Simplest skew products on \(n\)-dimensional \((n\geq 2 )\) cells, cylinders and tori
type: Item
instance of: scholarly article
title: Simplest skew products on \(n\)-dimensional \((n\geq 2 )\) cells, cylinders and tori
DOI: 10.1134/S1995080222100080
author: Yanyan Li
published in: Lobachevskii Journal of Mathematics
publication date: 2022-11-15T00:00:00Z
review text: The author considers nonwandering sets of \(n\)-dimensional cells, cylinders and tori (for \(n \geq 2\)) and investigates their structure for continuous skew product dynamical systems. The focus is on two particular cases. The first is where the set of periodic points is not empty (for self-maps of tori and cylinders) and closed for all maps. The second is the case where the set of periodic points is empty for self-maps of tori and cylinders and the nonwandering set is minimal. Using previous results found by the author and \textit{E. N. Makhrova} in [Russ. Math. Surv. 76, 821--881 (2021; Zbl 1496.37015)], some generalizations for skew products defined on compact plane rectangles are investigated. The paper includes proofs of the structure results and identification of the trajectories of points under the maps for both cases. In the first case, the maps are periodic; in the second, there is uniform recurrence but not periodicity. As the author notes, the topological entropy for the maps is zero in both cases.
Mathematics Subject Classification ID: 37E30
Mathematics Subject Classification ID: 37E25
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 37B40
zbMATH DE Number: 7617005
zbMATH Keywords: skew product
zbMATH Keywords: periodic point
zbMATH Keywords: nonwandering set
reviewed by: William J. Satzer Jr.
MaRDI profile type: Publication
full work available at URL: s1995080222100080
OpenAlex ID: W4312617374
cites work: Topological Entropy
cites work: Smooth triangular maps of the square with closed set of periodic points
cites work: Continuous maps of the circle without periodic points
cites work: A triangular map on \(I^2\) whose \(\omega\)-limit sets are all compact intervals of \(\{ 0\}\times I\).
cites work: On ω-limit Sets of Triangular Maps on the Unit Cube
cites work: On \(\Omega \)-blow-ups in the simplest \(C^{1}\)-smooth skew products of interval mappings
cites work: Morse-Smale Endomorphisms of the Circle
cites work: The Periodic Points of Morse-Smale Endomorphisms of the Circle
cites work: Homoclinic Points of Mappings of the Interval
cites work: Dynamics in one dimension
cites work: Q5608685
cites work: Entropy for Group Endomorphisms and Homogeneous Spaces
cites work: Q3796412
cites work: On the concept of \(\Omega\)-function for the skew product of interval mappings.
cites work: Q5228642
cites work: Differential properties and attracting sets of a simplest skew product of interval maps
cites work: Remarks on the Nonwandering Set of Skew Products with a Closed Set of Periodic Points of the Quotient Map
cites work: Absence of \(C^{1}\)-\(\Omega\)-explosion in the space of smooth simplest skew products
cites work: Boundary conditions for maps in fibers and topological transitivity of skew products of interval maps
cites work: Multivalued functions and nonwandering set of skew products of maps of an interval with complicated dynamics of quotient map
cites work: Nonwandering sets of \(C^1\)-smooth skew products of interval maps with complicated dynamics of quotient map
cites work: Dynamics of skew products of interval maps
cites work: One-dimensional dynamical systems
cites work: Geometrically integrable maps in the plane and their periodic orbits
cites work: The skew product on \(n\)-dimensional cell with transitive but not totally transitive \(n\)-dimensional attractor
cites work: On a class of totally topologically transitive skew products defined on cells in \(\mathbb{R}^n\), \(n\geq2\)
cites work: Nonwandering set of points of skew-product maps with base having closed set of periodic points
cites work: Rotation sets are closed
cites work: Q4840149
cites work: On Sharkovsky's cycle coexistence ordering
cites work: The triangular maps with closed sets of periodic points
cites work: Q5541832
cites work: Q3264374
cites work: Bifurcations and stability of families of diffeomorphisms
cites work: Q5658860
cites work: Maps of the Interval with Closed Periodic Set
cites work: Q5553902
cites work: Q5560901
cites work: Q3135828
description: scientific article; zbMATH DE number 7617005