Effective high-temperature estimates ensuring a spectral gap

Abstract: The main goal of the talk shall be to explain a few ideas from two classical theories : the thermodynamical formalism, and the perturbation of linear operators. The "thermodynamical formalism" is a framework to describe particular invariant measures of dynamical systems, called "equilibrium states", parametrized by functions on the phase space, called "potentials". This formalism is based on the "transfer operator"; when this operator has a spectral gap, the equilibrium state exists, is unique, and has very good statistical properties (exponential mixing, Central Limit Theorem, etc.) If one perturbs slightly the potential, the corresponding transfer operator is also perturbed. The classical theory of perturbation of operators ensures that the spectral gap property is an open condition and that under bounded pertubration, the eigendata of an operator depends analytically on the perturbation. It turns out that using the Implicit Function Theorem, this theory can be made effective with explicit bounds on the size of a neighborhood where the spectral gap persists. Using this effective perturbation theory, we show completely explicit bound on the potential ensuring the spectral gap property for transfert operators of classical families of dynamical systems.

differential geometry

Audience: researchers in the topic

Pangolin seminar

Series comments: TIME HAS CHANGED: 15:30 Paris 10:30AM Rio de Janeiro

Description: Differential geometry seminar

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