# Examples in the entropy theory of countable group actions

## TL;DR

This paper surveys examples illustrating the differences and similarities between classical entropy and newer invariants like sofic and Rokhlin entropy in countable group actions, highlighting counterintuitive properties.

## Contribution

It provides new examples that clarify the behavior of modern entropy invariants in countable group actions, many of which are previously unpublished.

## Key findings

- Counterintuitive property: factor maps can increase entropy
- Examples highlight differences from classical entropy theory
- Many examples are novel and not previously published

## Abstract

Kolmogorov-Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.