Equilibrium states for the random β - transformation through g -measures

K. Dajani; K. Power

doi:10.1007/s10474-021-01196-w

Equilibrium states for the random β - transformation through g -measures

K. Dajani^*, K. Power

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We consider the random β-transformation K_β, defined on {0,1}N×[0,⌊β⌋]β-1]], that generates all possible expansions of the form x=∑i=0∞aiβi, wherea_i∈ { 0 , 1 , … , ⌊ β⌋ } }. This transformation was introduced in [3–5], where two naturalinvariant ergodic measures were found. The first is the unique measure ofmaximal entropy, and the second is a measure of the form m_p× μ_β, with m_pthe Bernoulli (p, 1 - p) product measure and μ_β is a measure equivalent to theLebesgue measure. In this paper, we give an uncountable family of K_β-invariantexact g-measures for a certain collection of algebraic β’s. The construction of theseg-measures is explicit and the corresponding potentials are not locally constant.

Original language	English
Pages (from-to)	70-91
Number of pages	22
Journal	Acta Mathematica Hungarica
Volume	166
Issue number	1
DOIs	https://doi.org/10.1007/s10474-021-01196-w
Publication status	Published - Feb 2022

Bibliographical note

Funding Information:
Kieran Power gratefully acknowledges the support of the EPSRC (grant EP/V520093/1). Acknowledgement

Publisher Copyright:
© 2021, Akadémiai Kiadó, Budapest, Hungary.

Funding

Kieran Power gratefully acknowledges the support of the EPSRC (grant EP/V520093/1). Acknowledgement

Keywords

equilibrium states
exactness
g-measures
random β-transformation

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10.1007/s10474-021-01196-w

Dajani-Power2022_Article_EquilibriumStatesForTheRandomBFinal published version, 396 KBLicence: Taverne

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title = "Equilibrium states for the random β - transformation through g -measures",

abstract = "We consider the random β-transformation Kβ, defined on {0,1}N×[0,⌊β⌋]β-1]], that generates all possible expansions of the form x=∑i=0∞aiβi, whereai∈ { 0 , 1 , … , ⌊ β⌋ } }. This transformation was introduced in [3–5], where two naturalinvariant ergodic measures were found. The first is the unique measure ofmaximal entropy, and the second is a measure of the form mp× μβ, with mpthe Bernoulli (p, 1 - p) product measure and μβ is a measure equivalent to theLebesgue measure. In this paper, we give an uncountable family of Kβ-invariantexact g-measures for a certain collection of algebraic β{\textquoteright}s. The construction of theseg-measures is explicit and the corresponding potentials are not locally constant.",

keywords = "equilibrium states, exactness, g-measures, random β-transformation",

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note = "Funding Information: Kieran Power gratefully acknowledges the support of the EPSRC (grant EP/V520093/1). Acknowledgement Publisher Copyright: {\textcopyright} 2021, Akad{\'e}miai Kiad{\'o}, Budapest, Hungary.",

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AB - We consider the random β-transformation Kβ, defined on {0,1}N×[0,⌊β⌋]β-1]], that generates all possible expansions of the form x=∑i=0∞aiβi, whereai∈ { 0 , 1 , … , ⌊ β⌋ } }. This transformation was introduced in [3–5], where two naturalinvariant ergodic measures were found. The first is the unique measure ofmaximal entropy, and the second is a measure of the form mp× μβ, with mpthe Bernoulli (p, 1 - p) product measure and μβ is a measure equivalent to theLebesgue measure. In this paper, we give an uncountable family of Kβ-invariantexact g-measures for a certain collection of algebraic β’s. The construction of theseg-measures is explicit and the corresponding potentials are not locally constant.

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Equilibrium states for the random β - transformation through g -measures

Abstract

Bibliographical note

Funding

Keywords

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