Topological dynamics on nets

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 contributes to the study of topological symplectic dynamical systems, and hence to the extension of smooth symplectic dynamical systems. Using the positivity result of symplectic displacement energy [4], we prove that any generator of a strong symplectic isotopy uniquely determine the latter. This yields a symplectic analogue of a result proved by Oh [12], and the converse of the main theorem found in [6]. Also, tools for defining and for studying the topological symplectic dynamical systems are provided: We construct a right-invariant metric on the group of strong symplectic homeomorphisms whose restriction to the group of all Hamiltonian homeomorphism is equivalent to Oh’s metric [12], define the topological analogues of the usual symplectic displacement energy for non-empty open sets, and we prove that the latter is positive. Several open conjectures are elaborated.

We prove that any uniformly exponentially stable linear cocycle of matrices defined over a topological dynamical system can be reduced via suitable change of variables to a linear cocycle whose generator has a spectral norm less than 1 at each point. Moreover, we establish an analogous result for nonuniformly exponentially stable linear cocycles, i.e. cocycles with negative Lyapunov exponents. In addition, by using tools from ergodic theory, we obtain converse results that give conditions for (non)uniform exponential stability of linear cocyles under the assumption that they admit a suitable reduction to a linear cocycle whose generator has a spectral norm less than 1 at points that form a ‘large’ subset of our topological dynamical system in the appropriate sense. Finally, we discuss the versions of our results in the case of (non)uniformly hyperbolic linear cocyles.

On the Banach ⁎-algebra crossed product associated with a topological dynamical system

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.

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.

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 paper analyzes dynamic epistemic logic from a topological perspective. The main contribution consists of a framework in which dynamic epistemic logic satisfies the requirements for being a topological dynamical system thus interfacing discrete dynamic logics with continuous mappings of dynamical systems. The setting is based on a notion of logical convergence, demonstratively equivalent with convergence in Stone topology. Presented is a flexible, parametrized family of metrics inducing the latter, used as an analytical aid. We show maps induced by action model transformations continuous with respect to the Stone topology and present results on the recurrent behavior of said maps.

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.

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weightA, denoted by DA(G), is defined to be the least positive integer t such that, for every sequence (x1,..., xt) with xi ∈ G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and ai ∈ A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, EA(G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some ai ∈ A. When G is of order n, one considers A to be a non-empty subset of {1,..., n − 1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote EA(G) and DA(G) by EA(n) and DA(n) respectively.

Topological pressure and large deviations for multivariate potentials

This chapter defines the (dynamical) core of a topological polynomial (and the associated lamination). This notion extends that of the core of a unimodal interval map. Two explicit descriptions of the core are given: one related to periodic objects and one related to critical objects. Topological polynomials are topological dynamical systems that generalize complex polynomials with locally connected Julia sets restricted to their Julia sets and considered up to topological conjugacy. This chapter aims to illustrate the analogy between the dynamics of topological polynomials on their cutpoints and cut-atoms and interval dynamics. For example, it is known that for interval maps, periodic points and critical points play a significant, if not decisive, role. This chapter thus attempts to establish similar facts for topological polynomials.

Mean 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.

In this paper the historic set is divided into different level sets and we use topological pressure to describe the size of these level sets. We give an application of these results to dimension theory. Our primary focus is using topological pressure to describe the relative multifractal spectrum of ergodic averages and to give a positive answer to the conjecture posed by Olsen (2003 J. Math. Pures Appl. 82 1591–649).

We show that every topological k-graph constructed from a locally compact Hausdorff space {\Omega} and a family of pairwise commuting local homeomorphisms on {\Omega} satisfying a uniform boundedness condition on the cardinalities of inverse images may be realized as a semigroup crossed product in the sense of Larsen.