Volker Mayer

Uniformly quasiregular mappings of Lattès type

Using an analogy of the Lattès’ construction of chaotic rational functions, we show that there are uniformly quasiregular mappings of the nsphere R whose Julia set is the whole sphere. Moreover there are analogues of power mappings, uniformly quasiregular mappings whose Julia set is Sn−1 and its complement in Sn consists of two superattracting basins. In the chaotic case we study the invariant conformal structures and show that Lattès type rational mappings are either rigid or form a 1-parameter family of quasiconformal deformations.

Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order

Working with well chosen Riemannian metrics and employing Nevanlinna’s theory, we make the thermodynamical formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosine–root family and also compositions of these functions with arbitrary polynomials). In particular, the existence of conformal (Gibbs) measures is established and then the existence of probability invariant measures equivalent to conformal measures is proven. As a geometric consequence of the developed thermodynamic formalism, a version of Bowen’s formula expressing the Hausdorff dimension of the radial Julia set as the zero of the pressure function and, moreover, the real analyticity of this dimension, is proved.

Rigidity in holomorphic and quasiregular dynamics

We consider rigidity phenomena for holomorphic functions and then more generally for uniformly quasiregular maps.

Comparing measures and invariant line fields

We give elementary proofs of two rigidity results. The first one asserts that the maximal entropy measure \mu_f of a rational map f is singular with respect to any given conformal measure excepted if f is a power, Tchebychev or Lattes map. This is a variation of a result of Zdunik. Our second result is an improvement of a theorem of Fisher and Urbanski. It gives a sharp description of the exceptional functions that admit invariant line fields which are defined with respect to certain invariant measures

Quasiregular analogues of critically finite rational functions with parabolic orbifold

AbstractWe study uniformly quasiregular mappings of

The generalized Lichnerowicz problem: Uniformly quasiregular mappings and space forms

\bar {\mathbb{R}}^n

Finer geometric rigidity of limit sets of conformal IFS

, i.e., quasiregular mappingsf with uniform control of the dilatation of all the iteratesfk, which are analogues of critically finite rational functions with parabolic orbifold. They form a rich family of non-injective uniformly quasiregular mappings. In our main result we characterize them among all uniformly quasiregular mappings as those which have an invariant conformal structure flat at a point of a repelling cycle.

Countable Alphabet Random Subhifts of Finite Type with Weakly Positive Transfer Operator

1 Introduction.- 2 Expanding Random Maps.- 3 The RPF-theorem.- 4 Measurability, Pressure and Gibbs Condition.- 5 Fractal Structure of Conformal Expanding Random Repellers.- 6 Multifractal Analysis.- 7 Expanding in the Mean.- 8 Classical Expanding Random Systems.- 9 Real Analyticity of Pressure.