Rotation set for maps of degree 1 on the graph sigma
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Abstract
For a continuous map on a topological graph containing a unique loop S it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and for every rational r in this interval there exists a periodic point of rotation number r. The whole rotation set (i.e., the set of all rotation numbers) may not be connected and it is not known in general whether it is closed.
The graph sigma is the space consisting in an interval attached by one of its endpoints to a circle. We show that, for a map of degree 1 on the graph sigma, the rotation set is closed and has finitely many connected components. Moreover, for all rational numbers r in the rotation set, there exists a periodic point of rotation number r.
Keywords
Rational Number Universal Covering Periodic Point Rotation Number Compact IntervalReferences
- [1]Ll. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics, Vol. 5, World Scientific Publications, River Edge, NJ, 1993.MATHGoogle Scholar
- [2]Ll. Alsedà and S. Ruette, Rotation sets for graph maps of degree 1, Annales de l’Institut Fourier 58 (2008), 1233–1294.MATHGoogle Scholar
- [3]L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one dimensional maps in Global Theory of Dynamical Systems, Lecture Notes in Mathematics, Vol. 819, Springer-Verlag, Berlin, 1980, 18–34.CrossRefGoogle Scholar
- [4]M. Misiurewicz, Horseshoes for mappings of the interval, L’Académie Polonaise des Sciences. Bulletin. Série des Sciences Mathématiques 27 (1979), 167–169.MathSciNetMATHGoogle Scholar