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Abstract In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval$$[0,1]$$in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure$$\nu $$. We consider a class of non-square-integrable observables$$\phi $$, mostly of form$$\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$$, where$$x_0$$is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index$$\alpha \in (0,2)$$. The two types of maps we concatenate are a class of piecewise$$C^2$$expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and$$\alpha $$, we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by$$\nu $$. This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.
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Statistical determinism in non-Lipschitz dynamical systems
Abstract We study a class of ordinary differential equations with a non-Lipschitz point singularity that admits non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on a parameter$$\nu $$: the regularized dynamics is globally defined for each$$\nu> 0$$, and the original singular system is recovered in the limit of vanishing$$\nu $$. We prove that this limit yields aunique statistical solutionindependent of regularization when the deterministic system possesses a chaotic attractor having a physical measure with the convergence to equilibrium property. In this case, solutions become spontaneously stochastic after passing through the singularity: they are selected randomly with an intrinsic probability distribution.
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- Award ID(s):
- 2235395
- PAR ID:
- 10494723
- Publisher / Repository:
- ErgodicTheoryandDynamicalSystems
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 1 to 29
- Subject(s) / Keyword(s):
- statistical determinism
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/etds.2023.74
