On the size of the resonant set for the products of 2 × 2 matrices

Abstract

We would like to thank Professors Serguei Denissov and Alexander Kiselev for their constant support throughout the research and writing processes. We are grateful to them for suggesting the topic and providing us with the opportunity to apply the concepts we have learned throughout our mathematical careers. We thank them both for their excellent advice and encouragement.Abstract. For θ ∈ [0, 2π), consider the rotation matrix R θ and h = λ 0 0 0 , λ > 1.Let Wn(θ) denote the product of m R θ 's and n h's with the condition m ≤ [ǫn] (0 < ǫ < 1). We analyze the measure of the set of θ for which Wn(θ) ≥ λ δn (0 < δ < 1). This can be regarded as a model problem for the so-called Bochi-Fayad conjecture.

On Falconer's formula for the generalised Rényi dimension of a self-affine measure

“…, A N , s) is continuous, and then deduce the continuity of s via the formula (2). The argument which we employ in proving Theorem 1.3 essentially converse to this: we demonstrate the existence of discontinuities in the function (A 1 , A 2 ) → R q (A 1 , A 2 , p, s) and show that they induce discontinuities in r q via the equation (1).…”

“…see [7]; the bound on the norm was subsequently relaxed to 1 2 by B. Solomyak [24], and to 1 by T. Jordan, K. Simon and M. Pollicott for a notion of "almost self-affine set" which incorporates additional random translations [18]. While it is well-known that the Hausdorff dimension of Z T can fail to depend continuously on the affinites T 1 , .…”

“…Then the set of resistant pairs (A 1 , A 2 ) ∈ H × E has full Lebesgue measure. Some partial results in the direction of Conjecture 1.2 may be found in [1,2,13]. C. Bonatti has constructed explicit examples of resistant pairs in which A 2 is a rational rotation, and these examples are described in [6].…”

Generic properties of the lower spectral radius for some low-rank pairs of matrices

“…Proof. It is clear that the quantities appearing in the statement of the lemma are homogenous in H and R and are unaffected by a change-of-basis transformation, so by Lemma 2.1 it is sufficient to consider the case where (H, R) has the form (1).…”

“…Then the set of all resistant pairs (H, R) ∈ H × R has full Lebesgue measure. Some partial results in the direction of this conjecture may be found in [1,2,13]; we note that resistant pairs (H, R) ∈ H × R can easily be constructed explicitly in the special case where R n is equal to the identity for some integer n ≥ 1, see [8]. Let M 2 (R) denote the set of all 2 × 2 real matrices.…”