Equilibrium measures for uniformly quasiregular dynamics

We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism $f$ of degree at least 2 on a closed Riemannian manifold admits an equilibrium measure $\mu_f$, which is balanced and invariant under $f$ and non-atomic, and whose support agrees with the Julia set of $f$. Furthermore we show that $f$ is strongly mixing with respect to the measure $\mu_f$. We also characterize the measure $\mu_f$ using an approximation property by iterated pullbacks of points under $f$ up to a set of exceptional initial points of Hausdorff dimension at most $n-1$. These dynamical mixing and approximation results are reminiscent of the Mattila-Rickman equidistribution theorem for quasiregular mappings. Our methods are based on the existence of an invariant measurable conformal structure due to Iwaniec and Martin and the \cA-harmonic potential theory.Comment: 17 page

Entropy in uniformly quasiregular dynamics

Let M be a closed, oriented, and connected Riemannian nmanifold, for n ≥ 2, which is not a rational homology sphere. We show that, for a non-constant uniformly quasiregular self-map f : M → M, the topological entropy h(f) is log deg f. This proves Shub’s entropy conjecture in this case.Peer reviewe

nn-harmonic coordinates and the regularity of conformal mappings

This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harmonic coordinate system for $1 < p < \infty$ on any Riemannian manifold.Comment: 20 pages, updated referenc

Obstructions for automorphic quasiregular maps and Latt\`es-type uniformly quasiregular maps

Suppose that $M$ is a closed, connected, and oriented Riemannian $n$-manifold, $f \colon \mathbb{R}^n \to M$ is a quasiregular map automorphic under a discrete group $\Gamma$ of Euclidean isometries, and $f$ has finite multiplicity in a fundamental cell of $\Gamma$. We show that if $\Gamma$ has a sufficiently large translation subgroup $\Gamma_T$, then $\dim \Gamma \in \{0, n-1, n\}$. If $f$ is strongly automorphic and induces a non-injective Latt\`es-type uniformly quasiregular map, then the same holds without the assumption on the size of $\Gamma_T$. Moreover, an even stronger restriction holds in the Latt\`es case if $M$ is not a rational cohomology sphere.Comment: Version 2 includes a short yet significant improvement to the original results, pointed out to the author by Sylvester Eriksson-Biqu

Uniform cohomological expansion of uniformly quasiregular mappings

Let f:M -> M be a uniformly quasiregular self-map of a compact, connected, and oriented Riemannian n-manifold M without boundary, n > 2. We show that, for k is an element of{0, horizontal ellipsis ,n}, the induced homomorphism f*:Hk(M;R)-> Hk(M;R), where Hk(M;R) is the kth singular cohomology of M, is complex diagonalizable and the eigenvalues of f* have absolute value (degf)k/n. As an application, we obtain a degree restriction for uniformly quasiregular self-maps of closed manifolds. In the proof of the main theorem, we use a Sobolev-de Rham cohomology based on conformally invariant differential forms and an induced push-forward operator.Peer reviewe

Funktionentheorie

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Trends and Developments in Complex Dynamics

[no abstract available