A survey of some aspects of dynamical topology: Dynamical compactness and Slovak spaces

Sergiĭ Kolyada

doi:10.3934/dcdss.2020074

Article Contents

2020, Volume 13, Issue 4: 1291-1317. Doi: 10.3934/dcdss.2020074

This issue Previous Article Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem Next Article Coordinate-independent criteria for Hopf bifurcations

A survey of some aspects of dynamical topology: Dynamical compactness and Slovak spaces

Sergiĭ Kolyada^†

Institute of Mathematics, NASU, Tereshchenkivs'ka 3, 01601 Kyiv, Ukraine

Dedicated to Professor Jürgen Scheurle on the occasion of his 65th birthday
^† Editors' note: Professor Sergiĭ Kolyada passed away on May 16, 2018. He will be missed by the mathematical community, as a mathematician and as a person. Due to his untimely death, Professor Kolyada could not implement the changes to the first version of the manuscript as suggested in the (positive) reviews. With the consent of Professor Kolyada's family, Professor L'ubomír Snoha (Matej Bel University, Banská Bystrica, Slovakia), a friend and colleague of Professor Kolyada, assumed the responsibility of carrying out the revision. The editors thank Professor Snoha for this invaluable contribution

Received: January 2018

Revised: September 2018

Published: April 2020

This survey is based on lectures given by the author at the Max Planck Institute for Mathematics, Technical University of Munich, Paris-Sud University, Luminy Institute of Mathematics, Institute of Mathematics of Jussieu and several other mathematical departments in 2017

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Abstract

The area of dynamical systems where one investigates dynamical properties that can be described in topological terms is "Topological Dynamics". Investigating the topological properties of spaces and maps that can be described in dynamical terms is in a sense the opposite idea. This area has been recently called "Dynamical Topology". As an illustration, some topological properties of the space of all transitive interval maps are described. For (discrete) dynamical systems given by compact metric spaces and continuous (surjective) self-maps we survey some results on two new notions: "Slovak Space" and "Dynamical Compactness". A Slovak space, as a dynamical analogue of a rigid space, is a nontrivial compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Dynamical compactness is a new concept of chaotic dynamics. The omega-limit set of a point is a basic notion in the theory of dynamical systems and means the collection of states which "attract" this point while going forward in time. It is always nonempty when the phase space is compact. By changing the time we introduced the notion of the omega-limit set of a point with respect to a Furstenberg family. A dynamical system is called dynamically compact (with respect to a Furstenberg family) if for any point of the phase space this omega-limit set is nonempty. A nice property of dynamical compactness is that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.

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Mathematics Subject Classification: Primary: 37B05; Secondary: 54H20, 37B40.

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Figure 1. The space $\mathcal{TCS}_1\cup\mathcal{TCS}_2$

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Figure 2. Box map

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Figure 3. Boxes $I_i\times I_j$

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Figure 4. Basic loop $L_2$. It consists of four arcs represented by the four rows in this picture (instead of all elements of such an arc only five of them are shown)

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Figure 5. Deformation of the 1st arc of $L_2$

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Figure 6. Deformation of the 2nd arc of $L_2$

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Figure 7. Deformation of the 3rd arc of $L_2$

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Figure 8. Deformation of the 4th arc of $L_2$

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Figure 9. From $L_2$ to auxiliary loop consisting of two arcs

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Figure 10. The 1st arc of the auxiliary loop obtained from $L_2$

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Figure 11. The 2nd arc of the auxiliary loop obtained from $L_2$

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Figure 12. Topologically transitive systems

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Figure 13. Topologically transitive, non-proximal systems

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Figure 14. The first 4 steps in the construction of the Sierpinski carpet

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Figure 15. Steps in the construction of the De Groot - Wille rigid plane continuum

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Figure 16. The Julia set for the map $z \mapsto (z^2+0.3+0.05i)/(z^2-1)$

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Figure 17. The first 5 steps in the construction of the solenoid called the Smale-Williams attractor

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Figure 18. Composant $\bar{\gamma}$ of the Slovak space

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Table 1. Transitive maps in $\mathcal{CS}_2 $

$\mathcal{CS}_2$
code	picture	condition equivalent to transitivity
$ (a, 1, 0, d) $		$ d\leq a-4 + \frac{2}{a} \quad \text{or} \quad 1-a \leq (1-d) - 4 + \frac{2}{1-d} $
$ (a, 0, 1, d) $		$ a>d $
$ (1, 0, c, d) $		$ d \leq 2 + 2c - \frac{1}{c} $
$ (a, b, 1, 0) $		$ 1-a \leq 2 + 2(1-b) - \frac{1}{1-b} $

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