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

TL;DR

This paper characterizes when negative beta transformations share the same invariant measure, identifies the matching property for generalized multinacci numbers, and shows the density of simple -beta numbers with subshifts of finite type.

Contribution

It provides a complete characterization of pairs of non-integer beta values with identical invariant measures and establishes the density of simple -beta numbers with subshifts of finite type.

Findings

Invariant measures coincide for beta values satisfying a quadratic equation and differing by 1.

Negative beta transformations have the matching property for generalized multinacci numbers.

Simple -beta numbers with subshifts of finite type are dense in (1, ∞).

Abstract

We study the negative beta transformations $T_{-\beta}:=-\beta x +\lfloor\beta x\rfloor+1$ for $x\in(0,1]$ and $\beta>1$. We present a complete characterization of pairs of dstinct non-integers with the same $T_{-\beta}$-invariant measure: for two non-integers $\beta_1 ,\beta_2 >1$, the invariant measures of negative $\beta$-transformation coincide if and only if $\beta_1$ is the root of equation $x^2-qx-p=0$, where $p,q\in\mathbb{N}$ with $p\leq q$, and $\beta_2 = \beta_1 + 1$. Furthermore, we show that $T_{-\beta}$ has matching property for all $\beta$ being generalized multinacci numbers. We also prove that the set of simple $-\beta$ numbers, whose $-\beta$-shifts are subshifts of finite type, is dense in the parameter interval $(1,\infty)$.