Positive Lyapunov exponent for some Schr\"odinger cocycles over strongly expanding circle endomorphisms

We show that for a large class of potential functions and big coupling constant $\lambda$ the Schr\"odinger cocycle over the expanding map $x\mapsto bx ~( \text{mod} 1)$ on $\mathbb{T}$ has a Lyapunov exponent $>(\log\lambda)/4$ for all energies, provided that the integer $b\geq \lambda^3$.


Introduction
Let T = R/Z and let T : T → T be the expanding map T (x) = bx (mod1), where b ≥ 2 is an integer. In this note we consider the Schrödinger cocycle on T × R 2 defined by and v : T → R is a continuous function, λ ∈ R is a coupling constant and E ∈ R is the energy parameter. We let 1 A n E (x) = A E (T n−1 (x)) · · · A E (x), n ≥ 1, and define the (maximal) Lyapunov exponent by Recall that the Lebesgue measure on T is an invariant measure for T . Since T is ergodic with respect to this measure we have (1.1) lim n→∞ 1 n log A n E (x) = L(E) for a.e. x ∈ T. For the important connection to the discrete Schrödinger operator we refer to the articles [4,5,6,12] and references therein.
A natural question is to ask under which conditions (on λ, v, b and E) we have L(E) > 0. Here we are especially interested in conditions on λ, v and b which guarantees L(E) > const. > 0 for all E ∈ R. Besides the problem in itself, which has a general interest in the theory of non-uniformly hyperbolic dynamical systems, such uniform lower bounds are many times important for deriving finer properties of the associated Schrödinger operator (see, e.g., [3]).
Next follows a brief summary of previous results. It should be stressed that all the results hold for any b ≥ 2.
In [6] it is shown that if v is measurable, bounded and non-constant, and λ > 0, then L(E) > 0 for a.e. E ∈ R. Moreover, for small λ and smooth non-constant v one has 4]; and for large λ, and under quite general conditions on v, one has L(E) log λ for all E outside an exponentially small (in λ) set [10,12].
Furthermore, from Herman's subharmonic argument [7] it follows that if v is a nonconstant trigonometric polynomial, then L(E) log λ for all E ∈ R and all large λ. However, whether the corresponding result holds for v a non-constant real-analytic function does not seem to be known (if instead T (x) = x + ω, ω ∈ R \ Q, this is a well-known result [11]). Steps towards a proof of this result are taken in [9]. Another situation where one has L(E) log λ for all E ∈ R and all large λ is when v is C 1 and monotone on (0, 1), with a discontinuity at x = 0 [15].
On the other hand, if ϕ : T → R is any bounded and measurable function such that T ϕ(x)dx = 0, and we let v(x) = exp(ϕ(T (x))) + exp(−ϕ(x)) and take λ = 1, then L(0) = 0 (see [2]). Thus there are obstacles for obtaining uniform (in E) lower bounds on L(E). The aim of this paper is to extend the above results to new situations. We first define the collection of potential functions v with which we shall work. Definition 1.1. Let V 1 (T, R) denote the class of C 1 -functions v : T → R which satisfy the following condition: there exist ε 0 > 0, β > 0 and an integer s ≥ 1 such that for all 0 < ε ≤ ε 0 and all a ∈ R the set {x ∈ T : |v(x) − a| < ε} consists of at most s intervals, each of length at most ε β . Remark 1. It is easy to check that all non-constant real-analytic functions v : T → R belongs to V 1 (T, R). Note also that the assumption on v is very similar to [8, Definition 2.2].
Our main result is the following: Remark 2. (a) We do not aim for optimal conditions on any of the constants or required size of b (as function of λ).
(b) It would be very interesting to know if the statement of Theorem 1 holds true for a fixed (large) b independent of λ. Unfortunately the method we use in the proof requires b to be (much) larger than λ. (Heuristically it should be more likely to have L(E) > 0 the bigger is λ.) (c) We would like to stress that the proof of Theorem 1 does not use the fact that we have a linear system. In fact one can extend it to non-linear systems (as, e.g., we did in [1]) since what we analyze is the dynamics of forced circle diffeomorphisms (see the map (3.1)). However, since there is an interest in the Schrödinger cocycle, and for (hopefully!) transparency, we perform our analysis for this explicit system.
(d) The proof is based on ideas developed in [13]. Related problems (for systems which are not homotopic to the identity) are investigated in [1,14].

Preliminaries
We adopt the following convention on the coupling constant λ: In the statements of the lemmas below (where applicable) we always assume (without explicitly stating so) that λ > 0 is sufficiently large. There are only finitely many largeness conditions on λ, and they only depend on v. This will yield the constant λ 0 in the statement of Theorem 1.

Assumption on v.
We assume from now on that v ∈ V(T, R) is fixed, and that ε 0 , β and s are as in Definition 1.1. Without loss of generality we assume, for simplicity, that |v(x)| ≤ 1/3 for all x ∈ T (this only scales λ).

Projective action. Since
Thus, if there for some parameter E ∈ R exists a set X ⊂ T of positive measure such that for each x ∈ X there is an r ∈ R such that lim sup n→∞ 1 n log r n · · · r ≥ C, then it follows from (1.1) and (2.1) that L(E) ≥ C.
Proof. Take any x ∈ T and |r| > √ λ, and define r n = π 2 (G n E (x, r)), n ≥ 1. We note that Thus, in order to prove Theorem 1, we only need to consider |E| < λ/3 + 2 √ λ (which of course is the cumbersome region).

2.4.
Elementary probability. We begin by defining some natural partitions of T (relative the transformation T ). First, let Note that I 1 ∪ I 2 ∪ · · · ∪ I b = T and T (I j ) = T for all j.
In the following lemma we use the word "bad" to indicate that an interval does not have a certain property: Proof. Easy combinatorics. From the assumption it follows that there are n m q m (b − q) n−m n-tuples (j 1 , . . . , j n ) ∈ {1, . . . , b} n such that exactly m of the intervals I j 1 , I j 1 j 2 , . . . , I j 1 ...jn are bad. Using the fact that |I j 1 ...jn | = b −n yields the result. Proof. By the previous lemma we have (provided that [2(q/b)n] ≤ n) Applying the de Moivre-Laplace theorem yields the result.
The following lemma is crucial for the proof of Theorem 1 (compare with "admissible curves" in [13]; and also the idea in [13] that the image of an admissible curve "spreads out" in the y-direction). Recall that the constants β and s come from the assumption on v.
The statement thus says that we have a bound of the number of indices j for which the graph {(x, ϕ j (x)) : x ∈ [0, 1)} intersects the "bad region" [0, 1] × (− arctan √ λ, arctan √ λ); and the derivative estimate shows that we can iterate this process (iterate each of the j graphs) and still have a good control on the derivative (provided that b is large enough, independently of m).
We turn to the derivative estimate in (2). From the assumptions on v and E we have |λv(t) − E| < λ for all t. An easy computation shows that |(ϕ j ) ′ (x)| ≤ (λ v ′ + ϕ ′ 2λ 2 )/b, from which the desired bound follows. To obtain the estimate in the second term we have used the fact that if |a| ≤ λ, and λ is sufficiently large (larger than a numerical constant), then sin 2 t + (a sin t − cos t) 2 > 1/(2λ 2 ) for all t.