scholarly journals Smooth Triangular Maps of the Square with Closed Set of Periodic Points

1995 ◽  
Vol 196 (3) ◽  
pp. 987-997 ◽  
Author(s):  
C. Arteaga
Keyword(s):  
Periodic Points ◽  
Closed Set ◽  
Triangular Maps

scholarly journals Graph maps whose periodic points form a closed set

2010 ◽  
Vol 371 (2) ◽  
pp. 649-654
Author(s):  
Jie-Hua Mai ◽  
Song Shao
Keyword(s):  
Periodic Points ◽  
Closed Set

On dynamics of triangular maps of the square

1992 ◽  
Vol 12 (4) ◽  
pp. 749-768 ◽  
Author(s):  
S. F. Kolyada
Keyword(s):  
Periodic Points ◽  
Triangular Maps

AbstractThe paper is devoted to the triangular maps of the square into itself. The results presented were recently obtained by the author and are briefly stated (in Russian) in a difficult paper as well as those (jointly published with A. N. Sharkovsky) published in ECIT-89 (abstract). All these results are systematized and extended by the new ones. The more detailed proofs of all statements are given. It is shown, for example, that triangular maps exist such that their minimal attraction centres do not coincide with the centres, as well as such ones exist that the Milnor attractor is not contained in the closure of the set of periodic points.

scholarly journals On skew-product maps with the base having a closed set of periodic points

2008 ◽  
Vol 85 (3-4) ◽  
pp. 441-445 ◽  
Author(s):  
Juan Luis García Guirao ◽  
Fernando López Pelayo
Keyword(s):  
Periodic Points ◽  
Skew Product ◽  
Closed Set ◽  
Product Maps

scholarly journals Jungck theorem for triangular maps and related results

2000 ◽  
Vol 1 (1) ◽  
pp. 83 ◽  
Author(s):  
M. Grinc ◽  
L. Snoha
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Periodic Points ◽  
Fixed Point Property ◽  
Compact Interval ◽  
Invariance Property ◽  
Point Property ◽  
Dimensional Cube ◽  
Triangular Maps

<p>We prove that a continuous triangular map G of the n-dimensional cube I<sup>n</sup> has only fixed points and no other periodic points if and only if G has a common fixed point with every continuous triangular map F that is nontrivially compatible with G. This is an analog of Jungck theorem for maps of a real compact interval. We also discuss possible extensions of Jungck theorem, Jachymski theorem and some related results to more general spaces. In particular, the spaces with the fixed point property and the complete invariance property are considered.</p>

Embedding topological dynamical systems with periodic points in cubical shifts

2015 ◽  
Vol 37 (2) ◽  
pp. 512-538 ◽  
Author(s):  
YONATAN GUTMAN
Keyword(s):  
Stokes Equations ◽  
Main Tool ◽  
Periodic Points ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Closed Set ◽  
Finite Dimensional ◽  
Topological Version ◽  
Mean Dimension ◽  
Topological Dynamical Systems

According to a conjecture of Lindenstrauss and Tsukamoto, a topological dynamical system $(X,T)$ is embeddable in the $d$-cubical shift $(([0,1]^{d})^{\mathbb{Z}},\text{shift})$ if both its mean dimension and periodic dimension are strictly bounded by $d/2$. We verify the conjecture for the class of systems admitting a finite-dimensional non-wandering set and a closed set of periodic points. This class of systems is closely related to systems arising in physics. In particular, we prove an embedding theorem for systems associated with the two-dimensional Navier–Stokes equations of fluid mechanics. The main tool in the proof of the embedding result is the new concept of local markers. Continuing the investigation of (global) markers initiated in previous work it is shown that the marker property is equivalent to a topological version of the Rokhlin lemma. Moreover, new classes of systems are found to have the marker property, in particular, extensions of aperiodic systems with a countable number of minimal subsystems. Extending work of Lindenstrauss we show that, for systems with the marker property, vanishing mean dimension is equivalent to the small boundary property.

scholarly journals On Diffeomorphisms of Compact 2-Manifolds with All Nonwandering Points Being Periodic

2015 ◽  
Vol 25 (14) ◽  
pp. 1540020 ◽  
Author(s):  
Suzanne Boyd ◽  
Juan L. G. Guirao ◽  
Michael Hero
Keyword(s):  
Dynamical System ◽  
Positive Result ◽  
Compact Manifold ◽  
Discrete Dynamical System ◽  
Periodic Points ◽  
Closed Set ◽  
Open Question ◽  
Future Work ◽  
Nonwandering Points

The aim of the present paper is to study conditions under which all the nonwandering points are periodic points, for a discrete dynamical system of two variables defined on a compact manifold. We include a survey of known results in all dimensions, and study the remaining open question in dimension two. We present two results, one negative and one positive. The negative result: we construct a Kupka–Smale diffeomorphism in [Formula: see text] (which can be extended to a diffeomorphism of the sphere) with a closed set of periodic points that differs from the set of nonwandering points. The positive result: we present a condition on the widely studied Hénon family which guarantees that all nonwandering points are periodic. Finally, we close by describing what future work may be needed to resolve our broad goals.

scholarly journals Triangular maps with all periods and no infinite ω-limit set containing periodic points

2005 ◽  
Vol 153 (5-6) ◽  
pp. 818-832 ◽  
Author(s):  
G.-L. Forti ◽  
L. Paganoni ◽  
J. Smítal
Keyword(s):  
Periodic Points ◽  
Limit Set ◽  
Triangular Maps
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