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Statistical properties of dynamical systems via induced Weak Gibbs Markov maps

Publication . Ullah, Asad; Vilarinho, Hélder Soares

This thesis addresses the statistical properties of dynamical systems that admit an induced Weak Gibbs Markov map (not necessarily full branch). More precisely, it provides estimates on the decay of correlations, a Central Limit Theorem and Large Deviations results for such dynamical systems. These estimates for the decay of correlations and the Central Limit Theorem are obtained for H¨older observables as well as larger classes of observables with weaker regularity than H¨older. The results on Large Deviations are a consequence of the estimates on the decay of correlations in this thesis. To deal with the estimates on the decay of correlations and the Central Limit Theorem for H¨older observables, our approach generalizes L.-S. Young’s coupling arguments. Initially, we discuss how to ensure that the invariant probability measure for the tower system of an inducedWeak Gibbs Markov map, is mixing. Then, we estimate the decay of correlations in terms of the tail of the return time function and derive the Central Limit Theorem for the tower system. Finally, using a semiconjugacy, we obtain results on the decay of correlations and Central Limit Theorem for H¨older observables. The results for larger classes of observables follow a similar strategy. Specifically, we extend the decay of correlation estimates, the Central Limit Theorem and the Large Deviations estimates for H¨older observables to larger classes of observables.

2024-10-16Doctoral thesis

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