SRB Measures For Certain Markov Processes

TL;DR

This paper investigates SRB measures in Markov processes generated by monotonic iterated function systems, providing bounds and conditions for the existence of SRB measures, with applications to asset market models.

Contribution

It introduces bounds on the number of SRB measures for certain IFS and characterizes conditions for fixed point measures to be SRB, extending understanding of invariant measures in these systems.

Findings

Upper bound on the number of SRB measures for the IFS

Conditions for fixed point measures to be SRB or not

Application of results to asset market games

Abstract

We study Markov processes generated by iterated function systems (IFS). The constituent maps of the IFS are monotonic transformations of the interval. We first obtain an upper bound on the number of SRB (Sinai-Ruelle-Bowen) measures for the IFS. Then, when all the constituent maps have common fixed points at 0 and 1, theorems are given to analyze properties of the ergodic invariant measures $\delta_0$ and $\delta_1$. In particular, sufficient conditions for $\delta_0$ and/or $\delta_1$ to be, or not to be, SRB measures are given. We apply some of our results to asset market games.