Sharp cohomological bound for uniformly quasiregularly elliptic manifolds

TL;DR

This paper establishes a cohomological upper bound for compact manifolds admitting non-injective uniformly quasiregular self-maps, linking dynamical properties with topological invariants and addressing a conjecture in quasiregular dynamics.

Contribution

It proves a sharp cohomological bound for such manifolds, confirming a dynamical version of the Bonk-Heinonen conjecture and connecting Julia sets with cohomology.

Findings

Cohomology dimension is at most 2^n for manifolds with non-injective uniformly quasiregular maps.

If not a rational homology sphere, the Julia set has positive Lebesgue measure.

Provides a positive answer to a conjecture relating dynamics and topology in quasiregular maps.

Abstract

We show that if a compact, connected, and oriented $n$-manifold $M$ without boundary admits a non-constant non-injective uniformly quasiregular self-map, then the dimension of the real singular cohomology ring $H^*(M; \mathbb{R})$ of $M$ is bounded from above by $2^n$. This is a positive answer to a dynamical counterpart of the Bonk-Heinonen conjecture on the cohomology bound for quasiregularly elliptic manifolds. The proof is based on an intermediary result that, if $M$ is not a rational homology sphere, then each such uniformly quasiregular self-map on $M$ has a Julia set of positive Lebesgue measure.