# Entropy in uniformly quasiregular dynamics

## TL;DR

This paper proves that for certain non-injective uniformly quasiregular maps on specific manifolds, the topological entropy equals the logarithm of the map's degree, confirming Shub's entropy conjecture in this setting.

## Contribution

It establishes the equality of topological entropy and degree logarithm for non-injective uniformly quasiregular maps on manifolds not homologically trivial.

## Key findings

- Topological entropy equals log degree for these maps.
- Confirms Shub's entropy conjecture in this context.
- Applicable to manifolds with non-trivial rational homology.

## Abstract

Let $M$ be a closed, oriented, and connected Riemannian $n$-manifold, for $n\ge 2$, which is not a rational homology sphere. We show that, for a non-constant and non-injective uniformly quasiregular self-map $f\colon M\to M$, the topological entropy $h(f)$ is $\log \mathrm{deg}( f )$. This proves Shub's entropy conjecture in this case.