Limit theorems for suspension flows over Vershik automorphisms

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

“…Consider our surface M of genus g ≥ 2. By the results of [6,Section 4] there is a correspondence between almost every translation flow with respect to the Masur-Veech measure and a natural flow on a "random" 2-sided Markov compactum, which is, in turn, measurably isomorphic to the special flow over a one-sided Markov compactum, or, equivalently, an S-adic system X a + for a + ∈ A. The roof function of the special flow is piecewise constant, depending only on the first symbol, and we can express it as a vector of "heights".…”

“…Veech [26] constructed, for any connected component of a stratum H, a measurable map from the space V(R) of zippered rectangles corresponding to the Rauzy class R, to H, which intertwines the Teichmüller flow on H and a renormalization flow P t that Veech defined on V(R). Our convention here follows that of [6]. [28,29,30] ; for their spectral theory see [25].…”

“…Weakly Lipschitz functions. Following Bufetov [7,6], we consider the space of weakly Lipschitz functions on the space X s a + , except here we do everything in the S-adic framework. This is the class of functions that we obtain from Lipschitz functions on the translation surface M under the symbolic representation of the translation flow, for almost every Abelian differential.…”

“…Next we explain how to achieve the combinatorial properties (b) and (c) of a word q ∈ A k . To this end, we need to recall the construction of the Markov compactum and Bratteli-Vershik realization of the translation flow from [6]. The symbolic representation of the translation flow on the 2-sided Markov compactum is obtained as the natural extension of the 1-sided symbolic representation for the IET which we now describe.…”

Singular substitutions of constant length

“…Denote by E + be the linear span of generalized eigenvectors for S t corresponding to eigenvalues greater than one in absolute value. Let {Φ + v,x : v ∈ E + , x ∈ X ζ } be the Hölder cocycle (family of finitely-additive measures), defined on the algebra generated by line segments in R (see [7,8,9]). We refer to these papers for the definition of this cocycle, but collect their properties that we need in the lemma below.…”