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Russian Mathematical Surveys, 2021, Volume 76, Issue 5, Pages 821–881
DOI: https://doi.org/10.1070/RM9998 (Mi rm9998) 		 
This article is cited in 6 scientific papers (total in 6 papers)

One-dimensional dynamical systems

L. S. Efremovaab, E. N. Makhrovaa

a Lobachevsky State University of Nizhny Novgorod
b Moscow Institute of Physics and Technology
English version PDF (1109 kB) Citations (6)
References:
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DOI: https://doi.org/10.1070/RM9998
Abstract: 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.
Keywords: one-dimensional continuum, degree of a circle map, rotation set, finite graph, dendrite, periodic point, homoclinic point, horseshoe, topological entropy.
Received: 23.02.2021
Russian version:
Uspekhi Matematicheskikh Nauk, 2021, Volume 76, Issue 5(461), Pages 81–146
DOI: https://doi.org/10.4213/rm9998
Bibliographic databases:
Document Type: Article
UDC: 517.938.5
MSC: Primary 37B45, 37E05, 37E10, 37E25, 37E99; Secondary 37B40, 37E45
Language: English
Original paper language: Russian
Citation: L. S. Efremova, E. N. Makhrova, “One-dimensional dynamical systems”, Uspekhi Mat. Nauk, 76:5(461) (2021), 81–146; Russian Math. Surveys, 76:5 (2021), 821–881
Citation in format AMSBIB
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    • This publication is cited in the following 6 articles:
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