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Nonstationary Markov Partitions and Multidimensional Continued Fraction Algorithms

Pierre Arnoux, Valérie Berthé, Milton Minervino, Wolfgang Steiner, Jörg M. Thuswaldner

TL;DR

This work develops a nonstationary generalization of Markov partitions for sequences of toral automorphisms arising from unimodular multidimensional continued fraction maps under a Pisot-type convergence condition. By combining linear Anosov mapping families, nonstationary Markov partitions, $ ext{S}$-adic substitutions, and Rauzy fractals, it constructs explicit symbolic models as nonstationary edge shifts and shows how MD CF dynamics renormalize into induced toral rotations. The main contributions include a local/strong Pisot theory for bi-infinite matrix sequences, a metric framework via Oseledets-type splittings, and a nonstationary Markov-partition machinery with Rauzy boxes that yields generating partitions and tilings; these are applied to the Brun algorithm as a running example. The results provide a robust bridge between MD CF algorithms, fractal tilings, and symbolic dynamics, enabling a rigorous, computable symbolic description of a broad class of nonstationary hyperbolic systems and their induced rotations. This advances the understanding of higher-dimensional Diophantine approximation through a comprehensive nonstationary dynamical-systems framework with potential applications to ergodic theory and spectral problems.

Abstract

It is well known from results of Sinaĭ 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.

Nonstationary Markov Partitions and Multidimensional Continued Fraction Algorithms

TL;DR

This work develops a nonstationary generalization of Markov partitions for sequences of toral automorphisms arising from unimodular multidimensional continued fraction maps under a Pisot-type convergence condition. By combining linear Anosov mapping families, nonstationary Markov partitions, -adic substitutions, and Rauzy fractals, it constructs explicit symbolic models as nonstationary edge shifts and shows how MD CF dynamics renormalize into induced toral rotations. The main contributions include a local/strong Pisot theory for bi-infinite matrix sequences, a metric framework via Oseledets-type splittings, and a nonstationary Markov-partition machinery with Rauzy boxes that yields generating partitions and tilings; these are applied to the Brun algorithm as a running example. The results provide a robust bridge between MD CF algorithms, fractal tilings, and symbolic dynamics, enabling a rigorous, computable symbolic description of a broad class of nonstationary hyperbolic systems and their induced rotations. This advances the understanding of higher-dimensional Diophantine approximation through a comprehensive nonstationary dynamical-systems framework with potential applications to ergodic theory and spectral problems.

Abstract

It is well known from results of Sinaĭ 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 . 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 -adic shifts, has to be superimposed on the Anosov mapping family. In particular, the atoms of the Markov partitions are geometric realizations of -adic shifts, defined by suspensions of so-called -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.

Paper Structure

This paper contains 75 sections, 86 theorems, 338 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Markov partitions for hyperbolic toral automorphisms
  3. Dynamical interpretations of the classical continued fraction algorithm
  4. Generalization to multidimensional continued fraction algorithms
  5. Discussion of the main results
  6. Convergence results
  7. Pisot conjecture, pure discrete spectrum, and induced rotations
  8. Results on nonstationary Markov partitions
  9. The Brun algorithm, our running example
  10. Outline
  11. Sequences of integer matrices and mapping families
  12. Mapping families
  13. Eventually Anosov mapping families
  14. Nonstationary Markov partitions
  15. Symbolic representation of a mapping family
  16. ...and 60 more sections

Key Result

Proposition 6.8

Let $(X,f)$ be a mapping family and assume that the sequence of partitions $(\mathcal{P}_n)_{n\in\mathbb{Z}}$ has the nonstationary Property M for $(X,f)$. Then $(\mathcal{P}_n)_{n\in\mathbb{Z}}$ is a nonstationary Markov partition for $(X,f)$.

Figures (17)

  • Figure 1: The Markov partition $\{R_1,R_2\}$ for the cat map and its image.
  • Figure 2: The Markov partition $\{R_1,R_2\}$ and its image under $f$.
  • Figure 3: Illustration of the intersection properties of the Markov partition $\{R_1,R_2\}$.
  • Figure 4: Illustration of the restacking and renormalization process on the atoms of the nonstationary Markov partition furnished by the classical continued fraction algorithm. The "recoloring" is just done in order to visualize the Markov partition of the restacked $L$-shaped region.
  • Figure 5: The axis-parallel version of the restacking process of Figure \ref{['fig:Lshaped']}.
  • ...and 12 more figures

Theorems & Definitions (234)

  • Remark 6.1
  • Remark 6.2
  • Example 6.3
  • Definition 6.4: Nonstationary Markov partition
  • Remark 6.5
  • Definition 6.6: Transverse partitions; cf. A98
  • Definition 6.7: Nonstationary Property M
  • Proposition 6.8
  • Example 6.9
  • Proposition 6.10
  • ...and 224 more