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Genus g Cantor sets and germane Julia sets
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 02 December 2025, pp. 1-37
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- Article
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The primary aim of this paper is to give topological obstructions to Cantor sets in
$\mathbb{R}^3$ being Julia sets of uniformly quasiregular mappings. Our main tool is the genus of a Cantor set. We give a new construction of a genus g Cantor set, the first for which the local genus is g at every point, and then show that this Cantor set can be realized as the Julia set of a uniformly quasiregular mapping. These are the first such Cantor Julia sets constructed for 
$g\geq 3$. We then turn to our dynamical applications and show that every Cantor Julia set of a hyperbolic uniformly quasiregular map has a finite genus g; that a given local genus in a Cantor Julia set must occur on a dense subset of the Julia set; and that there do exist Cantor Julia sets where the local genus is non-constant.
Entropy in uniformly quasiregular dynamics
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 41 / Issue 8 / August 2021
- Published online by Cambridge University Press:
- 22 June 2020, pp. 2397-2427
- Print publication:
- August 2021
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Let
$M$ be a closed, oriented, and connected Riemannian 
$n$-manifold, for 
$n\geq 2$, which is not a rational homology sphere. We show that, for a non-constant and non-injective uniformly quasiregular self-map 
$f:M\rightarrow M$, the topological entropy 
$h(f)$ is 
$\log \deg f$. This proves Shub’s entropy conjecture in this case.
