# American Institute of Mathematical Sciences

September  2019, 39(9): 5319-5337. doi: 10.3934/dcds.2019217

## Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets

 National Institute of Technology, Ichinoseki College, Takanashi, Hagisho, Ichinoseki, Iwate 021-8511, Japan

Received  October 2018 Published  May 2019

We prove that for any infinite collection of quadratic Julia sets, there exists a transcendental entire function whose Julia set contains quasiconformal copies of the given quadratic Julia sets. In order to prove the result, we construct a quasiregular map with required dynamics and employ the quasiconformal surgery to obtain the desired transcendental entire function. In addition, the transcendental entire function has order zero.

Citation: Koh Katagata. Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5319-5337. doi: 10.3934/dcds.2019217
##### References:

show all references

##### References:
The definition of the quasiregular map g near infinity
The definition of the quasiregular map $g$ near $R_{m(j)}$
 [1] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [2] Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

2019 Impact Factor: 1.338