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.
Domaines
Systèmes dynamiques [math.DS]
Origine : Fichiers produits par l'(les) auteur(s)