mu ,\nu \in {\mathcal M}_\infty$ then

$$\begin{aligned} u_s(\mu ,\nu ) \equiv \Phi _s\left( \frac{1}{2}(\mu +\nu )\right) \!-\!\Phi _s(\mu *\nu )\!=\!\int _{{\mathbb D}\times {\mathbb D}} u_s(z,w) d\mu (z)d\nu (w) \ge 0,\nonumber \\ \end{aligned}$$

(5.4)

with equality if and only if $\mu =\nu$ is a Dirac mass. Notice that $u_s:{\mathcal M}\times {\mathcal M}\rightarrow [0,\infty ]$ is lower semicontinuous, so if $\mu _k \rightarrow \mu$ and $u_s(\mu _k,\mu _k) \rightarrow 0$ then $\mu$ is a Dirac mass. If $\mu _k \rightarrow \delta _s$ we have

$$\begin{aligned} \limsup _{k \rightarrow \infty } u_s(\mu _k,\mu _k) \ge \limsup _{k \rightarrow \infty } u_s(\mu _k,\delta _s) \ge \limsup _{k \rightarrow \infty } \int _{\mathbb D}\Phi _s(\delta _{z*s})-1 d\mu _k(z),\nonumber \\ \end{aligned}$$

(5.5)

and in particular if additionally $\lim u_s(\mu _k,\mu _k)=0$ then $\lim \Phi _s(\mu _k*\delta _s)=1$.

Given $\mu \in {\mathcal M}$, define $\mu ^{(k)}$ inductively by $\mu ^{(0)}=\mu$ and $\mu ^k=\mu ^{(k-1)}*\mu ^{(k-1)}$. If $\mu \in {\mathcal M}_\infty$ then $\mu ^{(k)} \in {\mathcal M}_\infty$ and we have $\Phi (\mu ^{(k+1)})=\Phi (\mu ^{(k)})-u_{s}(\mu ^{(k)},\mu ^{(k)})$. Thus $u_s(\mu ^{(k)},\mu ^{(k)}) \rightarrow 0$, and any limit of $\mu ^{(k)}$ (which exists by compactness) must be a Dirac mass. Moreover, if $\mu ^{(n_k)} \rightarrow \delta _s$ then $\Phi _s(\mu ^{(n_k)}*\delta _s) \rightarrow 1$, so $\Phi _s(\mu ^{(n)}*\delta _s) \rightarrow 1$ as well and $\delta _s$ must be the unique limit of $\mu ^{(n)}$. Now we can set ${\mathcal B}(\mu )=s$, which is clearly Borelian.¹⁷ $\square$

The estimates above allow us to obtain compactness result for invariant sections of cocycles. For instance, we have the following.

Proposition 5.2

Let $f_k:X \rightarrow X$ be a sequence of homeomorphisms of $X$ preserving a probability measure $\mu$ and converging uniformly to a homeomorphism $f:X \rightarrow X$. Let $A_k \in C^0(X,\Upsilon )$ be a sequence converging to $A \in C^0(X,\Upsilon )$. Assume there exists measurable $m_k:X \rightarrow {\mathbb D}$ satisfying $A_k(x) \cdot m_k(x)=m_k(f_k(x))$, such that

$$\begin{aligned} H \equiv \liminf _{K,k \rightarrow \infty } \int _X \min \left\{ K,\frac{1}{1-|m_k(x)|^2} \right\} d\mu (x)<\infty . \end{aligned}$$

(5.6)

Then there exists a measurable $m:X \rightarrow {\mathbb D}$ such that $A(x) \cdot m(x)=m(f(x))$ and $\int _X \frac{1}{1-|m(x)|^2} d\mu (x) \le H$.

Proof

Let $X_{K,k}=\{x \in X,\, \frac{1}{1-|m_k(x)|^2}<K\}$, and let

$\nu _{K,k}=\int _{X_{K,k}} \delta _{m_k(x)} d\mu (x)$. Let $\nu$ be any limit of $\nu _{K,k}$ along a sequence $K_i \rightarrow \infty$, $k_i \rightarrow \infty$ attaining the $\liminf$ in (5.6). Then $\nu$ is a probability measure which projects onto $\mu$ and satisfies $\int _{X \times {\mathbb D}} \frac{1}{1-|z|^2} d\nu (x,z) \le H$. Let $\nu _x$, $x \in {\mathbb R}/{\mathbb Z}$ be a disintegration of $\nu$: $\int _{X \times {\mathbb D}} \phi (x,z) d\nu (x,z)=\int _X (\int _{\mathbb D}\phi (x,z) d\nu _x(z)) d\mu (x)$. Then $\nu _{f(x)}$ is the pushforward of $\nu _x$ by $w \mapsto \mathring{{A}}(x) \cdot w$, and $\nu _x \in {\mathcal M}_\infty$ for $\mu$-almost every $x$. Let $m(x)={\mathcal B}(\nu _x)$. Then $m(f(x))=\mathring{{A}}(x) \cdot m(x)$ and we have $\int \frac{1}{1-|m(x)|^2} d\mu (x) \le \int \int \frac{1}{1-|z|^2} d\nu _x(z) \ d\mu (x) \le H$. $\square$

Appendix B: Transitivity of the projective action

We follow the notation of Sect. 3.3. Our goal is to show the transitivity of the projective action of multidimensional quasiperiodic cocycles which are not homotopic to the identity. The one-dimensional case was considered in [30], and in fact the topological ideas that make the one-dimensional argument work are easily implemented in the multidimensional case as well. Let $f_\alpha :x \mapsto x+\alpha$ be an ergodic translation in ${\mathbb R}^d/{\mathbb Z}^d$.

Proposition 6.1

Let $A \in C^0({\mathbb R}^d/{\mathbb Z}^d,{\mathrm {SL}}(2,{\mathbb R}))$ be non-homotopic to the identity. If $f_\alpha :x \mapsto x+\alpha$ is an ergodic translation in ${\mathbb R}^d/{\mathbb Z}^d$ then $(f_\alpha ,A)$ is transitive on ${\mathbb R}^d/{\mathbb Z}^d \times \partial {\mathbb D}$.

Proof

Up to change of coordinate, we may assume that $x_1 \mapsto A(x_1,\dots ,x_d)$ has positive degree $\deg \ge 1$. To prove transitivity, it is enough to show that for any open set $U \subset {\mathbb R}^d/{\mathbb Z}^d \times \partial {\mathbb D}$, the set $\cup _{k \ge 0} (f_\alpha ,A)^k(U)$ is dense in ${\mathbb R}^d/{\mathbb Z}^d \times \partial {\mathbb D}$.

We will actually show a stronger statement. Let $\Pi _1:{\mathbb R}^d/{\mathbb Z}^d \times \partial {\mathbb D}\rightarrow {\mathbb R}/{\mathbb Z}$, $\Pi _2:{\mathbb R}^d/{\mathbb Z}^d \times \partial {\mathbb D}\rightarrow {\mathbb R}^{d-1}/{\mathbb Z}^{d-1}$ and $\Pi _3:{\mathbb R}^d/{\mathbb Z}^d \times \partial {\mathbb D}\rightarrow \partial {\mathbb D}$ be given by $\Pi _1(x_1,\dots ,x_d,z)=x_1$, $\Pi _2(x_1,\dots ,x_d,z)=(x_2,\dots ,x_d)$ and $\Pi _3(x_1,\dots ,x_d,z)=z$.

Let $0<\epsilon <1/10$. Let us say that a point $x \in {\mathbb R}^d/{\mathbb Z}^d$ is $\epsilon$-short if there exists a sequence of paths $\gamma _n:[0,1] \rightarrow {\mathbb R}^d/{\mathbb Z}^d \times \partial {\mathbb D}$ such that $\Pi _1 \circ \gamma _n(t)$ converges uniformly to $\Pi _1(x)+\epsilon t$, $\Pi _2 \circ \gamma _n(t)$ converges uniformly to $\Pi _2(x)$, and $\Pi _3((f_\alpha ,A)^k(\gamma _n(t)))$ has (algebraic) length at most $2 \pi -1/10$ for every $k \ge 0$. It is clear that the set of $\epsilon$-short $x$ is forward invariant and closed, so for each $\epsilon$, either every point is $\epsilon$-short or no point is $\epsilon$-short.

If for every $\epsilon >0$ there is no point which is $\epsilon$-short, then for any $(x,z)=(x_1,\dots ,x_d,z)$ and for every $\delta >0$, there exists $k \ge 0$ such that, letting $J_\delta (x,z)=[x_1,x_1+\delta ] \times \{(x_2,\dots ,x_d,z)\}$, we have $|\Pi _3(f_\alpha ,A)^k(J_\delta (x,z))|>2 \pi -\delta$. It follows that for every $\delta _0$, the closure of $\cup _{k \ge 0} (f_\alpha ,A)^k(J_{\delta _0}(x,z))$ contains some circle $\{y\} \times \partial {\mathbb D}$. Since this set is also forward invariant, it must contain also the circles $\{y+l \alpha \} \times \partial {\mathbb D}$ for every $l \ge 0$, and hence, by minimality of $x \mapsto x+\alpha$, the whole ${\mathbb R}^d/{\mathbb Z}^d \times \partial {\mathbb D}$. Since $(x,z)$ and $\delta _0>0$ are arbitrary, transitivity follows.

Assume now that there exists $\epsilon >0$ such that every point is $\epsilon$-short. We may assume that $\epsilon =\frac{1}{k}$ for some $k \ge 2$. Let $\gamma _{n,i}$, $1 \le i \le k$, $n \ge 1$, be the sequences of paths associated to $(\frac{i-1}{k},0,\dots ,0)$. We define a sequence of paths $\tilde{\gamma }_n:{\mathbb R}/{\mathbb Z}\rightarrow {\mathbb R}^d/{\mathbb Z}^d \times \partial {\mathbb D}$ so that

• (1) $\tilde{\gamma }_n|[(i-1)/k,(3i-2)/3k]$ is given by $\tilde{\gamma }_n(t)=\gamma _{n,i}(3kt-3i+3)$,

• (2) $\tilde{\gamma }_n|[(3i-2)/3k,(3i-1)/3k]$ is such that $\Pi _1 \circ \tilde{\gamma }_n$ and $\Pi _2 \circ \tilde{\gamma }_n$ are constant and $\Pi _3 \circ \tilde{\gamma }_n$ is a homeomorphism,

• (3) the diameter of the image of $\tilde{\gamma }_n|[(3i-1)/3k,i/k]$ converges to $0$.

One readily checks that these properties imply that for every $l \ge 0$, if $n$ is sufficiently large, then $\Pi _1 \circ (f_\alpha ,A)^l \circ \tilde{\gamma }_n$ has topological degree $1$, $\Pi _2 \circ (f_\alpha ,A)^l \circ \tilde{\gamma }_n$ is homotopic to a constant and $\Pi _3 \circ (f_\alpha ,A)^l \circ \tilde{\gamma }_n$ has topological degree $\deg _{l,n}$ satisfying $|\deg _{l,n}| \le 2k-1$. But $\deg _{l+1,n}=\deg _{l,n}+\deg \ge \deg _{l,n}+1$ for every $l$ and $n$, since $A$ is not homotopic to the identity. Thus for large $n$ we have both $\deg _{4k,n}-\deg _{0,n} \ge 4k$ and $|\deg _{4k,n}|,|\deg _{0,n}| \le 2k-1$, a contradiction. $\square$