Nonstationary Markov Partitions and Multidimensional Continued Fraction Algorithms

TL;DR

This paper generalizes Markov partitions to nonstationary sequences of toral automorphisms generated by multidimensional continued fraction algorithms, providing a symbolic and geometric framework for understanding their hyperbolic dynamics.

Contribution

It introduces a method to construct explicit nonstationary Markov partitions for sequences of toral automorphisms derived from multidimensional continued fractions, extending classical hyperbolic theory.

Findings

Constructed bi-infinite Markov partitions for nonstationary automorphisms.

Linked Markov partitions to $ ext{S}$-adic Rauzy fractals and symbolic dynamics.

Provided a renormalization interpretation of multidimensional continued fractions.

Abstract

It is well known from results of Sina\u{\i} and Bowen that a hyperbolic toral automorphism admits a Markov partition. Our aim is to generalize this concept to the nonstationary case, i.e., we associate Markov partitions to nonstationary sequences of toral automorphisms. Special emphasis is placed on sequences of toral automorphisms produced by strongly convergent multidimensional continued fraction algorithms. The convergence of the algorithms is expressed in terms of a Pisot type condition which yields hyperbolicity for the nonstationary dynamics with a splitting into two subspaces of dimension 1 and codimension 1, respectively. For a multidimensional continued fraction map, we first consider its natural extension, whose orbits are given by bi-infinite sequences of matrices with determinant $\pm 1$. The Pisot type condition allows us to interpret an orbit of this natural extension as an Anosov mapping family, i.e., as a bi-infinite sequence of toral automorphisms with well-defined stable and unstable manifolds. We prove that this Anosov mapping family admits a bi-infinite sequence of explicit nonstationary Markov partitions. To obtain the atoms of the Markov partitions, a combinatorial structure, expressed in terms of symbolic dynamical systems, namely substitutive and $\mathcal{S}$-adic shifts, has to be superimposed on the Anosov mapping family. In particular, the atoms of the Markov partitions are geometric realizations of $\mathcal{S}$-adic shifts, defined by suspensions of so-called $\mathcal{S}$-adic Rauzy fractals. These Markov partitions then provide a symbolic model as a nonstationary edge shift for the Anosov mapping family. Restacking of the Markov partition yields a renormalization process that allows us to interpret a multidimensional continued fraction algorithm as a sequence of iteratively induced toral rotations.