Differentiability of thermodynamical quantities in non uniformly expanding dynamics. (English) Zbl 1359.37071
Summary: In this paper we study the ergodic theory of a robust non-uniformly expanding maps where no Markov assumption is required. We prove that the topological pressure is differentiable as a function of the dynamics and analytic with respect to the potential. Moreover we not only prove the continuity of the equilibrium states and their metric entropy as well as the differentiability of the maximal entropy measure and extremal Lyapunov exponents with respect to the dynamics. We also prove a local large deviations principle and central limit theorem and show that the rate function, mean and variance vary continuously with respect to observables, potentials and dynamics. Finally, we show that the correlation function associated to the maximal entropy measure is differentiable with respect to the dynamics and it is \(C^1\)-convergent to zero. In addition, precise formulas for the derivatives of thermodynamical quantities are given.
MSC:
| 37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |
| 37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |
| 37C30 | Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 60F05 | Central limit and other weak theorems |
| 37F15 | Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems |
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
Keywords:
nonuniform hyperbolicity; regularity of topological pressure; linear response formula; regularity of limit theoremsReferences:
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