Exchange of Intervals and Intrinsic Ergodicity of the Negative Beta shift
Florent Nguema Ndong, Anne Bertrand-Mathis
TL;DR
The paper investigates negative $\beta$-shifts, establishing that the bi-infinite shift is coded precisely when $|\beta|>\frac{1+\sqrt{5}}{2}$. It introduces a decreasing sequence of constants $(-\gamma_n)$ and a morphism $\phi$ with $\phi(k)=1(00)^k$ that links $\beta$-expansions across parameter intervals via $d(l_{\beta},\beta)=\phi^{n+1}(d(l_{x},x))$, thereby transferring maximal-entropy structure between shifts. A central contribution is proving intrinsic ergodicity of the bi-sided negative $\beta$-shift by constructing a positive recurrent prefix code $P_{\beta}$, for which the maximal-entropy measure is the Champernowne measure on $P_{\beta}^{\mathbb{Z}}$ and is unique; this extends known results for one-sided shifts to the bi-sided case. The work also clarifies the “exchange of intervals” phenomenon, showing how the maximal-entropy support for $S_{\beta}$ is coded by morphic images of $S_x$ and is stable under the $\phi^{n+1}$ transformation, with gaps in the language shown to be negligible for maximal entropy. Overall, the results illuminate the structure of negative $\beta$-shifts, linking coding theory, interval exchanges, and intrinsic ergodicity in a unified framework with explicit constructions of maximal-entropy measures.
Abstract
This work highlights a peculiar phenomenon of interval exchange. Considering a real number beta less than -1, the negative beta-shift is coded if and only if its absolute value is greater than the golden ratio. We study an increasing sequence of algebraic integers with limit-1 and the absolute value of the first term equals to the golden ratio such that for a base x taken in the interval of consicutive terms of this sequence, the measure of the maximal entropy is carried by the image of a beta-shift, with the golden ratio les than the absolute value of beta, under the mapping of an injective substitution.