# An effective version of Katok's horseshoe theorem for conservative $C^2$ surface diffeomorphisms

## TL;DR

This paper provides an explicit finite condition based on Bowen's coverings that guarantees positive topological entropy for area-preserving $C^2$ surface diffeomorphisms, extending Katok's horseshoe theorem effectively.

## Contribution

It introduces an explicit finite information criterion for positive entropy in conservative surface diffeomorphisms, and demonstrates its limitations in higher dimensions.

## Key findings

- Finite information condition guarantees positive entropy
- Effective version of Katok's horseshoe theorem for surfaces
- Counterexample in dimensions greater than 3

## Abstract

For area preserving $C^2$ surface diffeomorphisms, we give an explicit finite information condition, on the exponential growth of the number of Bowen's $(n,\delta)-$balls needed to cover a positive proportion of the space, that is sufficient to guarantee positive topological entropy. This can be seen as an effective version of Katok's horseshoe theorem in the conservative setting. We also show that the analogous result is false in dimension larger than $3$.