Benjamin Weiss
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View article: The isomorphism problem for group actions
The isomorphism problem for group actions Open
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.
View article: Poisson genericity in numeration systems with exponentially mixing probabilities
Poisson genericity in numeration systems with exponentially mixing probabilities Open
We define Poisson genericity for infinite sequences in any finite or countable alphabet with an invariant exponentially-mixing probability measure. A sequence is Poisson generic if the number of occurrences of blocks of symbols asymptotica…
View article: Lifting generic points
Lifting generic points Open
Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems, where $(X,T)$ has the weak specification property. Let $ξ$ be an invariant measure on the product system $(X\times Y, T\times S)$ with marginals $μ$ on $X$ and $ν$ on $Y$, with …
View article: Distal strongly ergodic actions
Distal strongly ergodic actions Open
Let \eta be an arbitrary countable ordinal. Using results of Bourgain, Gamburd, and Sarnak on compact systems with spectral gap, we show the existence of an action of the free group on three generators F_3 on a compact metric space X , adm…
View article: Measure preserving diffeomorphisms of the torus are unclassifiable
Measure preserving diffeomorphisms of the torus are unclassifiable Open
In 1932 von Neumann proposed classifying the statistical behavior of differentiable systems. In modern language this is interpreted as classifying diffeomorphisms of compact manifolds up to measure isomorphism. This paper proves that this …
View article: An ergodic system is dominant exactly when it has positive entropy
An ergodic system is dominant exactly when it has positive entropy Open
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 …
View article: Intermittent estimation for finite alphabet finitarily Markovian processes with exponential tails
Intermittent estimation for finite alphabet finitarily Markovian processes with exponential tails Open
We give some estimation schemes for the conditional distribution and conditional expectation of the the next output following the observation of the first n outputs of a stationary process where the random variables may take finitely many …
View article: Pure strictly uniform models of non-ergodic measure automorphisms
Pure strictly uniform models of non-ergodic measure automorphisms Open
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 me…
View article: Bernoulli disjointness
Bernoulli disjointness Open
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View article: On universal algorithms for classifying and predicting stationary processes
On universal algorithms for classifying and predicting stationary processes Open
This is a survey of results on universal algorithms for classification and prediction of stationary processes. The classification problems include discovering the order of a k-step Markov chain, determining memory words in finitarily Marko…
View article: A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system
A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system Open
We show the existence, over an arbitrary infinite ergodic -dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotient…
View article: Topological characteristic factors and nilsystems
Topological characteristic factors and nilsystems Open
We prove that the maximal infinite step pro-nilfactor $X_\infty$ 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 $π:X \rightarr…
View article: On weak rigidity and weakly mixing enveloping semigroups
On weak rigidity and weakly mixing enveloping semigroups Open
The question we deal with here, which was presented to us by Joe Auslander and Anima Nagar, is whether there is a nontrivial cascade (X,T) whose enveloping semigroup, as a dynamical system, is topologically weakly mixing (WM). After an int…
View article: When all points are generic for ergodic measures
When all points are generic for ergodic measures Open
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 …
View article: Predictive Sets
Predictive Sets Open
A set $P\subset \mathbb N$ is called predictive if for any zero entropy finite-valued stationary process $(X_i)_{i\in \mathbb Z}$, $X_0$ is measurable with respect to $(X_i)_{i\in P}$. We know that $\mathbb N$ is a predictive set. In this …
View article: From odometers to circular systems: A global structure theorem
From odometers to circular systems: A global structure theorem Open
The main result of this paper is that two large collections of ergodic measure preserving systems, the Odometer Based and the Circular Systems have the same global structure with respect to joinings. The classes are canonically isomorphic …
View article: On the disjointness property of groups and a conjecture of Furstenberg
On the disjointness property of groups and a conjecture of Furstenberg Open
In his seminal 1967 paper "Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation" Furstenberg introduced the notion of disjointness of dynamical systems, both topological and measure preserving. In this p…
View article: On minimal actions of countable groups
On minimal actions of countable groups Open
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 Lovasz local lemma (LLL). For a…
View article: Spectra of stationary processes on Z
Spectra of stationary processes on Z Open
We will discuss a somewhat striking spectral property of finitely valued\nstationary processes on Z that says that if the spectral measure of the process\nhas a gap then the process is periodic. We will give some extensions of this\nresult…
View article: Symmetric Birkhoff sums in infinite ergodic theory
Symmetric Birkhoff sums in infinite ergodic theory Open
We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity. Also, we cons…
View article: Affinely prime dynamical systems
Affinely prime dynamical systems Open
We study representations of groups by "affine" automorphisms of compact, convex spaces, with special focus on "irreducible" representations: equivalently "minimal" actions. When the group in question is PSL(2,R), we exhibit a one-one corre…
View article: On doubly minimal systems and a question regarding product recurrence
On doubly minimal systems and a question regarding product recurrence Open
We show that a doubly minimal system $X$ has the property that for every minimal system $Y$ the orbit closure of any pair $(y,x) \in Y \times X$ is either $Y \times X$ or it has the form $Γ_π= \{(π(x),x) : x \in X\}$ for some factor map $π…
View article: The emergence of 4-cycles in polynomial maps over the extended integers
The emergence of 4-cycles in polynomial maps over the extended integers Open
Let $f(x) \in \mathbb{Z}[x]$; for each integer $α$ it is interesting to consider the number of iterates $n_α$, if possible, needed to satisfy $f^{n_α}(α) = α$. The sets $\{α, f(α), \ldots, f^{n_α - 1}(α), α\}$ generated by the iterates of …
View article: Weak mixing properties for non-singular actions
Weak mixing properties for non-singular actions Open
For a general group $G$ we consider various weak mixing properties of non-singular actions. In the case where the action is actually measure preserving all these properties coincide, and our purpose here is to check which implications pers…
View article: Entropy and actions of sofic groups
Entropy and actions of sofic groups Open
In recent years there has been a great deal of progress in the study of actions of countable groups. In particular, the concept of the entropy of an action has been extended to all sofic groups following the seminal work of Lewis Bowen. Th…