Multiorders in amenable group actions

The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group we mean any probability measure $ u$ on the collection $mathcal O$ of linear orders of type $mathbb Z$ on $G$, invariant under the natural action of $G$ on such orders. Every free measure-preserving $G$-action $(X,mu,G)$ has a multiorder $(mathcal O, u,G)$ as a factor and has the same orbits as the $mathbb Z$-action $(X,mu,S)$, where $S$ is the successor map determined by the multiorder factor. The sub-sigma-algebra $Sigma_{mathcal O}$ associated with the multiorder factor is invariant under $S$, which makes the corresponding $mathbb Z$-action $(mathcal O, u,tilde S)$ a factor of $(X,mu,S)$. We prove that the entropy of any $G$-process generated by a finite partition of $X$, conditional with respect to $Sigma_{mathcal O}$, is preserved by the orbit equivalence with $(X,mu,S)$. Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to the one known for $mathbb Z$-actions. This fact is applied to prove a variant of a result by Rudolph and Weiss. The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a sub-sigma-algebra $Sigma$, as soon as the orbit change is $Sigma$-measurable. In our variant, we replace the measurability assumption by a simpler one: $Sigma$ should be invariant under both actions and the actions on the resulting factor should be free. In conclusion we prove that the Pinsker sigma-algebra of any $G$-process can be identified (with probability 1) using the following algorithm: (1) fix an arbitrary multiorder on $G$, (2) select any order from the support of that multiorder, (3) in the process, find the remote past along the selected order.

Pure strictly uniform models of non-ergodic measure automorphisms

The classical theorem of Jewett and Krieger gives a strictly ergodic model for any ergodic measure preserving system. An extension of this result for non-ergodic systems was given many years ago by George Hansel. He constructed, for any measure preserving system, a strictly uniform model, i.e. a compact space which admits an upper semicontinuous decomposition into strictly ergodic models of the ergodic components of the measure. In this note we give a new proof of a stronger result by adding the condition of purity, which controls the set of ergodic measures that appear in the strictly uniform model.

When all points are generic for ergodic measures

We establish connections between several properties of topological dynamical systems, such as: - every point is generic for an ergodic measure, - the map sending points to the measures they generate is continuous, - the system splits into uniquely (alternatively, strictly) ergodic subsystems, - the map sending ergodic measures to their topological supports is continuous, - the Cesaro means of every continuous function converge uniformly.

The Symbolic Extension Theory in Topological Dynamics

In this survey we will present the symbolic extension theory in topological dynamics, which was built over the past twenty years.

Uniform continuity of entropy rate with respect to the $bar f$-pseudometric

Assume that a sequence $x=x_0x_1ldots$ is frequency-typical for a finite-valued stationary stochastic process $textbf X$. We prove that the function associating to $x$ the entropy-rate $bar H(textbf X)$ of $textbf X$ is uniformly continuous when one endows the set of all frequency-typical sequences with the $bar f$ pseudometric. As a consequence, we obtain the same result for the $bar d$ pseudometric. We also give an alternative proof of the Abramov formula for the Kolmogorov-Sinai entropy of the induced measure-preserving transformation.

A fresh look at the notion of normality

Let $G$ be a countable cancellative amenable semigroup and let $(F_n)$ be a (left) F{o}lner sequence in $G$. We introduce the notion of an $(F_n)$-normal element of ${0,1}^G$. When $G$ = $(mathbb N,+)$ and $F_n = {1,2,...,n}$, the $(F_n)$-normality coincides with the classical notion. We prove that: $bullet$ If $(F_n)$ is a F{o}lner sequence in $G$, such that for every $alphain(0,1)$ we have $sum_n alpha^{|F_n|}<infty$, then almost every $xin{0,1}^G$ is $(F_n)$-normal. $bullet$ For any F{o}lner sequence $(F_n)$ in $G$, there exists an Cham-per-nowne-like $(F_n)$-normal set. $bullet$ There is a natural class of nice F{o}lner sequences in $(mathbb N,times)$. There exists a Champernowne-like set which is $(F_n)$-normal for every nice F{o}lner sq. $bullet$ Let $Asubsetmathbb N$ be a classical normal set. Then, for any F{o}lner sequence $(K_n)$ in $(mathbb N,times)$ there exists a set $E$ of $(K_n)$-density $1$, such that for any finite subset ${n_1,n_2,dots,n_k}subset E$, the intersection $A/{n_1}cap A/{n_2}capldotscap A/{n_k}$ has positive upper density in $(mathbb N,+)$. As a consequence, $A$ contains arbitrarily long geometric progressions, and, more generally, arbitrarily long geo-arithmetic configurations of the form ${a(b+ic)^j,0le i,jle k}$. $bullet$ For any F{o}lner sq $(F_n)$ in $(mathbb N,+)$ there exist uncountably many $(F_n)$-normal Liouville numbers. $bullet$ For any nice F{o}lner sequence $(F_n)$ in $(mathbb N,times)$ there exist uncountably many $(F_n)$-normal Liouville numbers.

Decomposition of a symbolic element over a countable amenable group into blocks approximating ergodic measures

Consider a subshift over a finite alphabet, $Xsubset Lambda^{mathbb Z}$ (or $XsubsetLambda^{mathbb N_0}$). With each finite block $BinLambda^k$ appearing in $X$ we associate the empirical measure ascribing to every block $CinLambda^l$ the frequency of occurrences of $C$ in $B$. By comparing the values ascribed to blocks $C$ we define a metric on the combined space of blocks $B$ and probability measures $mu$ on $X$, whose restriction to the space of measures is compatible with the weak-$star$ topology. Next, in this combined metric space we fix an open set $mathcal U$ containing all ergodic measures, and we say that a block $B$ is ergodic if $Binmathcal U$. In this paper we prove the following main result: Given $varepsilon>0$, every $xin X$ decomposes as a concatenation of blocks of bounded lengths in such a way that, after ignoring a set $M$ of coordinates of upper Banach density smaller than $varepsilon$, all blocks in the decomposition are ergodic. The second main result concerns subshifts whose set of ergodic measures is closed. We show that, in this case, no matter how $xin X$ is partitioned into blocks (as long as their lengths are sufficiently large and bounded), after ignoring a set $M$ of upper Banach density smaller than $varepsilon$, all blocks in the decomposition are ergodic. The first half of the paper is concluded by examples showing, among other things, that the small set $M$, in both main theorems, cannot be avoided. The second half of the paper is devoted to generalizing the two main results described above to subshifts $XsubsetLambda^G$ with the action of a countable amenable group $G$. The role of long blocks is played by blocks whose domains are members of a Fo lner sequence while the decomposition of $xin X$ into blocks (of which majority is ergodic) is obtained with the help of a congruent system of tilings.

Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation

A classical Kamae-Weiss theorem states that an increasing sequence $(n_i)_{iinmathbb N}$ of positive lower density is emph{normality preserving}, i.e. has the property that for any normal binary sequence $(b_n)_{ninmathbb N}$, the sequence $(b_{n_i})_{iinmathbb N}$ is normal, if and only if $(n_i)_{iinmathbb N}$ is a deterministic sequence. Given a countable cancellative amenable semigroup $G$, and a Fo lner sequence $mathcal F=(F_n)_{ninmathbb N}$ in $G$, we introduce the notions of normality preservation, determinism and subexponential complexity for subsets of $G$ with respect to $mathcal F$, and show that for sets of positive lower $mathcal F$-density these three notions are equivalent. The proof utilizes the apparatus of the theory of tilings of amenable groups and the notion of tile-entropy. We also prove that under a natural assumption on $mathcal F$, positive lower $mathcal F$-density follows from normality preservation. Finally, we provide numerous examples of normality preserving sets in various semigroups

A strictly ergodic, positive entropy subshift uniformly uncorrelated to the Moebius function

A recent result of Downarowicz and Serafin (DS) shows that there exist positive entropy subshifts satisfying the assertion of Sarnaks conjecture. More precisely, it is proved that if $y=(y_n)_{nge 1}$ is a bounded sequence with zero average along every infinite arithmetic progression (the Mobius function is an example of such a sq $y$) then for every $Nge 2$ there exists a subshift $Sigma$ over $N$ symbols, with entropy arbitrarily close to $log N$, uncorrelated to $y$. In the present note, we improve the result of (DS). First of all, we observe that the uncorrelation obtained in (DS) is emph{uniform}, i.e., for any continuous function $f:Sigmato {mathbb R}$ and every $epsilon>0$ there exists $n_0$ such that for any $nge n_0$ and any $xinSigma$ we have $$ left|frac1nsum_{i=1}^{n}f(T^ix),y_iright|<epsilon. $$ More importantly, by a fine-tuned modification of the construction from (DS) we create a emph{strictly ergodic} subshift, with all the desired properties of the example in (DS) (uniformly uncorrelated to $y$ and with entropy arbitrarily close to $log N$). The question about these two additional properties (uniformity of uncorrelation and strict ergodicity) has been posed by Mariusz Lemanczyk in the context of the so-called strong MOMO (Mobius Orthogonality on Moving Orbits) property. Our result shows, among other things, that strong MOMO is essentially stronger than uniform uncorrelation, even for strictly ergodic systems.

Symbolic extensions of amenable group actions and the comparison property

Symbolic Extension Entropy Theorem (SEET) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift. It gives an estimate on the entropy of symbolic extensions (and the necessary number of symbols). Unlike in the measure-theoretic case, where Kolmogorov--Sinai entropy is the estimate, in the topological setup the task reaches beyond the classical theory of entropy. Tools from an extended theory of entropy structures are needed. The main goal of this paper is to prove the SEET for actions of countable amenable groups: Let a countable amenable group $G$ act by homeomorphisms on a compact metric space $X$ and let $mathcal M_G(X)$ denote the simplex of $G$-invariant probability measures on $X$. A function $E $ on $mathcal M_G(X)$ equals the extension entropy function $h^pi$ of a symbolic extension $pi:(Y,G)to (X,G)$, where $h^pi(mu)=sup{h_ u(Y,G): uinpi^{-1}(mu)}$ ($muinmathcal M_G(X)$), if and only if $E $ is an affine superenvelope of the entropy structure of $(X,G)$. The statement is preceded by presentation of the concepts of an entropy structure and superenvelopes, adapted from $mathbb Z$-actions. In full generality we prove a slightly weaker version of SEET, in which symbolic extensions are replaced by quasi-symbolic extensions, i.e., extensions in form of a joining of a subshift with a zero-entropy tiling system. The notion of a tiling system is a subject of earlier works and in this paper we review and complement the theory developed there. The full version of the SEET is proved for groups which are either residually finite or enjoy the comparison property. In order to describe the range of our theorem, we devote a large portion of the paper to the comparison property. Our main result in this aspect shows that all subexponential groups have the comparison property (and thus satisfy the SEET).