One-dimensional dynamical systems
L. S. Efremovaab, E. N. Makhrovaa
a Lobachevsky State University of Nizhny Novgorod
b Moscow Institute of Physics and Technology
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.
one-dimensional continuum, degree of a circle map, rotation set, finite graph, dendrite, periodic point, homoclinic point, horseshoe, topological entropy.
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Russian Mathematical Surveys, 2021, 76:5, 821–881
MSC: Primary 37B45, 37E05, 37E10, 37E25, 37E99; Secondary 37B40, 37E45
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
\by L.~S.~Efremova, E.~N.~Makhrova
\paper One-dimensional dynamical systems
\jour Uspekhi Mat. Nauk
\jour Russian Math. Surveys
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