Quasi Markovian behavior in mixing maps

TL;DR

This paper investigates how mixing maps lead to Markovian behavior in coarse-grained observables, demonstrating exponential convergence of transition probabilities to a Markov chain and providing explicit calculations for the baker map.

Contribution

It provides a detailed analysis of the convergence of time-dependent distributions to Markov chains in mixing maps, including explicit computations for the baker map.

Findings

Transition probabilities converge exponentially fast to a Markov chain.

Exact agreement for uniform rectangles forming a Markov partition.

Explicit computation of the baker map's nth iterate and its implications.

Abstract

We consider the time dependent probability distribution of a coarse grained observable Y whose evolution is governed by a discrete time map. If the map is mixing, the time dependent one-step transition probabilities converge in the long time limit to yield an ergodic stochastic matrix. The stationary distribution of this matrix is identical to the asymptotic distribution of Y under the exact dynamics. The nth time iterate of the baker map is explicitly computed and used to compare the time evolution of the occupation probabilities with those of the approximating Markov chain. The convergence is found to be at least exponentially fast for all rectangular partitions with Lebesgue measure. In particular, uniform rectangles form a Markov partition for which we find exact agreement.