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Pré-Publication, Document De Travail Année : 2023

Thermodynamic Formalism and Perturbation Formulae for Quenched Random Open Dynamical Systems

Résumé

We develop a quenched thermodynamic formalism for open random dynami- cal systems generated by finitely branched, piecewise-monotone mappings of the interval. The openness refers to the presence of holes in the interval, which terminate trajectories once they enter; the holes may also be random. Our random driving is generated by an invertible, ergodic, measure-preserving transformation σ on a probability space (Ω, F , m). For each ω ∈ Ω we associate a piecewise-monotone, surjective map Tω : I → I, and a hole Hω ⊂ [0, 1]; the map Tω , the random potential φω , and the hole Hω generate the corresponding open transfer operator Lω . The paper is divided into two chapters. In the first chapter we prove, for a contracting potential, that there exists a unique random prob- ability measure νω supported on the survivor set Xω,∞ satisfying νσ(ω)(Lω f ) = λω νω (f ). Correspondingly, we also prove the existence of a unique (up to scaling and modulo ν) random family of functions φω that satisfy Lω φω = λω φσ(ω). Together, these provide an ergodic random invariant measure μ = νφ supported on the global survivor set X∞, while φ combined with the random closed conformal measure yields a random absolutely continuous conditional invariant measure (RACCIM) η supported on [0, 1]. Further, we prove quasi-compactness of the transfer operator cocycle generated by Lω and exponential decay of correlations for μ. The escape rates of the random closed conformal measure and the RACCIM η coincide, and are given by the difference of the expected pressures for the closed and open random systems. Finally, we prove that the Hausdorff dimension of the surviving set Xω,∞ is equal to the unique zero of the expected pressure function for almost every fiber ω ∈ Ω. We provide examples, including a large class of random Lasota-Yorke maps with holes, for which the above results apply. In the second chapter of the paper we consider quasi-compact linear operator cocycles Ln ω,0 := Lσn−1ω,0 ◦ · · · ◦ Lσω,0 ◦ Lω,0, and their small perturbations Ln ω,ε. The operators Lω,0 and Lω,ε need not be transfer operators. We prove an abstract ω-wise first-order formula for the leading Lyapunov multipliers λω,ε = λω,0 − θω ∆ω,ε + o(∆ω,ε), where ∆ω,ε quantifies the closeness of Lω,ε and Lω,0. We then consider the situation where Ln ω,0 is a transfer operator cocycle for a closed random map cocycle T n ω and the perturbed transfer operators Lω,ε are defined by the introduction of small random holes Hω,ε in [0, 1], cre- ating a random open dynamical system. We obtain a first-order perturbation formula in this setting, which reads λω,ε = λω,0 − θω μω,0(Hω,ε) + o(μω,0(Hω,ε)), where μω,0 is the unique equivariant random measure (and equilibrium state) for the original closed ran- dom dynamics. Our new machinery is then deployed to create a spectral approach for a quenched extreme value theory that considers random dynamics and random observations. An extreme value law is derived using the first-order terms θω . Further, in the setting of random piecewise expanding interval maps, we establish the existence of random equi- librium states and conditionally invariant measures for random open systems with small holes via a random perturbative approach, in contrast to the cone-based arguments of the first chapter. Finally we prove quenched statistical limit theorems for random equilibrium states arising from contracting potentials. We illustrate all of the above theory with a variety of examples.
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Dates et versions

hal-03942121 , version 1 (16-01-2023)

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  • HAL Id : hal-03942121 , version 1

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Jason Atnip, Gary Froyland, Cecilia González-Tokman, Sandro Vaienti. Thermodynamic Formalism and Perturbation Formulae for Quenched Random Open Dynamical Systems. 2023. ⟨hal-03942121⟩
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