On combinatorial types of periodic orbits of the map x↦kx (mod Z)

Carsten L. Petersen, Saeed Zakeri*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review


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 q-cycle 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 contributionOm kombinatoriske typer af periodiske baner of afbildningen x -> kx mod Z for k>1
Original languageEnglish
Article number106953
JournalAdvances in Mathematics
Issue number361
Number of pages38
Publication statusPublished - 12 Feb 2020


  • Combinatorial Encoding
  • periodic orbits of multiplication by k on the circle
  • Dynamical Systems
  • Complex dynamics
  • External rays
  • Circle maps
  • rotation number

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