On piecewise-monotone mappings with closed set of periodic points on dendrites

2005 ◽  
Vol 126 (5) ◽  
pp. 1419-1435
Author(s):  
L. S. Efremova ◽  
E. N. Makhrova
Keyword(s):  
Periodic Points ◽  
Monotone Mappings ◽  
Closed Set ◽  
Piecewise Monotone

scholarly journals ω-limit sets for maps of the interval

1986 ◽  
Vol 6 (3) ◽  
pp. 335-344 ◽  
Author(s):  
Louis Block ◽  
Ethan M. Coven
Keyword(s):  
Periodic Points ◽  
Compact Interval ◽  
Minimal Set ◽  
Limit Sets ◽  
Piecewise Monotone ◽  
Continuous Map ◽  
Limit Points

AbstractLet f denote a continuous map of a compact interval to itself, P(f) the set of periodic points of f and Λ(f) the set of ω-limit points of f. Sarkovskǐi has shown that Λ(f) is closed, and hence ⊆Λ(f), and Nitecki has shown that if f is piecewise monotone, then Λ(f)=. We prove that if x∈Λ(f)−, then the set of ω-limit points of x is an infinite minimal set. This result provides the inspiration for the construction of a map f for which Λ(f)≠.

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

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 On the uniqueness of equilibrium states for piecewise monotone mappings

1990 ◽  
Vol 97 (1) ◽  
pp. 27-36 ◽  
Author(s):  
Manfred Denker ◽  
Gerhard Keller ◽  
Mariusz Urbański
Keyword(s):  
Equilibrium States ◽  
Monotone Mappings ◽  
Piecewise Monotone ◽  
Uniqueness Of Equilibrium

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.

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