Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence and combinatorics of the Reverse algorithm

Hiroaki Ito, Niels Langeveld, Jörg Thuswaldner

Abstract

We study the Reverse algorithm, a multidimensional continued fraction algorithm, which is not unimodular. We show that the Reverse algorithm is ergodic and, by proving that its second Lyapunov exponent is negative, that it is a.e. exponentially convergent. In addition to that, we attach substitutions to this algorithm and study the $S$-adic languages generated by sequences of these substitutions. The negativity of the second Lyapunov exponent implies that almost all of these languages are balanced. By a thorough study of the combinatorics of the substitutions, we are even able to obtain a concrete generic family of balanced languages that is characterized in terms of a simple condition on the underlying sequence of substitutions.

Convergence and combinatorics of the Reverse algorithm

Abstract

We study the Reverse algorithm, a multidimensional continued fraction algorithm, which is not unimodular. We show that the Reverse algorithm is ergodic and, by proving that its second Lyapunov exponent is negative, that it is a.e. exponentially convergent. In addition to that, we attach substitutions to this algorithm and study the -adic languages generated by sequences of these substitutions. The negativity of the second Lyapunov exponent implies that almost all of these languages are balanced. By a thorough study of the combinatorics of the substitutions, we are even able to obtain a concrete generic family of balanced languages that is characterized in terms of a simple condition on the underlying sequence of substitutions.
Paper Structure (6 sections, 25 theorems, 95 equations)

This paper contains 6 sections, 25 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. $2$-dimensional continued fraction algorithms
  3. Ergodicity
  4. The second Lyapunov exponent
  5. Balancedness
  6. The sorted version of the Reverse algorithm and its second Lyapunov exponent

Key Result

Theorem 1.3

The Reverse algorithm $(\Delta, f_{\mathrm{R}})$ is ergodic with respect to the invariant measure $\mu$ with density $h$ defined in eq:density.

Theorems & Definitions (47)

  • Definition 1.1: Reverse algorithm; see AL18
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: $2$-dimensional continued fraction algorithm
  • Lemma 2.2: see thuswaldner2019boldsymbolsadic
  • Definition 2.3: Topological convergence
  • Definition 2.4: Exponential convergence
  • Proposition 2.5: Schweiger:00
  • ...and 37 more