Rauzy induction of polygon partitions and toral $ \mathbb{Z}^2 $-rotations

<p style='text-indent:20px;'>We extend the notion of Rauzy induction of interval exchange transformations to the case of toral <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation, i.e., <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action defined by rotations on a 2-torus. If <inline-formula><tex-math id="M3">\begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document}</tex-math></inline-formula> denotes the symbolic dynamical system corresponding to a partition <inline-formula><tex-math id="M4">\begin{document}$ \mathscr{P} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M7">\begin{document}$ R $\end{document}</tex-math></inline-formula> is Cartesian on a sub-domain <inline-formula><tex-math id="M8">\begin{document}$ W $\end{document}</tex-math></inline-formula>, we express the 2-dimensional configurations in <inline-formula><tex-math id="M9">\begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document}</tex-math></inline-formula> as the image under a <inline-formula><tex-math id="M10">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-dimensional morphism (up to a shift) of a configuration in <inline-formula><tex-math id="M11">\begin{document}$ \mathscr{X}_{\widehat{\mathscr{P}}|_W, \widehat{R}|_W} $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M12">\begin{document}$ \widehat{\mathscr{P}}|_W $\end{document}</tex-math></inline-formula> is the induced partition and <inline-formula><tex-math id="M13">\begin{document}$ \widehat{R}|_W $\end{document}</tex-math></inline-formula> is the induced <inline-formula><tex-math id="M14">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action on <inline-formula><tex-math id="M15">\begin{document}$ W $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We focus on one example, <inline-formula><tex-math id="M16">\begin{document}$ \mathscr{X}_{\mathscr{P}_0, R_0} $\end{document}</tex-math></inline-formula>, for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift <inline-formula><tex-math id="M17">\begin{document}$ X_0 $\end{document}</tex-math></inline-formula> of the Jeandel–Rao Wang shift computed in an earlier work by the author. As a consequence, <inline-formula><tex-math id="M18">\begin{document}$ {\mathscr{P}}_0 $\end{document}</tex-math></inline-formula> is a Markov partition for the associated toral <inline-formula><tex-math id="M19">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation <inline-formula><tex-math id="M20">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula>. It also implies that the subshift <inline-formula><tex-math id="M21">\begin{document}$ X_0 $\end{document}</tex-math></inline-formula> is uniquely ergodic and is isomorphic to the toral <inline-formula><tex-math id="M22">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation <inline-formula><tex-math id="M23">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> which can be seen as a generalization for 2-dimensional subshifts of the relation between Sturmian sequences and irrational rotations on a circle. Batteries included: the algorithms and code to reproduce the proofs are provided.</p>

Symbolic dynamics for non-uniformly hyperbolic systems

This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature

It is well known that C2-transformation φ of the unit interval into itself with a Markov partition (2.1) π = {Ik : k ∈ K} admits φ-invariant density g (g ≥ 0, ∥g∥ = 1) if: (2.2) ∣(φn)′∣ ≥ C1 > 1 for some n (expanding condition); (2.3) ∣φ″(x)/(φ′(y))2∣ ≤ C2 < ∞ (second derivative condition); and (2.4) #π < ∞ or φ (Ik) = [0, 1], for each Ik ∈ π. If (2.4) is deleted, then the situation dramatically changes. The cause of this fact was elucidated in connection with so-called Adler’s Theorem ([1] and [2]). However after that time in the literature occur claims and opinions concerning the existence of invariant densities and their properties for Markov Maps, which satisfy (2.2), (2.3) and do not satisfy (2.4), revealing unacquaintance with that question. In this note we discuss the problems arising from the mentioned claims and opinions. Some solutions of that problems are given, in a systematic way, on the base of the already published results and by providing appropriate examples.

Non-monotone periodic orbits of a rotational horseshoe

In this paper, we present results for the forcing relation on the set of braid types of periodic orbits of a rotational horseshoe on the annulus. Precisely, we are concerned with a family of periodic orbits, called the Boyland family, and we prove that for each pair $(r,s)$ of rational numbers with $r<s$ in $(0,1)$, there exists a non-monotone orbit $B_{r,s}$ in this family which has pseudo-Anosov type and rotation interval $[r,s]$. Furthermore, the forcing relation among these orbits is given by the inclusion order on their rotation sets. It is also proved that the Markov partition associated to each Boyland orbit comes from a pruning map which projects to a bimodal circle map. This family also contains the Holmes orbits $H_{p/q}$, which are the largest for the forcing order among all the $(p,q)$-orbits of the rotational horseshoe.

On The Minimal Number of Elements of Markov Partitions for Pseudo-Anosov Homeomorpisms

In the paper an estimation of the minimal number of elements of Markov partition for generalized pseudo-Anosov homeomorphism of closed non necessary orientable surface is given. It is formulated in terms of characteristic of invariant foliation of generalized pseudo-Anosov homeomorphism.

On t-conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes

In this paper we consider horseshoes with homoclinic tangencies inside the limit set. For a class of such maps, we prove the existence of a unique equilibrium state μt, associated to the (non-continuous) potential −tlog Ju. We also prove that the Hausdorff dimension of the limit set, in any open piece of unstable manifold, is the unique number t0 such that the pressure of μt0 is zero. To deal with the discontinuity of the jacobian, we introduce a countable Markov partition adapted to the dynamics, and work with the first return map defined in a rectangle of it.