Topological entropy for appropriately approximated C*-algebrasa)

This work delves into the theory of dynamic systems, focusing on the analysis of entropy in both classical and topological contexts. Beginning with an exposition of fundamental concepts in dynamical systems theory, particular attention is given to topological dynamical systems (TDS). The discussion progresses to explore discrete topological entropy and its significance within dynamical systems, culminating in the introduction of topological entropy pressure as a nuanced form of this concept. The study then investigates various applications of topological entropy within dynamic systems, emphasizing its utility in understanding chaotic systems and its role in ergodic theory. A novel theory, termed Topological Ergodic Entropy Theory (TEET), is presented, offering a fresh perspective on the analysis of ergodic dynamical systems. Furthermore, the work introduces the Ergodic Theory of Turbulent Flow (ETTF), which probes the interplay between topological entropy and the ergodic properties of dynamic systems governed by the Navier-Stokes equations. Through these explorations, the findings contribute significantly to the comprehension of the intricate nature of dynamical systems and their diverse applications across mathematics and physics. By scrutinizing topological entropy and its implications in dynamical systems, this research offers novel insights into the chaotic and stochastic behaviors exhibited by these systems. Additionally, the introduction of pioneering theories like ETTF opens up new avenues for understanding and modeling turbulent phenomena, thereby enriching our understanding of complex dynamical processes.

By using the Carathéodory-Pesin structure (C-P structure), the topological entropy on the whole space introduced for a proper map, is generalized to the cases of arbitrary subset, i.e., we introduce three notions of topological entropy. Some of the properties of these notions are provided. As some applications, for the proper map of locally compact separable metric space, we prove the following variational principles: (1) The upper capacity topological entropy on any subset and the minimum of the Bowen-Dinaburg entropies always coincide; (2) For any invariant probability measure, the measure-theoretic entropy and the infimum of the topological entropies on all sets which are of full measures always coincide; (3) The relationship between the topological entropies of level sets of the ergodic average of some continuous functions and the measure-theoretic entropies are given. These are the extensions of results of Patrão and Pesin, etc.

In the present paper, we introduce a natural extension of Adler, Konheim and MacAndrew topological entropy for proper maps of locally compact separable metrizable spaces and prove a variational principle which states that this topological entropy, the supremum of the Kolmogorov–Sinai entropies and the minimum of the Bowen entropies always coincide. We apply this variational principle to show that the topological entropy of automorphisms of simply connected nilpotent Lie groups always vanishes. This shows that the Bowen formula for the Bowen entropy of an automorphism of a non-compact Lie group with respect to some invariant metric is just an upper bound for its topological entropy.

Topological entropy for locally linearly compact vector spaces

In this paper, we consider two questions about topological entropy of dynamical systems. We propose to resolve these questions by the same approach of using étale analogs of topological and algebraic dynamical systems. The first question is to define topological entropy for a topological dynamical system ( f , X , Ω ) (f,X,\Omega ) . The main idea is to make use of, in addition to invariant compact subspaces of ( X , Ω ) (X,\Omega ) , compactifications of étale covers π : ( f ′ , X ′ , Ω ′ ) → ( f , X , Ω ) \pi :(f’,X’,\Omega ’)\rightarrow (f,X,\Omega ) ; that is, π ∘ f ′ = f ∘ π \pi \circ f’=f\circ \pi and the fibers of π \pi are all finite. We prove some basic results and propose a conjecture, whose validity allows us to prove further results. The second question is to define topological entropy for algebraic dynamical systems, with the requirement that it should be as close to the pullback on cohomology groups as possible. To this end, we develop an étale analog of algebraic dynamical systems.

The relationship between a description of a multicomponent mixture based on the entropy functional with the terms quadratic in the density gradients of components and in the temperature on the one hand, and a description within the framework of classical and quantum statistical mechanics on the other hand, is investigated. Explicit expressions for the entropy functional are written in the context of classical and quantum theories. A quadratic approximation is then calculated for the case of small perturbations of a uniform state. The terms quadratic in the gradients are separated in this approximation. This allows ab initio calculations of the corresponding phenomenological coefficients to be done.

This thesis is dedicated to studying the theory of entropy and its relation to the Kolmogorov complexity. Originating in physics, the notion of entropy was introduced to mathematics by C. E. Shannon as a way of measuring the rate at which information is coming from a data source. There are, however, a few different ways of telling how much information there is: an alternative approach to quantifying the amount of information is the Kolmogorov complexity, which was proposed by A. N. Kolmogorov. The Shannon entropy is the key ingredient in the definition of the Kolmogorov-Sinai entropy of a measure-preserving systems. Roughly speaking, the Kolmogorov-Sinai entropy is the expected amount of information in `Shannon sense' that one obtains per unit of time by observing the evolution of a measure-preserving system. In topological dynamics, the topological entropy takes place of the Kolmogorov-Sinai entropy. For metrizable systems, the topological entropy measures the exponential growth rate of the number of distinguishable partial orbits of length n as n tends to infinity. Originally defined for Z-actions, the 'classical' theories of entropy were later extended to actions of amenable groups. We provide a necessary background on amenable groups, topological/measurepreserving dynamics and the entropy theory in Chapters 1, 2, 3 and 5. The main focus of this thesis is extending the following results. First of all, a common generalization of the topological and the Kolmogorov-Sinai entropy theories for Z-systems was suggested by G. Palm. We provide an abstract generalization of the work of Palm for actions of discrete amenable groups in the language of measurement functors in Chapter 6. Secondly, we investigate the connection of entropy and Kolmogorov complexity. Originally, the equality between the topological entropy and a certain quantity measuring maximal asymptotic Kolmogorov complexity of the trajectories was established by A. A. Brudno for subshifts over Z. Later, he proved the equality of the Kolmogorov-Sinai entropy and the asymptotic Kolmogorov complexity of almost every trajectory for ergodic subshifts over Z. We provide a generalization of these results as follows. Firstly, in Chapter 4 we give a background on computability and Kolmogorov complexity and, further, introduce computable Folner monotilings, which are central in our extensions of Brudno's results. We treat the 'first' and the 'second' theorems of Brudno in Chapter 7. The first theorem is generalized for subshifts over computable groups admitting computable Folner monotilings, while the second theorem is proved under the assertion of regularity of the monotiling, which we introduce in Chapter 7 as well.

Topological entropy for discontinuous semiflows

We define the concept of continuum-wise expansiveness for set-valued functions. We prove that for a compact metric space X X with positive topological dimension, if a set-valued function is c w cw -expansive, then its topological entropy is positive. We also introduce the notion of point-wise specification property for set-valued functions and prove that set-valued functions with this property have positive entropy.

Topological R-entropy and topological entropy of free semigroup actions

About 5 years ago, Dai, Zhou and Geng proved the following result. If $X$ is a metric compact space and $f:X\to X$ a Lipschitz continuous map, then the Hausdorff dimension of $X$ is bounded from below by the topological entropy of $f$ divided by the logarithm of its Lipschitz constant. We show that this is a simple consequence of a 30 years old Bowen's definition of topological entropy for noncompact sets. Moreover, a modification of this definition provides a new insight into the entropy of subshifts of finite type.

We introduce a notion of entropy for automorphisms of discrete groups which admit amenable actions on a compact space. This entropy is dual to classical topological entropy in the sense that if G is discrete and abelian then our notion of entropy agrees with the topological entropy of the induced automorphism on the (compact) dual group of G. We prove a number of basic properties of this dual entropy and give a few calculations. In particular, we are able to give precise calculations for arbitrary automorphisms of crystallographic groups.

The literature contains several extensions of the standard definitions of topological entropy for a continuous self-map $f: X \rightarrow X$ from the case when $X$ is a compact metric space to the case when $X$ is allowed to be noncompact. These extensions all require the space $X$ to be totally bounded, or equivalently to have a compact completion, and are invariants of uniform conjugacy. When the map $f$ is uniformly continuous, it extends continuously to the completion, and the various notions of entropy reduce to the standard ones (applied to this extension). However, when uniform continuity is not assumed, these new quantities can differ. We consider extensions proposed by Bowen (maximizing over compact subsets and a definition of Hausdorff dimension type) and Friedland (using the compactification of the graph of $f$) as well as a straightforward extension of Bowen and Dinaburg's definition from the compact case, assuming that $X$ is totally bounded, but not necessarily compact. This last extension agrees with Friedland's, and both dominate the one proposed by Bowen (Theorem 6). Examples show how varying the metric outside its uniform class can vary both quantities. The natural extension of Adler--Konheim--McAndrew's original (metric-free) definition of topological entropy beyond compact spaces dominates these other notions, and is unfortunately infinite for a great number of noncompact examples.

Topological sequence entropy and topological dynamics of tree maps

Topological entropy of nonautonomous dynamical systems