Étale dynamical systems and topological entropy

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.

The boundary quotient for algebraic dynamical systems

The paper introduces a new method for finding optimal control of algebraic dynamic systems. The structure of algebraic dynamical systems is nonlinear with quadratic and bilinear terms. A new hybrid extended Fourier series is introduced, and state and control variables of the system are expanded by this series. Moreover, properties of new series are presented, and integration and product operational matrices are obtained. Using operational matrices, optimal control of the systems is converted to a set of simultaneous nonlinear algebraic relations. An illustrative example is included to compare our results with those in the literature.

Existence of a measure with maximal entropy (that is, an invariant measure whose measure-theoretic entropy is equal to the topological entropy of the system) is an important subject in dynamical system theory. Weak expansiveness of the dynamical system, including h-expansiveness and asymptotic h-expansiveness, provides a non-trivial sufficient condition for existence of such measures. In this paper we give new characterizations for weak expansiveness of actions by amenable groups, by interpreting equivalently topological conditional entropy of such actions using either topological entropy of subsets in the sense of separated and spanning sets or ideas of Bowen’s dimensional entropy of subsets. Our results extend not only the well-known results about weak expansiveness by Bowen (1972, 1973), Marczyńska (1977) and Misiurewicz (1976) from single transformations to amenable group actions, but also a recent result relating topological entropy to Bowen’s dimensional entropy for amenable group actions which was proved first by Zheng and Chen (2016) using measure-theoretic methods and then re-proved by Dou and Zhang (2018) via purely topological tools.

‘‘Classical’’ topological entropy is one of the main numerical invariants in topological dynamics on compact spaces. Here, the author’s recent development of a noncommutative generalization of topological entropy, in the natural setting of general C*-algebras as the noncommutative counterpart of continuous function algebras on compact spaces, is presented in a slightly modified and improved form. This includes both a survey of earlier results with some important corrections, and also new general results in response to (and inspired by) a more recent counterproposal for a noncommutative topological entropy by Thomsen. Finally, some partially new examples for the calculation of the defined topological entropy are shown. The rather self-evident physical interpretation in the framework of (operator-algebraic) quantum statistical mechanics and of ‘‘chaotic’’ quantum dynamical systems is briefly touched upon.

A Fej\'{e}r-type theorem is proved within the framework of $C^*$-algebras associated with certain irreversible algebraic dynamical systems. This makes it possible to strengthen a result on the structure of the relative commutant of a family of generating isometries in a boundary quotient.

In Chapter 10 we showed how a topological dynamical system \( (K;\varphi) \) gives rise to a measure-preserving system by choosing a \( \varphi \)-invariant measure on K. The existence of such a measure is guaranteed by the Krylov–Bogoljubov Theorem 10.2. In general there can be many invariant measures, and we also investigated how minimality of the topological system is reflected in properties of the associated measure-preserving system. It is now our aim to go in the other direction: starting from a given measure-preserving system we shall construct some topological system (sometimes called topological model) and an invariant measure so that the resulting measure-preserving system is isomorphic to the original one. By doing this, methods from the theory of topological dynamical system will be available, and we may gain further insights into measure-preserving systems. Thus, by switching back and forth between the measure theoretic and the topological situation, we can deepen our understanding of dynamical systems. In particular, this procedure will be carried out in Chapter 17

This paper deals with the relation between topological entropy and e-entropy of the space of segments of trajectories of dynamical systems, as well as the relation between topological and metrical entropies. The results obtained are applied to some classical dynamical systems.

Traces of algebraic integers and dynamical systems

Given a matroid M on the ground set E, the Bergman fan e B(M), or space of M- ultrametrics, is a polyhedral complex in R E which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics. Motivated by the phylogenetic situ- ation, we study the following problem: Given a point ω in R E , we wish to nd an M-ultrametric which is closest to it in the '¥-metric. The solution to this problem follows easily from the existence of the subdominant M-ultramet- ric: a componentwise maximum M-ultrametric which is componentwise smaller than ω. A pro- cedure for computing it is given, which brings together the points of view of matroid theory and tropical geometry. When the matroid in question is the graphical matroid of the complete graph Kn, the Bergman fan e B(Kn) parameterizes the equidistant phylogenetic trees with n leaves. In this case, our results provide a conceptual explanation for Chepoi and Fichet's method for computing the tree that most closely matches measured data.

The equidistribution conjecture is proved for general semiabelian varieties over number fields. Previously, this conjecture was only known in the special case of almost split semiabelian varieties through work of Chambert-Loir. The general case has remained intractable so far because the height of a semiabelian variety is negative unless it is almost split. In fact, this places the conjecture outside the scope of Yuan’s equidistribution theorem on algebraic dynamical systems. To overcome this, an asymptotic adaption of the equidistribution technique invented by Szpiro, Ullmo, and Zhang is used here. It also allows a new proof of the Bogomolov conjecture and hence a self-contained proof of the strong equidistribution conjecture in the same general setting.

We prove a Hilbert-Samuel type result of arithmetic big line bundles in Arakelov geometry, which is an analogue of a classical theorem of Siu. An application of this result gives equidistribution of small points over algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also generalize Chambert-Loir's non-archimedean equidistribution.

The aim of this course is to present methods coming from quasiconformal geometry in metric spaces which can be used to characterize conformal dynamical systems. We will focus on some specific classes of rational maps and of Kleinian groups (semi-hyperbolic rational maps and convex-cocompact Kleinian groups). These classes can be characterized among conformal dynamical systems by topological properties, which will enable us to define classes of dynamical systems on the sphere (coarse expanding conformal maps and uniform convergence groups). It turns out that these topological dynamical systems carry some non-trivial geometric information enabling us to associate a coarse conformal structure invariant by their dynamics. This conformal structure will be derived from hyperbolic geometry in the sense Gromov. We associate to this conformal structure a numerical invariant, the Ahlfors regular conformal dimension, which will contain the information that such topological dynamical systems are conjugate to genuine conformal dynamical systems.

Topological entropy measures the number of distinguishable orbits in a dynamical system, thereby quantifying the complexity of chaotic dynamics. One approach to computing topological entropy in a two-dimensional space is to analyze the collective motion of an ensemble of system trajectories taking into account how trajectories "braid" around one another. In this spirit, we introduce the Ensemble-based Topological Entropy Calculation, or E-tec, a method to derive a lower-bound on topological entropy of two-dimensional systems by considering the evolution of a "rubber band" (piece-wise linear curve) wrapped around the data points and evolving with their trajectories. The topological entropy is bounded below by the exponential growth rate of this band. We use tools from computational geometry to track the evolution of the rubber band as data points strike and deform it. Because we maintain information about the configuration of trajectories with respect to one another, updating the band configuration is performed locally, which allows E-tec to be more computationally efficient than some competing methods. In this work, we validate and illustrate many features of E-tec on a chaotic lid-driven cavity flow. In particular, we demonstrate convergence of E-tec's approximation with respect to both the number of trajectories (ensemble size) and the duration of trajectories in time.

Let (X, T) be a topological dynamical system (TDS), Ω(T) be the set of non-wondering points and \({h_{top}^{s}(T)=\sup_{A} h_{top}^A(T)}\) be the topological sequence entropy. In this paper, an example on a countable compactum X with \({h_{top}^{s}(T)\neq h_{top}^{s}(T|_{\Omega(T)})}\) is given. Then for TDSs on countable compacta X, it is proved that when d(X) ≤ 1, \({h_{top}^{s}(T)=0}\); and when d(X) ≥ 2, there exists a homeomorphism T on X such that Xd is the sequence entropy set of (X, T), where d(X) and Xd are the derived degree of X and the set of all accumulation points of X respectively.