Abstract
We consider the random \(\beta\)-transformation \(K_{\beta}\), defined on \(\{0,1\}^{\mathbb{N}}\times[0,\frac{\lfloor\beta\rfloor]}{\beta-1}]\)], that generates all possible expansions of the form \(x=\sum_{i=0}^{\infty}\frac{a_i}{\beta^i}\), where \(a_i\in \{0,1 , \ldots,\lfloor\beta\rfloor\}\)}. This transformation was introduced in [3–5], where two natural invariant ergodic measures were found. The first is the unique measure of maximal entropy, and the second is a measure of the form \(m_p\times \mu_{\beta}\), with \(m_p\) the Bernoulli \((p,1-p)\) product measure and \(\mu_{\beta}\) is a measure equivalent to the Lebesgue measure. In this paper, we give an uncountable family of \(K_{\beta}\)-invariant exact \(g\)-measures for a certain collection of algebraic \(\beta\)’s. The construction of these \(g\)-measures is explicit and the corresponding potentials are not locally constant.
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The authors thank the editor, as well as the anonymous referees for their valuable comments.
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Kieran Power gratefully acknowledges the support of the EPSRC (grant EP/V520093/1).
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Dajani, K., Power, K. Equilibrium states for the random \(\beta\)- transformation through \(g\)-measures. Acta Math. Hungar. 166, 70–91 (2022). https://doi.org/10.1007/s10474-021-01196-w
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DOI: https://doi.org/10.1007/s10474-021-01196-w