Lift and Synchronization

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 $μ$ to be $F$-liftable as well as a condition for the lift to be an ergodic measure. Moreover, we show that every lift of $μ$ is a weighted average of the restriction of $μ$ 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 $μ$, showing that we can synchronize `Pliss times' $μ$ 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.

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