2021/03/08 by Atnip, Jason, Froyland, Gary, González-Tokman, Cecilia +1
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2103.04712
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,\mathscrF,m). For each ω∈Ω we associate a piecewise-monotone, surjective map Tω:I→ I, and a hole Hω⊂ I; the map Tω, the random potential φω, and the hole Hω generate the corresponding open transfer operator Lω. For a contracting potential, under a condition on the open random dynamics in the spirit of Liverani--Maume-Deschamps, we prove there exists a unique random probability measure νω supported on the survivor set Xω,∞ satisfying νσ(ω)(Lωf)=λωνω(f). We also prove the existence of a unique random family of functions qω that satisfy Lωqω=λωqσ(ω). These yield an ergodic random invariant measure μ=νq supported on the global survivor set, while q combined with the random closed conformal measure yields a unique random absolutely continuous conditional invariant measure (RACCIM) η supported on I. We prove quasi-compactness of the transfer operator cocycle and exponential decay of correlations for μ. Finally, the escape rates of the random closed conformal measure and the RACCIM η coincide, and are given in terms of the expected pressure, as is the Hausdorff dimension of the surviving set Xω,∞. We provide examples of our general theory, including random β-transformations and random Lasota-Yorke maps.