Non-Gibbsianness of SRB measures for the natural extension of intermittent systems. (English) Zbl 1098.37022
Summary: For countable-to-one transitive Markov maps, we show that the natural extensions of invariant ergodic weak Gibbs measures, absolutely continuous with respect to weak Gibbs conformal measures, possess a version of the \(u\)-Gibbs property. In particular, if dynamical potentials admit generalized indifferent periodic points, then the natural extensions exhibit a non-Gibbsian character in statistical mechanics. Our results can be applicable to certain non-hyperbolic number-theoretical transformations of which natural extensions possess unstable (respectively stable) leaves with subexponential expansion (respectively contraction).
MSC:
37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
60J05 | Discrete-time Markov processes on general state spaces |
28A80 | Fractals |
37A45 | Relations of ergodic theory with number theory and harmonic analysis (MSC2010) |
37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
37D99 | Dynamical systems with hyperbolic behavior |