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Regularity conditions and Bernoulli properties of equilibrium states and $g$-measures

By Peter Walters


When T : X -> X is a one-sided topologically mixing subshift of finite type and {varphi} : X -> R is a continuous function, one can define the Ruelle operator L{varphi} : C(X) -> C(X) on the space C(X) of real-valued continuous functions on X. The dual operator Formula always has a probability measure {nu} as an eigenvector corresponding to a positive eigenvalue (Formula = {lambda}{nu} with {lambda} > 0). Necessary and sufficient conditions on such an eigenmeasure {nu} are obtained for {varphi} to belong to two important spaces of functions, W(X, T) and Bow (X, T). For example, {varphi} isin Bow(X, T) if and only if {nu} 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{varphi} of {varphi} isin 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

Topics: QA
Publisher: Cambridge University Press
Year: 2005
OAI identifier: oai:wrap.warwick.ac.uk:735

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  1. (2003). A necessary and sufficient condition for a two-sided continuous function to be cohomologous to a one-sided continuous function’, doi
  2. (1982). An introduction to ergodic theory, doi
  3. (2001). Convergence of the Ruelle operator for a function satisfying Bowen’s condition’,
  4. (1970). Friedman,a n dD. Ornstein, ‘On the isomorphism of weak Bernoulli transformations’, doi
  5. (1978). Invariant measures and equilibrium states for some mappings which expand distances’, doi
  6. (2001). La condition de Walters’, doi
  7. (1993). Non-uniqueness of g-functions’, doi
  8. (1983). O nt h eψ-mixing condition for stationary random sequences’, doi
  9. (1972). On the root problem in ergodic theory’,
  10. (1973). Principe variationel et syst` emes symbolique’,
  11. (1990). Random Markov processes and uniform martingales’, doi
  12. (1997). Rigidity of continuous coboundaries’, doi
  13. (1975). Ruelle’s operator theorem and g-measures’, doi
  14. (1974). Some systems with unique equilibrium states’, doi
  15. (1972). Strongly mixing g-measures’, doi

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