On transfer operators and maps with random holes

Bahsoun, Wael; Schmeling, Jörg; Vaienti, Sandro

doi:10.1088/0951-7715/28/3/713

Cited by 2 publications

(2 citation statements)

References 33 publications

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“…Furthermore, the same holds for ν ω,c ; see Lemma 9.2 of [1]. 3 Any κ > ϑ will work for n sufficiently large.…”

Section: Decay Of Correlationsmentioning

confidence: 87%

“…fashion, [4] consider escape rates for the annealed (averaged) transfer operator in the small hole limit (the Lebesgue measure of the H ω goes to zero). In a similar setting, now assuming T to be Markov and considering non-vanishing holes, [3] show existence of equilibrium states, again for the annealed transfer operator. In [2], the authors consider random, fullbranched interval maps with negative Schwarzian derivative.…”

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confidence: 92%

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Thermodynamic Formalism for Random Interval Maps with Holes

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Froyland,

González-Tokman

et al. 2021

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We develop a quenched thermodynamic formalism for open random dynamical 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 T ω : I → I, and a hole H ω ⊂ [0, 1]; the map T ω , the random potential ϕ ω , and the hole H ω generate the corresponding open transfer operator L ω . For a contracting potential, under a condition on the open random dynamics in the spirit of Liverani-Maume-Deschamps [29], we prove there exists a unique random probability measure ν ω supported on the survivor set X ω,∞ satisfying ν σ(ω) (L ω f ) = λ ω ν ω (f ). Correspondingly, we also prove the existence of a unique (up to scaling and modulo ν) random family of functions q ω that satisfy L ω q ω = λ ω q σ(ω) . Together, these provide an ergodic random invariant measure µ = νq supported on the global survivor set X ∞ , while q combined with the random closed conformal measure yields a unique random absolutely continuous conditional invariant measure (RACCIM) η supported on [0, 1]. Further, we prove quasi-compactness of the transfer operator cocycle generated by L ω 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 X ω,∞ is equal to the unique zero of the expected pressure function for almost every fiber ω ∈ Ω. We provide several examples of our general theory. In particular, we apply our results to random β-transformations and random Lasota-Yorke maps, but our results also apply to the random non-uniformly expanding maps that are treated in [1], such as intermittent maps and maps with contracting branches.

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“…Furthermore, the same holds for ν ω,c ; see Lemma 9.2 of [1]. 3 Any κ > ϑ will work for n sufficiently large.…”

Section: Decay Of Correlationsmentioning

confidence: 87%

mentioning

confidence: 92%