Uniform enveloping semigroupoids for groupoid actions

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

Given a topological dynamical system that admits mistakes and an invariant measure, this paper proves the Brin-Katok formula in this case. Consequently, this paper shows the Brin-Katok formula for the mean Bowen ball which replacing the Bowen metrics with the mean Bowen metrics.

Entropy sets are defined both topologically and for ameasure. The set of topological entropy sets isthe union of the sets of entropy sets for all invariant measures.For a topological system $(X,T)$ and an invariant measure $\mu$ on$(X,T)$, let $H(X,T)$ (resp. $H^\mu(X,T)$) be the closure of theset of all entropy sets (resp. $\mu$- entropy sets) in thehyperspace $2^X$. It is shown that if $h_{\text{top}}(T)>0$ (resp.$h_\mu(T)>0$), the subsystem $(H(X,T),\hat{T})$ (resp.$(H^\mu(X,T),\hat{T}))$ of $(2^X,\hat{T})$ has aninvariant measure with full support and infinite topologicalentropy. Weakly mixing sets and partial mixing of dynamical systems are introduced and characterized. It is proved that if $h_{\text{top}}(T)>0$ (resp.$h_\mu(T)>0$) the set of all weakly mixing entropy sets (resp. $\mu$-entropysets) is a dense $G_\delta$ in$H(X,T)$ (resp. $H^\mu(X,T)$). A Devaney chaotic but not partly mixing system is constructed. Concerning entropy capacities, it is shown that when $\mu$ is ergodicwith $h_\mu(T)>0$, the set of all weakly mixing $\mu$-entropy sets $E$ suchthat the Bowen entropy $h(E)\ge h_\mu(T)$is residual in $H^\mu(X,T)$. When in addition$(X,T)$ is uniquely ergodic the set of all weakly mixing entropy sets $E$with $h(E)=h_{\text{top}}(T)$ is residual in $H(X,T)$.

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.

Let (M, g) be a complete Riemannian manifold, L=\Delta -\nabla \phi \cdot \nabla be a Markovian symmetric diffusion operator with an invariant measure d\mu(x)=e^{-\phi(x)}d\nu(x) , where \phi\in C^2(M) , \nu is the Riemannian volume measure on (M, g) . A fundamental question in harmonic analysis and potential theory asks whether or not the Riesz transform R_a(L)=\nabla(a-L)^{-1/2} is bounded in L^p(\mu) for all 1<p<\infty and for certain a\geq 0 . An affirmative answer to this problem has many important applications in elliptic or parabolic PDEs, potential theory, probability theory, the L^p -Hodge decomposition theory and in the study of Navier-Stokes equations and boundary value problems. Using some new interplays between harmonic analysis, differential geometry and probability theory, we prove that the Riesz transform R_a(L)=\nabla(a-L)^{-1/2} is bounded in L^p(\mu) for all a>0 and p\geq 2 provided that L generates a ultracontractive Markovian semigroup P_t=e^{tL} in the sense that P_t 1=1 for all t\geq 0 , \|P_t\|_{1, \infty} < Ct^{-n/2} for all t\in (0, 1] for some constants C>0 and n > 1 , and satisfies (K+c)^{-}\in L^{{n\over 2}+\epsilon}(M, \mu) for some constants c\geq 0 and \epsilon>0 , where K(x) denotes the lowest eigenvalue of the Bakry-Emery Ricci curvature Ric(L)=Ric+\nabla^2\phi on T_x M , i.e., K(x)=\inf\limits\{Ric(L)(v, v): v\in T_x M, \|v\|=1\}, \quad\forall\ x\in M. Examples of diffusion operators on complete non-compact Riemannian manifolds with unbounded negative Ricci curvature or Bakry-Emery Ricci curvature are given for which the Riesz transform R_a(L) is bounded in L^p(\mu) for all p\geq 2 and for all a>0 (or even for all a\geq 0 ).

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.

Let G be a topological locally compact, Hausdorff and second countable groupoid with a Haar system and K a proper subgroupoid of G with a Haar system too. (G, K) is an internally Gelfand pair if for any u in the unit space, the algebra of bi-K(u)-invariant functions on G(u) is commutative under convolution. In this work, we give some characterizations of these pairs and extend to this context some classical results of harmonic analysis.

We consider attractors for certain types of random dynamical systems. These are skew–product systems whose base transformations preserve an ergodic invariant measure. We discuss definitions of invariant sets, attractors and invariant measures for deterministic and random dynamical systems. Under assumptions that include, for example, iterated function systems, but that exclude stochastic differential equations, we demonstrate how random attractors can be seen as examples of Milnor attractors for a skew–product system. We discuss the minimality of these attractors and invariant measures supported by them. As a further connection between random dynamical systems and deterministic dynamical systems, we show how dynamical or D–bifurcations of random attractors with multiplicative noise can be seen as blowout bifurcations, and we relate the issue of branching at such D–bifurcations to branching at blowout bifurcations.

We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from zero, we obtain finitely many ergodic absolutely continuous invariant probability measures, describing the asymptotics of almost every point. We also prove a similar result for higher-dimensional random non-uniformly expanding dynamical systems. The results are consequences of the construction of such measures for skew-products with essentially arbitrary base dynamics and asymptotic expansion along the fibers. In both cases our method deals with either critical o singular points for the random maps.

We introduce two properties of dynamical systems on Polish metric spaces: closeability and linkability. We show that they imply density of ergodic measures in the space of invariant probability measures and the existence of a generic point for every invariant measure. In the compact case, it follows from our conditions that the set of invariant measures is either a singleton of a measure concentrated on a periodic orbit or the Poulsen simplex. We provide examples showing that closability and linkability are independent properties. Our theory applies to systems with the periodic specification property, irreducible Markov chains over a countable alphabet, certain coded systems including $\unicode[STIX]{x1D6FD}$-shifts and $S$-gap shifts, $C^{1}$-generic diffeomorphisms of a compact manifold $M$ and certain geodesic flows of a complete connected negatively curved manifold.

We use the concept of enveloping Ellis semigroups in order to classify the behaviour of dynamical systems on intervals. Several notions of chaos and non-chaos are expressed by means of algebraic properties of the corresponding Ellis semigroups.

Fourier–Stieltjes Algebras of Locally Compact Groupoids

Chapter 27 - Measures on Algebraic—Topological Structures

For minimal $\mathbb{Z}^{2}$-topological dynamical systems, we introduce a cube structure and a variation of the usual regional proximality relation for $\mathbb{Z}^2$ actions, which allow us to characterize product systems and their factors. We also introduce the concept of topological magic systems, which is the topological counterpart of measure theoretic magic systems introduced by Host in his study of multiple averages for commuting transformations. Roughly speaking, magic systems have less intricate dynamics, and we show that every minimal $\mathbb{Z}^2$ dynamical system has a magic extension. We give various applications of these structures, including the construction of some special factors in topological dynamics of $\mathbb{Z}^2$ actions and a computation of the automorphism group of the minimal Robinson tiling.

The dynamical system of ODEs is about closed-form researches into the field of ODEs from the perspective of dynamical systems. This paper, starting with the research of path in autonomous differential equations and discussion on Poincaré’s viewpoints, probes into the complicated topological dynamic system of ODEs.