Embedding ℤk-actions in cubical shifts and ℤk-symbolic extensions

2010 ◽  
Vol 31 (2) ◽  
pp. 383-403 ◽  
Author(s):  
YONATAN GUTMAN
Keyword(s):  
Dynamical Systems ◽  
Universal Constant ◽  
Boundary Property ◽  
Entropy Theory ◽  
Topological Version ◽  
Symbolic Extension ◽  
Mean Dimension ◽  
Topological Dynamical Systems ◽  
Entropy Structure

AbstractMean dimension is an invariant which makes it possible to distinguish between topological dynamical systems with infinite entropy. Extending in part the work of Lindenstrauss we show that if (X,ℤk) has a free zero-dimensional factor then it can be embedded in the ℤk-shift on ([0,1]d)ℤk, where d=[C(k) mdim(X,ℤk)]+1 for some universal constant C(k), and a topological version of the Rokhlin lemma holds. Furthermore, under the same assumptions, if mdim(X,ℤk)=0, then (X,ℤk) has the small boundary property. One of the applications of this theory is related to Downarowicz’s entropy structure, a master invariant for entropy theory, which captures the emergence of entropy on different scales. Indeed, we generalize this invariant and prove the Boyle–Downarowicz symbolic extension entropy theorem in the setting of ℤk-actions. This theorem describes what entropies are achievable in symbolic extensions.

scholarly journals Conditional mean dimension

2021 ◽  
pp. 1-15
Author(s):  
BINGBING LIANG
Keyword(s):  
Dynamical Systems ◽  
Conditional Mean ◽  
Extension System ◽  
Mean Dimension ◽  
Factor Map ◽  
The Mean ◽  
Topological Dynamical Systems ◽  
Dynamical Embedding

Abstract We introduce some notions of conditional mean dimension for a factor map between two topological dynamical systems and discuss their properties. With the help of these notions, we obtain an inequality to estimate the mean dimension of an extension system. The conditional mean dimension for G-extensions is computed. We also exhibit some applications in dynamical embedding problems.

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 Q-entropy for general topological dynamical systems

2019 ◽  
Vol 39 (4) ◽  
pp. 2059-2075 ◽  
Author(s):  
Yun Zhao ◽  
◽  
Wen-Chiao Cheng ◽  
Chih-Chang Ho ◽  
◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Topological Dynamical Systems

Almost topological dynamical systems

1979 ◽  
Vol 34 (1-2) ◽  
pp. 139-160 ◽  
Author(s):  
Manfred Denker ◽  
Michael Keane
Keyword(s):  
Dynamical Systems ◽  
Topological Dynamical Systems

On the dynamics of the Tent function - Phase diagrams

2016 ◽  
Vol 7 (4) ◽  
pp. 261
Author(s):  
Prince Amponsah Kwabi ◽  
William Obeng Denteh ◽  
Richard Kena Boadi
Keyword(s):  
Dynamical Systems ◽  
Phase Diagrams ◽  
Initial Conditions ◽  
Asymptotic Properties ◽  
Topological Dynamical System ◽  
Topological Transitivity ◽  
Tent Map ◽  
One Dimensional ◽  
Definition Of ◽  
Topological Dynamical Systems

This paper focuses on the study of a one-dimensional topological dynamical system, the tent function. We give a good background to the theory of dynamical systems while establishing the important asymptotic properties of topological dynamical systems that distinguishes it from other fields by way of an example - the tent function. A precise definition of the tent function is given and iterates are clearly shown using the phase diagrams. By this same method, chaos in the tent map is shown as iterates become higher. We also show that the tent map has infinitely many chaotic orbits and also express some important features of chaos such as topological transitivity, boundedness and sensitivity to change in initial conditions from a topological viewpoint.

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