# Combinatorial independence and naive entropy

## TL;DR

This paper explores the concept of naive entropy in group actions, demonstrating that positive naive entropy implies chaos and untameness, while distal actions have zero naive entropy, answering a question posed by Lewis Bowen.

## Contribution

It introduces the independence density for families of set tuples in group actions and establishes its implications for entropy and chaos.

## Key findings

- Actions with positive naive entropy are Li-Yorke chaotic.
- Distal actions have zero naive entropy.
- The results answer a question of Lewis Bowen.

## Abstract

We study the independence density for finite families of finite tuples of sets for continuous actions of discrete groups on compact metrizable spaces. We use it to show that actions with positive naive entropy are Li-Yorke chaotic and untame. In particular, distal actions have zero naive entropy. This answers a question of Lewis Bowen.