Topics in Dynamics and Ergodic Theory

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.

Chapter 27 - Measures on Algebraic—Topological Structures

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.

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.

We investigate some connections between the Fraiss? theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. We show, in particular, that results from the structural Ramsey theory can be quite useful in recognizing the universal minimal flows of this kind of groups. As result we compute universal minimal flows of several well known topological groups such as, for example, the automorphism group of the random graph, the automorphism group of the random triangle-free graph, the automorphism group of the ?-dimensional vector space over a finite field, the automorphism group of the countable atomless Boolean algebra, etc. So we have here a reversal in the traditional relationship between topological dynamics and Ramsey theory, the Ramsey-theoretic results are used in proving theorems of topological dynamics rather than vice versa.

In answer to a question of Macpherson and Neumann related to the classification of maximal subgroups of infinite symmetric groups, we characterize set ideals on an infinite set X whose stabilizers in the symmetric group Sym(X) are maximal subgroups. Specifically, for an ideal I containing all subsets of cardinality <∣X∣, the stabilizer S{I} is maximal if and only if the corresponding Boolean dynamical system (X, I, S{I}) is minimal, that in turn can be expressed as an intrinsic structural property of the Boolean space (X, I). The general case is reduced to a minimal dynamical system on a subset. This characterization clarifies many earlier results, and reveals an interesting link between the theory of infinite permutation groups, topological dynamics and ergodic theory. 1991 Mathematics Subject Classification 20B35, 20B27.

The paper introduces a cohomological approach to the outer conjugacy problem in ergodic theory. Specifically, the following fact is proved: up to an isomorphism of an approximately finite type II ergodic full group there exists only one cohomological class of cocycles with dense range in a given amenable group. This result is used to establish outer conjugacy for strictly outer actions of continuous unimodular amenable subgroups of the normalizer of the full group generated by an ergodic type II automorphism. As a special case we show that outer conjugacy of compact groups reduces merely to conjugation. Besides that the correlation is established between the automorphisms of a principal groupoid with continuous orbits and ones of its discrete reduction. Also the correspondence is found between the automorphisms of groupoids with continuous orbits and those of the associated von Neumann algebras.

In this paper we consider a bootstrap class $$\mathfrak {C}$$ of countable discrete groups, which is closed under countable unions and extensions by the integers, and we study actions of such groups on $$\mathrm {C}^*$$ -algebras. This class includes all torsion-free abelian groups, poly- $$\mathbb {Z}$$ -groups, as well as other examples. Using the interplay between relative Rokhlin dimension and semi-strongly self-absorbing actions established in prior work, we obtain the following two main results for any group $$\Gamma \in \mathfrak {C}$$ and any strongly self-absorbing $$\mathrm {C}^*$$ -algebra $$\mathcal {D}$$ : In fact we establish more general relative versions of these two results for actions of amenable groups that have a predetermined quotient in the class $$\mathfrak {C}$$ . For the monotracial case, the proof comprises an application of Matui–Sato’s equivariant property (SI) as a key method.

We introduce topological pressure for continuous actions of countable sofic groups on compact metrizable spaces. This generalizes the classical topological pressure for continuous actions of countable amenable groups on such spaces. We also establish the variational principle for topological pressure in this sofic context.

We show that if $M$ is a compact manifold, then there is a residual subset $\mathcal {N}$ of the set of homeomorphisms on $M$ with the property that if $f\in \mathcal {N}$, then the periodic points of $f$ are dense in its chain recurrent set. This result was first announced in [J. Palis, C. Pugh, M. Shub, M. Sullivan, Genericity theorems in topological dynamics, Dynamical Systems – Warwick 1974 (Springer Lect. Notes in Math. #468), Springer-Verlag, New York, 1975, pp. 241–250], but a flaw in that argument was noted in [E.M. Coven, J. Madden, Z. Nitecki, A note on generic properties of continuous maps, Ergodic Theory and Dynamical Systems II, Boston, Birkhäuser, 1982, pp. 97–101], where a different proof was given. It was recently noted in [S.Y. Pilyugin, The Space of Dynamical Systems with the $C^0$ Topology, (Springer Lect. Notes in Math #1571), Springer-Verlag, New York, 1994.] that this new argument only serves to show that the density of periodic points in the chain recurrent set is generic in the closure of the set of diffeomorphisms. We show how to patch the original argument from [J. Palis, C. Pugh, M. Shub, M. Sullivan, Genericity theorems in topological dynamics, Dynamical Systems – Warwick 1974 (Springer Lect. Notes in Math. #468), Springer-Verlag, New York, 1975, pp. 241–250] to prove the result.

Constraint Satisfaction Problems (CSPs) are natural computational problems that appear in many areas of theoretical computer science. Exploring which CSPs are solvable in polynomial time and which are NP-hard reveals a surprising link with central questions in universal algebra. This monograph presents a self-contained introduction to the universal-algebraic approach to complexity classification, treating both finite and infinite-domain CSPs. It includes the required background from logic and combinatorics, particularly model theory and Ramsey theory, and explains the recently discovered link between Ramsey theory and topological dynamics and its implications for CSPs. The book will be of interest to graduate students and researchers in theoretical computer science and to mathematicians in logic, combinatorics, and dynamics who wish to learn about the applications of their work in complexity theory.

Horocycles, at least in the hyperbolic plane, have been studied for many years. However, it was Hedlund who first used them in topological dynamics and ergodic theory ([9], [i0]), and it was he who had the courage actually to apply them in tile case of non-constant curvature. It is this assurance, subsequently exploited by E. Hopf, Anosov, and Sinai ([12, [13], [I], [2]), that has led me, through the back door, so to speak, to a viewpoint of Riemannian geometry about which I intend to speak today• The claim is that arbitrary Riemannian manifolds have much in common with Riemannian symmetric spaces, and that horocycles enable us to construct analogues of the Iwasawa and Bruhat decompositions of semisimple Lie groups when the curvature is strictly negative. In §i we establish the notation and express the Cartan structure equations in vector form. The relationship to the Cartan decomposition of a semisimple group is pointed out, and meanings attached to the assertion that the resulting flows act (pointwise) transitively in the bundle of frames. In §2 we recall the definition of horospheres and give growth estimates for certain functions associatz>d,~ith them. §3 is devoted to defining the horocycles. §4 is qualitative, pointing out the analogies to the Iwasawa and Bruhat decompositions connected with rank one symmetric spaces. In §S we give an application to ergodic theory of these ideas, namely, the Theorem. Let M be a compact Riemannian manifold with i/4-pinched negative curvature. Then any generalized geodesic flow in the bundle of orthonormal frames is ergodic.

In recent years, dynamical systems which are conjugate to their squares have been studied in ergodic theory. In this paper we study the consequences of groups having elements which are conjugate to their squares and consider examples arising from topological dynamics and more general dynamical systems

We show that stationary characters on irreducible lattices Gamma < G of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice Gamma < G, the left regular representation lambda _{Gamma } is weakly contained in any weakly mixing representation pi . We prove that for any such irreducible lattice Gamma < G, any Uniformly Recurrent Subgroup (URS) of Gamma is finite, answering a question of Glasner–Weiss. We also obtain a new proof of Peterson’s character rigidity result for irreducible lattices Gamma < G. The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.

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.