Let G be an infinite countable amenable group and let (X,G) be a G-subshift with specification, containing a free element. We prove that (X,G) is universal, i.e., has positive topological entropy and for any free ergodic (measure-preserving) G-action on a standard probability space, (Y,ν,G), with h(ν)<htop(X), there exists a shift-invariant measure μ on X such that the systems (Y,ν,G) and (X,μ,G) are isomorphic. In particular, any K-shift (consisting of the indicator functions of all maximal K-separated sets) containing a free element is universal.

Our purpose here is to review some recent developments in the theory of dynamical systems whose common theme is a link between minimal dynamical systems, certain Ramsey type combinatorial properties, and the Lov & aacute;sz local lemma (LLL). For a general countable group G the two classes of minimal systems we will deal with are (I) the minimal subsystems of the subgroup system (Sub(G), G), called URS's (uniformly recurrent subgroups), and (II) minimal subshifts; i.e. subsystems of the binary Bernoulli G-shift ({0, 1}G, {sigma g}g is an element of G).

We prove an analog of Rudolph's theorem for actions of countable amenable groups, which asserts that among invariant measures with entropy at least c on the G-shift (Λ^G,σ), a typical measure has entropy c and is Bernoulli. We also address a relative version of this theorem.

By using a similar pattern of arguments, we show that in three categories the collection of isomorphisms forms a residual subset of the space of morphisms. We first consider surjective continuous mappings on Cantor spaces. Next, we look at measure preserving maps on Polish measure spaces. Finally, we examine continuous, measure preserving maps on Cantor spaces equipped with so-called good measures.

We discuss the isomorphism problem for ergodic actions of locally compact groups. In particular we show that the conjugacy relation is not Borel for ergodic measure preserving actions of indicable groups.

We prove that the maximal infinite step pro-nilfactor X-infinity of a minimal dynamical system (X,T) is the topological characteristic factor in a certain sense. Namely, we show that by an almost one-to-one modification of pi:X -> X-infinity, the induced open extension pi*:X*-> X-infinity* has the following property: for x in a dense G(delta) subset of X*, the orbit closure L-x=O-((x,...,x),TxT(2)x...xT(d)) is (pi*)((d))-saturated, i.e., L-x=((pi*)((d))-1)(pi*)((d))(L-x). Using results derived from the above fact, we are able to answer several open questions: (1) if (X,T-k) is minimal for some k >= 2, then for any d is an element of N and any 0 <= j x,T(i)(2n)x -> x,...,T(i)(dn)x -> x for x in a dense G(delta) subset of X; (2) if (X,T) is totally minimal, then {T(n2)x:n is an element of Z} is dense in X for x in a dense G(delta) subset of X; (3) for any d is an element of N and any minimal t.d.s. which is an open extension of its maximal distal factor, RP[d]=AP([d]), where the former is the regionally proximal relation of order d and the latter is the regionally proximal relation of order d along arithmetic progressions.

For a given ergodic measure preserving transformation T of a standard measure space each finite labelled partition defines an ergodic stationary process. There is a complete metric on the space of partitions which is separable. Various generic properties of these processes will be given. For example: 1. The generic partition defines a process that is not Rosenblatt mixing. 2. If T is a K-automorphism that is not Bernoulli then the generic partition is also K but not Bernoulli. Extensions to the relative setting and to actions of amenable groups will also be discussed.

We consider an intermediate factor situation in two categories: probability measure preserving ergodic theory and compact topological dynamics. In the first we prove a master-key theorem and examine a wide range of applications. In the second we provide some counterexamples.

Twenty years ago Dan Rudolph gave a seminar talk at the University of Maryland in which he discussed his discovery that if one restricts the shift invariant measures on a finite alphabet shift space to have entropy greater than or equal to c > 0, then the generic measure there defines a process that has entropy c and is isomorphic to a Bernoulli shift. He never published a proof of this theorem. In the following note we will present a proof of this result.

Our main result is to show that every infinite, countable, residually finite group $G$ admits a Hausdorff group topology which is neither discrete nor precompact.

Let $\mathcal{E}$ denote the space of entire functions with the topology of uniform convergence on compact sets. The action of $\mathbb{C}$ by translations on $\mathcal{E}$ is defined by $T_{z}f(w) = f(w+z)$. Let $\mathcal{U}$ denote the set of entire functions whose orbit under $T$ is dense. Birkhoff showed, in 1929, that $\mathcal{U}$ is not empty. One of the problems in the collection by T.-C. Dinh and N. Sibony in 2020 asks whether there exists an invariant probability measure on $\mathcal{E}$ whose support is contained in $\mathcal U$. We will show how an old construction of the second author can be modified to provide a positive answer to their question. Furthermore, we modify the construction to produce a wealth of ergodic measures on the space of entire functions of several complex variables.

Up to now there has been no proof in the literature of the often quoted fact that the Jewett-Krieger theorem is valid for all countable amenable groups. In this brief note I will close this gap by applying a recent result of B. Frej and D. Huczek [FH].

We define Poisson genericity for infinite sequences in any countable alphabet with an invariant exponentially ψ \psi -mixing probability measure. A sequence is Poisson generic if the number of occurrences of blocks of symbols asymptotically follows a Poisson law as the block length increases. We prove that almost all sequences are Poisson generic. Our result generalizes Peres and Weiss’ theorem about Poisson genericity of integer bases numeration systems. In particular, we obtain that the continued fraction expansions of almost all real numbers are Poisson generic.

Kac's lemma determines the expected return time to a set of positive measure under iterations of an ergodic probability preserving transformations. We introduce the notion of an allocation for a probability preserving action of a countable group. Using this notion, we formulate and prove generalization of Kac's lemma for an action of a general countable group, and another generalization that applies to probability preserving equivalence relations. As an application, we provide a short proof for the existence of countable generating partitions for any ergodic action of a countable group.

We present here a number of results that provide universal rates of convergence for certain non parametric estimation problems. For example consider the class C of all finite order Markov chains on a countable alphabet and the problem of estimating the conditional distribution of Xn+1 given the first n outputs of the process. We will give a sequence of stopping times with density one and estimators at those times such that almost surely our estimators will eventually differ from the true conditional distribution by no more than a certain fixed sequence tending to zero. Similar results are given for estimating the conditional expectation of Xn+1 given the first n outputs, but here some additional moment conditions are required. An example shows that this is not possible in general.

Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems, where $(X,T)$ has the weak specification property. Let $\xi $ be an invariant measure on the product system $(X\times Y, T\times S)$ with marginals $\mu $ on X and $\nu $ on Y, with $\mu $ ergodic. Let $y\in Y$ be quasi-generic for $\nu $ . Then there exists a point $x\in X$ generic for $\mu $ such that the pair $(x,y)$ is quasi-generic for $\xi $ . This is a generalization of a similar theorem by T. Kamae, in which $(X,T)$ and $(Y,S)$ are full shifts on finite alphabets.

In this note we give a fairly direct proof of a recent theorem of Gorska, Lemanczyk and de la Rue which characterises the class of measure preserving transformations that are disjoint from every ergodic measure preserving transformation. Our proof works just as well for any countable acting group.

Construction sequences are a general method of building symbolic shifts that capture cut-and-stack constructions and are general enough to give symbolic representations of Anosov-Katok diffeomorphisms. We show here that any finite entropy system that has an odometer factor can be represented as the limit of a special class of construction sequences, the odometer based construction sequences. These naturally correspond to those cut-and-stack constructions that do not use spacers. The odometer based construction sequences can be constructed to have the small word property and every Choquet simplex can be realized as the simplex of invariant measures of the limit of an odometer based construction sequence.

An ergodic dynamical system $\mathbf{X}$ is called dominant if it is isomorphic to a generic extension of itself. It was shown in an earlier paper by Glasner, Thouvenot and Weiss that Bernoulli systems with finite entropy are dominant. In this work we show first that every ergodic system with positive entropy is dominant, and then that if $\mathbf{X}$ has zero entropy then it is not dominant.