Premium
Regularity Conditions and Bernoulli Properties of Equilibrium States and g ‐Measures
Author(s) -
Walters Peter
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704006076
Subject(s) - measure (data warehouse) , bernoulli's principle , mathematics , eigenvalues and eigenvectors , product (mathematics) , mixing (physics) , operator (biology) , probability measure , pure mathematics , bernoulli scheme , state (computer science) , function (biology) , combinatorics , type (biology) , continuous function (set theory) , mathematical analysis , discrete mathematics , physics , quantum mechanics , geometry , thermodynamics , computer science , repressor , algorithm , database , ecology , chemistry , biology , biochemistry , evolutionary biology , transcription factor , gene
When T : X → X is a one‐sided topologically mixing subshift of finite type and φ : X → R is a continuous function, one can define the Ruelle operator L φ : C ( X ) → C ( X ) on the space C ( X ) of real‐valued continuous functions on X . The dual operatorL φ *always has a probability measure ν as an eigenvector corresponding to a positive eigenvalue ( L φ * ν = λν with λ > 0). Necessary and sufficient conditions on such an eigenmeasure ν are obtained for φ to belong to two important spaces of functions, W ( X, T ) and Bow ( X, T ). For example, φ ∈ Bow( X, T ) if and only if ν is a measure with a certain approximate product structure. This is used to apply results of Bradley to show that the natural extension of the unique equilibrium state μ φ of φ ∈ Bow( X, T ) has the weak Bernoulli property and hence is measure‐theoretically isomorphic to a Bernoulli shift. It is also shown that the unique equilibrium state of a two‐sided Bowen function has the weak Bernoulli property. The characterizations mentioned above are used in the case of g ‐measures to obtain results on the ‘reverse’ of a g ‐measure.