Projects per year
Abstract
We study the combinatorial types of periodic orbits of the standard covering endomorphisms m_k(x)=kx (mod ℤ) of the circle for integers k≥2 and the frequency with which they occur. For any qcycle σ in the permutation group S_q, we give a full description of the set of period q orbits of m_k that realize σ and in particular count how many such orbits there are. The description is based on an invariant called the "fixed point distribution" vector and is achieved by reducing the realization problem to finding the stationary state of an associated Markov chain. Our results generalize earlier work on the special case where σ is a rotation cycle, and can be viewed as a missing combinatorial ingredient for a proper understanding of the dynamics of complex polynomial maps of degree ≥3 and the structure of their parameter spaces.
Translated title of the contribution  Om kombinatoriske typer af periodiske baner of afbildningen x > kx mod Z for k>1 

Original language  English 
Article number  106953 
Journal  Advances in Mathematics 
Volume  2020 
Issue number  361 
Number of pages  38 
ISSN  00018708 
DOIs  
Publication status  Published  12 Feb 2020 
Keywords
 Combinatorial Encoding
 periodic orbits of multiplication by k on the circle
 Dynamical Systems
 Complex dynamics
 External rays
 Circle maps
 rotation number
Projects
 1 Finished

Holomorpic Dynamics and Orthogonal Polynomials
Petersen, C. L., Petersen, H. L., Henriksen, C. & Christiansen, J. S.
01/11/2015 → 31/10/2018
Project: Research