Lift and Synchronization

TL;DR

This paper investigates conditions for lifting invariant measures to induced maps, introduces coherent schedules of events, and applies these concepts to synchronization and SRB measures in dynamical systems.

Contribution

It provides necessary and sufficient conditions for measure liftability, introduces coherent schedules, and connects these to synchronization and SRB measure existence.

Findings

Characterization of measure liftability conditions

Introduction of coherent schedules of events

Proof of synchronization at almost every point

Abstract

We study the problem of lifting a measure to an induced map $F(x)=f^{R(x)}(x)$. In particular, we give a necessary and sufficient condition for an ergodic $f$ invariant probability $\mu$ to be $F$-liftable as well as a condition for the lift to be an ergodic measure. Moreover, we show that every lift of $\mu$ is a weighted average of the restriction of $\mu$ to a countable number of $F$-ergodic components. We introduce the concept of a coherent schedule of events and relate it to the lift problem. As a consequence, we prove that we can always synchronize coherent schedules at almost every point with respect to a given invariant probability $\mu$, showing that we can synchronize `Pliss times' $\mu$ almost everywhere. We also provide a version of this synchronization to non-invariant measures and, from that, we obtain some results related to Viana's conjecture on the existence of SRB measures for maps with non-zero Lyapunov exponents for Lebesgue almost every point.