One-dimensional dynamical systems

The survey is devoted to the topological dynamics of maps defined on one-dimensional continua such as a closed interval, a circle, finite graphs (for instance, finite trees), or dendrites (locally connected continua without subsets homeomorphic to a circle). Connections between the periodic behaviou...

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Bibliographic Details
Published in:Russian mathematical surveys 2021-10, Vol.76 (5), p.821-881
Main Authors: Efremova, L. S., Makhrova, E. N.
Format: Article
Language:English
Subjects:
degree of a circle map
dendrite
Entropy
finite graph
homoclinic point
horseshoe
one-dimensional continuum
periodic point
rotation set
topological entropy
Topology
Trees (mathematics)
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Summary:The survey is devoted to the topological dynamics of maps defined on one-dimensional continua such as a closed interval, a circle, finite graphs (for instance, finite trees), or dendrites (locally connected continua without subsets homeomorphic to a circle). Connections between the periodic behaviour of trajectories, the existence of a horseshoe and homoclinic trajectories, and the positivity of topological entropy are investigated. Necessary and sufficient conditions for entropy chaos in continuous maps of an interval, a circle, or a finite graph, and sufficient conditions for entropy chaos in continuous maps of dendrites are presented. Reasons for similarities and differences between the properties of maps defined on the continua under consideration are analyzed. Extensions of Sharkovsky’s theorem to certain discontinuous maps of a line or an interval and continuous maps on a plane are considered. Bibliography: 207 titles.
ISSN:0036-0279
1468-4829
DOI:10.1070/RM9998