Outer Conjugacy for Actions of Continuous Amenable Groups

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.

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.

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.

This book contains a collection of survey papers by leading researchers in ergodic theory, low-dimensional and topological dynamics and it comprises nine chapters on a range of important topics. These include: the role and usefulness of ultrafilters in ergodic theory, topological dynamics and Ramsey theory; topological aspects of kneading theory together with an analogous 2-dimensional theory called pruning; the dynamics of Markov odometers, Bratteli-Vershik diagrams and orbit equivalence of non-singular automorphisms; geometric proofs of Mather's connecting and accelerating theorems; recent results in one dimensional smooth dynamics; periodic points of nonexpansive maps; arithmetic dynamics; the defect of factor maps; entropy theory for actions of countable amenable groups.

On the non-commutative Neveu decomposition and ergodic theorems for amenable group action

We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenable groups with "controlled" Folner sequences; for general amenable groups we prove the weaker result that 1-cocycles with sufficiently small growth are almost coboundaries. Besides, we show that there exist, on a-T-menable groups, proper cocycles with arbitrary small growth.

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.

In 1952 K. Roth showed that any subset of N having positive upper density contains arithmetic progressions of length 3, a result E. Szemer´edi extended to progressions of arbitrary length in 1975. An approach involving multiple recurrence for measure preserving systems developed by H. Furstenberg led to extensions to linear configurations in more general groups while demonstrating families of such configurations to be large in various senses. Using ultrafilter techniques for doing ergodic theory without averaging, we prove a version of Roth's theorem, in its ergodic-theoretic formulation, for general countable groups G. A combinatorial consequence, valid in amenable groups, is the presence of many configurations of the form {(a, b), (ag, b), (ag, bg)} in positive density subsets of G × G. Here "many" implies in particular "central* in the variable g".

We prove that for actions of a discrete countable residuallyfinite amenable group on a compact metric space with specification property, periodic measures are dense in theset of invariant measures. We also prove that certain expansiveactions of a countable discrete group by automorphisms of compact abelian groups have specification property.

Slow entropy type invariants and smooth realization of commuting measure-preserving transformations

A new approach to actions of countable amenable groups with completely positive entropy (cpe), allowing one to answer some basic questions in this field, was recently developed. The question of the existence of cpe actions which are not Bernoulli was raised. In this paper, we prove that every countable amenable groupG, which contains an element of infinite order, has non-Bernoulli cpe actions. In fact we can produce, for any$h \in (0, \infty ]$, an uncountable family of cpe actions of entropyh, which are pairwise automorphically non-isomorphic. These actions are given by a construction which we call co-induction. This construction is related to, but different from the standard induced action. We study the entropic properties of co-induction, proving that ifαGis co-induced from an actionαΓof a subgroup Γ, thenh(αG)=h(αΓ). We also prove that ifαΓis a non-Bernoulli cpe action of Γ, thenαGis also non-Bernoulli and cpe. Hence the problem of finding an uncountable family of pairwise non-isomorphic cpe actions of the same entropy is reduced to one of finding an uncountable family of non-Bernoulli cpe actions of$\mathbb Z$, which pairwise satisfy a property we call ‘uniform somewhat disjointness’. We construct such a family using refinements of the classical cutting and stacking methods.

The ergodic properties of Gibbs states for shift-invariant specifications on a sequence space indexed by an amenable group are studied. For continuous specifications satisfying Dobrushin’s condition, it is shown that the unique Gibbs state is Bernoulli. In addition, the extreme Gibbs states (with respect to the usual partial ordering) of attractive continuous specifications are shown to be Bernoulli; in particular, a unique Gibbs state is Bernoulli. (This result has been obtained previously for Ising spin systems and -actions.) The non-continuous case is also considered.

Abstract: This paper examines and strengthens the Cuntz--Thomsen picture of equivariant Kasparov theory for arbitrary second-countable locally compact groups, in which elements are given by certain pairs of cocycle representations between $\mathrm{C}^*$-dynamical systems. The main result is a stable uniqueness theorem that generalizes a fundamental characterization of ordinary $KK$-theory by Lin and Dadarlat--Eilers. Along the way, we prove an equivariant Cuntz--Thomsen picture analog of the fact that the equivalence relation of homotopy agrees with the (\emph{a priori} stronger) equivalence relation of stable operator homotopy. The results proved in this paper will be employed as the technical centerpiece in forthcoming work of the authors to classify certain amenable group actions on Kirchberg algebras by equivariant Kasparov theory.

Amenable group actions on infinite graphs

Ihara's zeta function for periodic graphs and its approximation in the amenable case