# Rotation sets for graph maps of degree 1

Lluís Alsedà[1]; Sylvie Ruette[2]

• [1] Universitat Autònoma de Barcelona Departament de Matemàtiques 08913 Cerdanyola del Vallès, Barcelona (Spain)
• [2] Université Paris-Sud 11 Laboratoire de Mathématiques CNRS UMR 8628 Bâtiment 425 91405 Orsay cedex (France)
• Volume: 58, Issue: 4, page 1233-1294
• ISSN: 0373-0956

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## Abstract

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For a continuous map on a topological graph containing a loop $S$ it is possible to define the degree (with respect to the loop $S$) and, for a map of degree $1$, rotation numbers. We study the rotation set of these maps and the periods of periodic points having a given rotation number. We show that, if the graph has a single loop $S$ then the set of rotation numbers of points in $S$ has some properties similar to the rotation set of a circle map; in particular it is a compact interval and for every rational $\alpha$ in this interval there exists a periodic point of rotation number $\alpha$.For a special class of maps called combed maps, the rotation set displays the same nice properties as the continuous degree one circle maps.

## How to cite

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Alsedà, Lluís, and Ruette, Sylvie. "Rotation sets for graph maps of degree 1." Annales de l’institut Fourier 58.4 (2008): 1233-1294. <http://eudml.org/doc/10348>.

@article{Alsedà2008,
abstract = {For a continuous map on a topological graph containing a loop $S$ it is possible to define the degree (with respect to the loop $S$) and, for a map of degree $1$, rotation numbers. We study the rotation set of these maps and the periods of periodic points having a given rotation number. We show that, if the graph has a single loop $S$ then the set of rotation numbers of points in $S$ has some properties similar to the rotation set of a circle map; in particular it is a compact interval and for every rational $\alpha$ in this interval there exists a periodic point of rotation number $\alpha$.For a special class of maps called combed maps, the rotation set displays the same nice properties as the continuous degree one circle maps.},
affiliation = {Universitat Autònoma de Barcelona Departament de Matemàtiques 08913 Cerdanyola del Vallès, Barcelona (Spain); Université Paris-Sud 11 Laboratoire de Mathématiques CNRS UMR 8628 Bâtiment 425 91405 Orsay cedex (France)},
author = {Alsedà, Lluís, Ruette, Sylvie},
journal = {Annales de l’institut Fourier},
keywords = {Rotation numbers; graph maps; sets of periods; rotation numbers; one-dimensional dynamics},
language = {eng},
number = {4},
pages = {1233-1294},
publisher = {Association des Annales de l’institut Fourier},
title = {Rotation sets for graph maps of degree 1},
url = {http://eudml.org/doc/10348},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Alsedà, Lluís
AU - Ruette, Sylvie
TI - Rotation sets for graph maps of degree 1
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 4
SP - 1233
EP - 1294
AB - For a continuous map on a topological graph containing a loop $S$ it is possible to define the degree (with respect to the loop $S$) and, for a map of degree $1$, rotation numbers. We study the rotation set of these maps and the periods of periodic points having a given rotation number. We show that, if the graph has a single loop $S$ then the set of rotation numbers of points in $S$ has some properties similar to the rotation set of a circle map; in particular it is a compact interval and for every rational $\alpha$ in this interval there exists a periodic point of rotation number $\alpha$.For a special class of maps called combed maps, the rotation set displays the same nice properties as the continuous degree one circle maps.
LA - eng
KW - Rotation numbers; graph maps; sets of periods; rotation numbers; one-dimensional dynamics
UR - http://eudml.org/doc/10348
ER -

## References

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