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Negative $β$-transformations: invariant measures, subshifts of finite type and matching property

Yan Huang, Yun Sun

Abstract

We study the negative beta transformations $T_{-β}:=-βx +\lfloorβx\rfloor+1$ for $x\in(0,1]$ and $β>1$. We present a complete characterization of pairs of dstinct non-integers with the same $T_{-β}$-invariant measure: for two non-integers $β_1 ,β_2 >1$, the invariant measures of negative $β$-transformation coincide if and only if $β_1$ is the root of equation $x^2-qx-p=0$, where $p,q\in\mathbb{N}$ with $p\leq q$, and $β_2 = β_1 + 1$. Furthermore, we show that $T_{-β}$ has matching property for all $β$ being generalized multinacci numbers. We also prove that the set of simple $-β$ numbers, whose $-β$-shifts are subshifts of finite type, is dense in the parameter interval $(1,\infty)$.

Negative $β$-transformations: invariant measures, subshifts of finite type and matching property

Abstract

We study the negative beta transformations for and . We present a complete characterization of pairs of dstinct non-integers with the same -invariant measure: for two non-integers , the invariant measures of negative -transformation coincide if and only if is the root of equation , where with , and . Furthermore, we show that has matching property for all being generalized multinacci numbers. We also prove that the set of simple numbers, whose -shifts are subshifts of finite type, is dense in the parameter interval .
Paper Structure (8 sections, 21 theorems, 87 equations)

This paper contains 8 sections, 21 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Symbolic Dynamics and $(-\beta)$-expansions
  3. Symbolic Dynamics Basics
  4. $(-\beta)$-expansions and $(-\beta)$-shifts
  5. The coincidence of invariant measures
  6. Matching property and simple $(-\beta)$ numbers
  7. Matching property
  8. Denseness of simple $(-\beta)$ numbers

Key Result

Theorem 1.1

(Hochman-Shmerkin-2015) Let $\beta_1, \beta_2 > 1$ with $\beta_1\nsim \beta_2$ and $\beta_1$ a Pisot number. If $\mu$ is jointly invariant under $T_{\beta_1},T_{\beta_2}$, and if all ergodic components of $\mu$ under $T_{\beta_2}$ have positive entropy, then $\mu$ is the common Rényi-Parry measure f

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • ...and 26 more