Intrinsic ergodicity for factors of $(-\beta)$-shifts

TL;DR

The paper proves that under certain conditions, all subshift factors of a ($-eta$)-shift are intrinsically ergodic with a Gibbs measure, extending the understanding of ergodic properties in negative beta-shifts.

Contribution

It establishes conditions for intrinsic ergodicity of subshift factors of ($-eta$)-shifts and applies advanced techniques beyond the specification property.

Findings

All subshift factors are intrinsically ergodic when $eta \\geq rac{1+\\sqrt{5}}{2}$ and the expansion of 1 is not periodic with odd period.

The unique measure of maximal entropy satisfies a Gibbs property.

Existence of non-intrinsically ergodic subshift factors outside the specified conditions.

Abstract

We show that every subshift factor of a ($-\beta$)-shift is intrinsically ergodic, when $\beta\geq \frac{1+\sqrt{5}}{2}$ and the ($-\beta$)-expansion of $1$ is not periodic with odd period. Moreover, the unique measure of maximal entropy satisfies a certain Gibbs property. This is an application of the technique established by Climenhaga and Thompson to prove intrinsic ergodicity beyond specification. We also prove that there exists a subshift factor of a ($-\beta$)-shift which is not intrinsically ergodic in the cases other than the above.