Some topological characterizations of rational maps and Kleinian groups

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

Dimension rigidity in conformal structures

Developing the pioneering work of Lars Olsen [14], we deal with the question of continuity of the numerical value of Hausdorff measures of some natural families of conformal dynamical systems endowed with an appropriate natural topology. In particular, we prove such continuity for hyperbolic polynomials from the Mandelbrot set, and more generally for the space of hyperbolic rational functions of a fixed degree. We go beyond hyperbolicity by proving continuity for maps including parabolic rational functions, for example that the parameter $1/4$ is such a continuity point for quadratic polynomials $z\mapsto z^2+c$ for $c\in [0,1/4]$. We prove the continuity of the numerical value of Hausdorff measures also for the spaces of conformal expanding repellers and parabolic ones, more generally for parabolic Walters conformal maps. We also prove some partial continuity results for all conformal Walters maps; these are in general of infinite degree. In order to do this, as one of our tools, we provide a detailed local analysis, uniform with respect to the parameter, of the behavior of conformal maps around parabolic fixed points in any dimension. We also establish continuity of numerical values of Hausdorff measures for some families of infinite $1$-dimensional iterated function systems.

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.

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.

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.

Topological dynamics on nets

In this dissertation we focus on the application of thermodynamic formalism to non-autonomous and random dynamical systems. Specifically we use the thermodynamic formalism to investigate the dimension of various fractal constructions via the, now standard, technique of Bowen which he developed in his 1979 paper on quasi-Fuchsian groups. Bowen showed, roughly speaking, that the dimension of a fractal is equal to the zero of the relevant topological pressure function. We generalize the results of Rempe-Gillen and Urbanski on non-autonomous iterated function systems to the setting of non-autonomous graph directed Markov systems and then show that the Hausdorff dimension of the fractal limit set is equal to the zero of the associated pressure function provided the size of the alphabets at each time step do not grow too quickly. In trying to remove these growth restrictions, we present several other systems for which Bowen's formula holds, most notably ascending systems. We then use these various constructions to investigate the Hausdorff dimension of various subsets of the Julia set for different large classes of transcendental meromorphic functions of finite order which have been perturbed non-autonomously. In particular we find lower and upper bounds for the dimension of the subset of the Julia set whose points escape to infinity, and in many cases we find the exact dimension. While the upper bound was known previously in the autonomous case, the lower bound was not known in this setting, and all of these results are new in the non-autonomous setting. We also use transfer operator techniques to prove an almost sure invariance principle for random dynamical systems for which the thermodynamical formalism has been well established. In particular, we see that if a system exhibits a fiberwise spectral gap property and the base dynamical system is sufficiently well behaved, i.e. it exhibits an exponential decay of correlations, then the almost sure invariance principle holds. We then apply these results to uniformly expanding random systems like those studied by Mayer, Skorulski, and Urbanski and Denker and Gordin.

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

p-adic dynamical systems of (2,2)-rational functions with unique fixed point

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.

In this chapter we will embark upon the task of systematically identifying important specific phenomena associated with the asymptotic behavior of smooth dynamical systems. We will build upon the results of our survey of specific examples in Chapter 1 as well as on the insights gained from the general structural approach outlined and illustrated in Chapter 2. Most of the properties discussed in the present chapter are in fact topological invariants and can be defined for broad classes of topological dynamical systems, including symbolic ones. The predominance of topological invariants fits well with the picture that emerges from the considerations of Sections 2.1, 2.3, 2.4, and 2.6. The considerations of the previous chapter make it very plausible that smooth dynamical systems are virtually never differentiably stable and can only rarely be classified locally up to smooth conjugacy. In contrast, structural and the related topological stability seem to be fairly widespread phenomena. We will consider three broad classes of asymptotic invariants: (i) growth of the numbers of orbits of various kinds and of the complexity of orbit families, (ii) types of recurrence, and (iii) asymptotic distribution and statistical behavior of orbits. The first two classes are of a purely topological nature; they are discussed in the present chapter. The last class is naturally related to ergodic theory and hence we will provide an introduction to key aspects of that subject. This will require some space so we put that material into a separate chapter. The two chapters are intimately connected.

The topological one-way waveguide mode of the acoustic wave has recently been demonstrated in various meta-structure systems. Here, we show that in a topological phononic crystal with a symmetry-broken acoustic unit cell, the topological state possesses not just a “spin-like” pseudospin mode but also real spin angular momentum. By rotating the double-square units, the band of the phononic crystal will become inverse and induce both topological phase transition and spin angular momentum reversal. As such, by putting two topologically different systems together, the spin angular momentum dependent one-way interface modes can be selectively excited by acoustic spin sources, exhibiting robust transport protected by tight spin-momentum locking. The spin angular momentum density distribution in the unit cell and edge states shows that in addition to the pseudospin, there is a strong correlation between the real spin angular momentum and topological properties in this acoustic system, producing the topologically selective acoustic torque. Revealing the real spin angular momentum and associated acoustic spin torque properties of these topological phononic and acoustic systems will give people a more general understanding about symmetric breaking wave systems and help people to explore more potential applications of acoustic spins in various topological systems.