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Distribution of the displacement sequence of an orientation preserving circle homeomorphism

Abstract

In some applications not only the knowledge of the behaviour of trajectories of a map is important, but also their displacements. We describe in detail the distribution of elements of the displacement sequence along a trajectory of an orientation preserving circle homeomorphism ϕ with irrational rotation number ϱ(ϕ). The values of displacement are dense in a set which depends on the map γ (semi-)conjugating ϕ with the rotation by ϱ(ϕ) and which is the support of the displacement distribution. We provide an effective formula for the density of this distribution if γ is a C1-diffeomorphism. Moreover, we show approximation of the displacement distribution by sample displacements measured for any other circle homeomorphism sufficiently close to the initial homeomorphism ϕ, which constitutes a rigorous proof of numerical results seen in certain integrate-and-fire models.

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Authors (2)

  • Photo of  Wacław Marzantowicz

    Wacław Marzantowicz

    • adam Mickiewicz University of Poznań .
  • Photo of dr inż. Justyna Signerska-Rynkowska

    Justyna Signerska-Rynkowska dr inż.

    • Polish Academy of Science Institute of Mathematics

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Details

Category:
Articles
Type:
artykuł w czasopiśmie wyróżnionym w JCR
Published in:
DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL no. 29, edition 2, pages 153 - 166,
ISSN: 1468-9367
Language:
English
Publication year:
2014
Bibliographic description:
Marzantowicz W., Signerska-Rynkowska J.: Distribution of the displacement sequence of an orientation preserving circle homeomorphism// DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL. -Vol. 29, iss. 2 (2014), s.153-166
DOI:
Digital Object Identifier (open in new tab) 10.1080/14689367.2013.868867
Verified by:
Gdańsk University of Technology

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