• Corpus ID: 248571921

Hausdorff dimension in quasiregular dynamics

@inproceedings{Bergweiler2022HausdorffDI,
  title={Hausdorff dimension in quasiregular dynamics},
  author={Walter Bergweiler and Athanasios Tsantaris},
  year={2022}
}
. It is shown that the Hausdorff dimension of the fast escaping set of a quasiregular self-map of R 3 can take any value in the interval [1 , 3]. The Hausdorff dimension of the Julia set of such a map is estimated under some growth condition. 

References

SHOWING 1-10 OF 40 REFERENCES

Non-escaping points of Zorich maps

We extend results about the dimension of the radial Julia set of certain exponential functions to quasiregular Zorich maps in higher dimensions. Our results improve on previous estimates of the

The growth rate of an entire function and the Hausdorff dimension of its Julia set

Let f be a transcendental entire function in the Eremenko–Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is

The Julia Set and the Fast Escaping Set of a Quasiregular Mapping

It is shown that for quasiregular maps of positive lower order, the Julia set coincides with the boundary of the fast escaping set.

Functions of genus zero for which the fast escaping set has Hausdorff dimension two

We study a family of transcendental entire functions of genus zero, for which all of the zeros lie within a closed sector strictly smaller than a half-plane. In general these functions lie outside

Fatou–Julia theory for non-uniformly quasiregular maps

Abstract Many results of the Fatou–Julia iteration theory of rational functions extend to uniformly quasiregular maps in higher dimensions. We obtain results of this type for certain classes of

Karpińska's paradox in dimension 3

For 0 < c < 1/e the Julia set of f(z) = c exp(z) is an uncountable union of pairwise disjoint simple curves tending to infinity [Devaney and Krych 1984], the Hausdorff dimension of this set is two

On the differentiability of hairs for Zorich maps

Devaney and Krych showed that for the exponential family λe, where 0 < λ < 1/e, the Julia set consists of uncountably many pairwise disjoint simple curves tending to ∞. Viana proved that these curves

Iteration of Quasiregular Mappings

We survey some results on the iteration of quasiregular mappings. In particular we discuss some recent results on the dynamics of quasiregular maps which are not uniformly quasiregular.

Hausdorff dimensions of escaping sets of transcendental entire functions

Suppose that f and g are transcendental entire functions, each with a bounded set of singular values, and that f and g are affinely equivalent. We show that the escaping sets of f and g have the same

A transcendental Julia set of dimension 1

We construct a non-polynomial entire function whose Julia set has finite 1-dimensional spherical measure, and hence Hausdorff dimension 1. In 1975, Baker proved the dimension of such a Julia set must