Entropy invariants of generic actions

TL;DR

This paper explores the behavior of typical dynamical systems, showing they can temporarily resemble non-deterministic systems with low entropy before decreasing entropy, and introduces a new entropy measure distinguishing generic transformations.

Contribution

It introduces the Kushnirenko entropy, which is zero for certain measure-preserving transformations but infinite for generic systems, highlighting new invariants in dynamical systems.

Findings

Typical systems behave like non-deterministic systems with small entropy for long periods.

Kushnirenko entropy is zero for specific transformations but infinite for generic ones.

The behavior of entropy in dynamical systems can be highly variable over time.

Abstract

We show that the typical dynamical system sometimes begins to behave like a non-deterministic system with a small classical entropy, and this behavior lasts an extremely long time, until the system starts decreasing entropy. Then again it will become almost non-deterministic for a very very long time, but with more smaller classical entropy. Playing on this fact and considering sigma-compact families of measure-preserving zero-entropy transformations, for example, the rectangle exchange transformations, we choose the Kushnirenko entropy so that it is equal to zero for the transformations under consideration, but is infinite for the generic transformation.