Entropy is the only finitely observable invariant
Donald Ornstein Benjamin Weiss
Journal of Modern Dynamics 2007, 1(1): 93-105 doi: 10.3934/jmd.2007.1.93
Our main purpose is to present a surprising new characterization of the Shannon entropy of stationary ergodic processes. We will use two basic concepts: isomorphism of stationary processes and a notion of finite observability, and we will see how one is led, inevitably, to Shannon's entropy. A function $J$ with values in some metric space, defined on all finite-valued, stationary, ergodic processes is said to be finitely observable (FO) if there is a sequence of functions $S_{n}(x_{1},x_{2},...,x_{n})$ that for all processes $\mathcal{X}$ converges to $J(\mathcal{X})$ for almost every realization $x_{1}^{\infty}$ of $\mathcal{X}$. It is called an invariant if it returns the same value for isomorphic processes. We show that any finitely observable invariant is necessarily a continuous function of the entropy. Several extensions of this result will also be given.
keywords: finitely observable entropy isomorphism invariants finitary isomorphism.
Entropy and actions of sofic groups
Benjamin Weiss
Discrete & Continuous Dynamical Systems - B 2015, 20(10): 3375-3383 doi: 10.3934/dcdsb.2015.20.3375
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. This survey is an invitation to these new developments. It includes a new proof of the analogue of Kolmogorov's theorem for sofic groups, namely that isomorphic Bernoulli shifts have the same base entropy.
keywords: Kolmogorov's theorem. Entropy amenable groups sofic groups Bernoulli shifts
Generating product systems
Nir Avni Benjamin Weiss
Journal of Modern Dynamics 2010, 4(2): 257-270 doi: 10.3934/jmd.2010.4.257
Generalizing Krieger's finite generation theorem, we give conditions for an ergodic system to be generated by a pair of partitions, each required to be measurable with respect to a given subalgebra, and also required to have a fixed size.
keywords: Slepian-Wolf Krieger's theorem. generating partition
Measured topological orbit and Kakutani equivalence
Andres del Junco Daniel J. Rudolph Benjamin Weiss
Discrete & Continuous Dynamical Systems - S 2009, 2(2): 221-238 doi: 10.3934/dcdss.2009.2.221
Suppose $X$ and $Y$ are Polish spaces each endowed with Borel probability measures $\mu$ and $\nu$. We call these Polish probability spaces. We say a map $\phi$ is a nearly continuous if there are measurable subsets $X_0\subseteq X$ and $Y_0\subseteq Y$, each of full measure, and $\phi:X_0\to Y_0$ is measure-preserving and continuous in the relative topologies on these subsets. We show that this is a natural context to study morphisms between ergodic homeomorphisms of Polish probability spaces. In previous work such maps have been called almost continuous or finitary. We propose the name measured topological dynamics for this area of study. Suppose one has measure-preserving and ergodic maps $T$ and $S$ acting on $X$ and $Y$ respectively. Suppose $\phi$ is a measure-preserving bijection defined between subsets of full measure on these two spaces. Our main result is that such a $\phi$ can always be regularized in the following sense. Both $T$ and $S$ have full groups ($FG(T)$ and $FG(S)$) consisting of those measurable bijections that carry a point to a point on the same orbit. We will show that there exists $f\in FG(T)$ and $h\in FG(S)$ so that $h\phi f$ is nearly continuous. This comes close to giving an alternate proof of the result of del Junco and Şahin, that any two measure-preserving ergodic homeomorphisms of nonatomic Polish probability spaces are continuously orbit equivalent on invariant $G_\delta$ subsets of full measure. One says $T$ and $S$ are evenly Kakutani equivalent if one has an orbit equivalence $\phi$ which restricted to some subset is a conjugacy of the induced maps. Our main result implies that any such measurable Kakutani equivalence can be regularized to a Kakutani equivalence that is nearly continuous. We describe a natural nearly continuous analogue of Kakutani equivalence and prove it strictly stronger than Kakutani equivalence. To do this we introduce a concept of nearly unique ergodicity.
keywords: Kakutani equivalence finitary orbit equivalence Measured Topological Dynamics

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