The Hölder property for the spectrum of translation flows in genus two

Abstract

The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus g ≥ 2. These flows are weakly mixing by the Avila-Forni theorem. In genus 2, the Hölder property for the spectral measures of these flows was established in [10,12]. Recently Forni [17], motivated by [10], obtained Hölder estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Forni's idea with the symbolic approach of [10] and prove Hölder regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera.

Absolute continuity of non-homogeneous self-similar measures

“…It is a typical "Erdős-Kahane" estimate, although we need to take extra care in our setting due to the fact that we deal not with a deterministic parametrized family, but with a random collection of parametrized families. We should note that the rest of the proof has many common features with another version of the Erdős-Kahane argument in a random setting, which appeared in [1,Section 10]. First we derive Theorem 5.1 from Proposition 5.4.…”

“…In the general case λ 1 = λ 2 , much less is known. For p = 1/2, Jörg Neunhäuserer [10] and Sze-Man Ngai and Yang Wang [11] proved that ν 1/2 λ 1 ,λ 2 is absolutely continuous for almost all λ 1 , λ 2 in a certain simply connected region which is very far from covering the whole super-critical parameter region λ 1 λ 2 > 1/4 (which corresponds to s 1/2 λ 1 ,λ 2 > 1), and in particular is disjoint from a neighborhood of (1,1). Ngai and Wang conjectured that, in fact, ν 1/2 λ 1 ,λ 2 is absolutely continuous for almost all λ 1 , λ 2 ∈ (0, 1) such that λ 1 λ 2 > 1/4.…”

“…As mentioned before, ν p λ 1 ,λ 2 is not a convolution. Indeed, if one wishes to choose a point x ∈ R at random according to ν p λ 1 ,λ 2 , then by the self-similarity relation in (1), it suffices to consider an i.i.d. sequenceω := (ω n ) n∈N ⊆ {1, 2} of Bernoulli random variables with parameter 1 − p and take x as the unique element in R satisfying (2) {x} = n∈N Bω |n whereω|n = (ω 1 , .…”

“…Lemma 4.9. If P verifies Assumptions 4.8, then there exist ρ > 0 and a full P-measure set Ω (1) ⊆ Ω, such that for any δ > 0 there is a constant C δ > 0 satisfying…”

“…Proof of Theorem 4.7. Consider the full P-measure set Ω ′ := Ω * ∩ Ω (1) ∩ Ω (2) given by Corollary 4.2 and Lemmas 4.9-4.10, take ε ∈ (0, 1) and fix an auxiliary ε ′ ∈ (0, ε 2 ). Let ρ be the number given by Lemma 4.9 and, given ω ∈ Ω ′ , choose C ε ′ as in Lemma 4.9 and m ′′ ε ′ := max{m ε ′ , m ′ ε ′ } where m ε ′ ∈ N is as in Lemma 4.10 and m ′ ε ′ as in Corollary 4.12.…”