Statistical properties of weak Gibbs measures for complex systems with nonhyperbolic periodic orbits

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The purpose of this project is to construct mathematical models which show phase transition, failure of the Gibbs property and slow decay of correlations, that can be observed for many complex systems exhibiting Intermittency. For this purpose, we establish Thermodynamic formalism for non-Holder potentials in nonhyperbolic situation. More specifically, for countable to one transitive Markov systems we construct conformal measures ν which are weak Gibbs measures for potentials φ of weak bounded variation, and show the existence of equilibrium states μ for φ equivalent to the weak Gibbs measures ν ([1]). We see that generalized indifferent periodic orbits cause phase transition, non-Gibbsianness and force the decay of correlations to be slow ([1], [4]). Applying these results allow us to establish a multifractal formalism of weak Gibbs measures associated to potentials of weak bounded variation ([3]). Furthermore, we establish a version of the local product structure (weak local product structure) for the invertible extension of ergodic weak Gibbs measures μ ([2]). About the decay of correlations of weak Gibbs equilibrium measures, polynomial upper bounds can be obtained for a class of functions which contains all piecewise Lipschitz functions ([4]), by clarifying the speed of uniform convergence of iterated transfer operators on compact sets excluding indifferent periodic points.