In this article we provide sufficient conditions on a self-similar interval exchange map, whose renormalization matrix has complex eigenvalues of modulus greater than one, for the existence of affine interval exchange maps with wandering intervals and semi-conjugate with it. These conditions are based on the algebraic properties of the complex eigenvalues and the complex fractals built from the natural substitution emerging from self-similarity. We show that the cubic Arnoux-Yoccoz interval exchange map satisfies these conditions.
A. We classify the Rauzy-Veech groups of all connected components of all strata of the moduli space of translation surfaces in absolute homology, showing, in particular, that they are commensurable to arithmetic lattices of symplectic groups. As a corollary, we prove a conjecture of Zorich about the Zariski-density of such groups. 1 ⊺ γ . In particular, if π ′ = π (that is, if γ is a cycle), one has that B γ (acting on row vectors) belongs to Sp(Ω π , Z). The Rauzy-Veech group of π is the group generated by matrices of this form:Definition 2.1. Let R be a Rauzy class and π ∈ R be a fixed vertex. We define the Rauzy-Veech group RV(π) of π as the set of matrices of the form B γ ∈ Sp(Ω π , Z) where γ is a cycle onR with endpoints at π. We will always consider the action of RV(π) on row vectors unless explicitly stated.
A. We prove that the "plus" Rauzy-Veech groups of all connected components of the strata of meromorphic quadratic di erentials de ned on Riemann surfaces of genus at least one having at most simple poles and at least three singularities (zeros or poles), not all of even order, are nite-index subgroups of their ambient symplectic groups. This shows that the "plus" Lyapunov spectrum of such strata is simple. Moreover, we show that the index of the "minus" Rauzy-Veech group is also nite for connected components of strata satisfying the same conditions and having exactly two singularities of odd order. This shows that the "minus" Lyapunov spectrum of such strata is simple.
The Rauzy gasket R is the maximal invariant set of a certain renormalization procedure for special systems of isometries naturally appearing in the context of Novikov's problem in conductivity theory for monocrystals.It was conjectured by Novikov and Maltsev in 2003 that the Hausdorff dimension dim H (R) of Rauzy gasket is strictly comprised between 1 and 2.In 2016, Avila, Hubert and Skripchenko confirmed that dim H (R) < 2. In this note, we use some results by Cao-Pesin-Zhao in order to show that dim H (R) > 1.19.
We give effective estimates for the number of saddle connections on a translation surface that have length ≤ L and are in a prescribed homology class modulo q. Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur-Veech measure on the stratum.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.