We establish a quenched Central Limit Theorem (CLT) for a smooth observable
of random sequences of iterated linear hyperbolic maps on the torus. To this
end we also obtain an annealed CLT for the same system. We show that, almost
surely, the variance of the quenched system is the same as for the annealed
system. Our technique is the study of the transfer operator on an anisotropic
Banach space specifically tailored to use the cone condition satisfied by the
maps.Comment: 18 page
This paper discusses the evolution of probability distributions for certain time-dependent dynamical systems. Exponential loss of memory is proved for expanding maps and for one-dimensional piecewise expanding maps with slowly varying parameters.
We propose a model of Sinai billiards with moving scatterers, in which the locations and shapes of the scatterers may change by small amounts between collisions. Our main result is the exponential loss of memory of initial data at uniform rates, and our proof consists of a coupling argument for non-stationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on the statistical properties of time-dependent dynamical systems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.