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There are numerous characterizations of Shannon entropy and Tsallis entropy as measures of information obeying certain properties. Using work by Faddeev and Furuichi, we derive a very simple characterization. Instead of focusing on the entropy of a probability measure on a finite set, this characterization focuses on the “information loss”, or change in entropy, associated with a measure-preserving function. Information loss is a special case of conditional entropy: namely, it is the entropy of a random variable conditioned on some function of that variable. We show that Shannon entropy gives the only concept of information loss that is functorial, convex-linear and continuous. This characterization naturally generalizes to Tsallis entropy as well.

The factorization problem was proposed during the Summer Symposium in Real Analysis XXXIII, which was held at Southeastern Oklahoma State University. Here, Grahl and Nishiura present a solution to the problem proposed at the symposium.

This comprehensive text on entropy covers three major types of dynamics: measure preserving transformations; continuous maps on compact spaces; and operators on function spaces. Part I contains proofs of the Shannon–McMillan–Breiman Theorem, the Ornstein–Weiss Return Time Theorem, the Krieger Generator Theorem and, among the newest developments, the ergodic law of series. In Part II, after an expanded exposition of classical topological entropy, the book addresses symbolic extension entropy. It offers deep insight into the theory of entropy structure and explains the role of zero-dimensional dynamics as a bridge between measurable and topological dynamics. Part III explains how both measure-theoretic and topological entropy can be extended to operators on relevant function spaces. Intuitive explanations, examples, exercises and open problems make this an ideal text for a graduate course on entropy theory. More experienced researchers can also find inspiration for further research.

We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in $L^{p}$ -spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.

A fundamental question in the field of cohomology of dynamical systems is to determine when there are solutions to the coboundary equation: $$ \\begin{align*} f = g - g \\circ T. \\end{align*} $$ In many cases, T is given to be an ergodic invertible measure-preserving transformation on a standard probability space $(X, {\\mathcal B}, \\mu )$ and is contained in $L^p$ for $p \\geq 0$ . We extend previous results by showing for any measurable f that is non-zero on a set of positive measure, the class of measure-preserving T with a measurable solution g is meager (including the case where $\\int _X f\\,d\\mu = 0$ ). From this fact, a natural question arises: given f, does there always exist a solution pair T and g? In regards to this question, our main results are as follows. Given measurable f, there exist an ergodic invertible measure-preserving transformation T and measurable function g such that $f(x) = g(x) - g(Tx)$ for almost every (a.e.) $x\\in X$ , if and only if $\\int _{f> 0} f\\,d\\mu = - \\int _{f < 0} f\\,d\\mu $ (whether finite or $\\infty $ ). Given mean-zero $f \\in L^p(\\mu )$ for $p \\geq 1$ , there exist an ergodic invertible measure-preserving T and $g \\in L^{p-1}(\\mu )$ such that $f(x) = g(x) - g( Tx )$ for a.e. $x \\in X$ . In some sense, the previous existence result is the best possible. For $p \\geq 1$ , there exists a dense $G_{\\delta }$ set of mean-zero $f \\in L^p(\\mu )$ such that for any ergodic invertible measure-preserving T and any measurable g such that $f(x) = g(x) - g(Tx)$ almost everywhere, then $g \\notin L^q(\\mu )$ for $q> p - 1$ . Finally, it is shown that we cannot expect finite moments for solutions g, when $f \\in L^1(\\mu )$ . In particular, given any such that $\\lim _{x\\to \\infty } \\phi (x) = \\infty $ , there exist mean-zero $f \\in L^1(\\mu )$ such that for any solutions T and g, the transfer function g satisfies: $$ \\begin{align*} \\int_{X} \\phi ( | g(x) | )\\,d\\mu = \\infty. \\end{align*} $$

This volume contains the proceedings of the 2016 Summer School on Fractal Geometry and Complex Dimensions, in celebration of Michel L. Lapidus's 60th birthday, held from June 21-29, 2016, at California Polytechnic State University, San Luis Obispo, California. The theme of the contributions is fractals and dynamics and content is split into four parts, centered around the following themes: Dimension gaps and the mass transfer principle, fractal strings and complex dimensions, Laplacians on fractal domains and SDEs with fractal noise, and aperiodic order (Delone sets and tilings).

We prove a generalized central limit theorem for dynamical systems with an infinite ergodic measure which induce a Gibbs–Markov map on some subset, provided the return time to this subset has regularly varying tails. We adapt a method designed by Csáki and Földes for observables of random walks to show that the partial sums of some functions of the system—the return time and the observable—are asymptotically independent. Some applications to random walks and Pomeau–Manneville maps are discussed.

Let XX be a standard probability space and TtT_t a measure-preserving semiflow on XX. We show that there exists a set X0X_0 of full measure in XX such that for any x∈X0x \\in X_0 and t≥0t \\geq 0 there are measures μx,t+\\mu _{x,t}^+ and μx,t−\\mu _{x,t}^- which for all but a countable number of tt give a distribution on the set of points yy such that Tt(y)=Tt(x)T_t(y) = T_t(x). These measures arise by taking weak∗−^*-limits of suitable conditional expectations. Say that a point xx has a measurable orbit discontinuity at time t0t_0 if either μx,t+\\mu _{x,t}^+ or μx,t−\\mu _{x,t}^- is weak∗−^*-discontinuous in tt at t0t_0. We show that there exists an invariant set of full measure in XX such that any point in this set has at most countably many measurable orbit discontinuities. Furthermore we show that if xx has a measurable orbit discontinuity at time 0, then xx has an orbit discontinuity at time 0 in the sense of Orbit discontinuities and topological models for Bordel semiflows, D. McClendon.

Existence of measure preserving maps has been discussed in books on real analysis where the Axiom of Choice is instrumental. In this note we introduce a method to construct such maps. For our construction we use the distribution function and elementary differential equations. [PUBLICATION ABSTRACT]

This volume contains the proceedings of the AMS Special Session on Unimodularity in Randomly Generated Graphs, held from October 8-9, 2016, in Denver, Colorado. Unimodularity, a term initially used in locally compact topological groups, is one of the main examples in which the generalization from groups to graphs is successful. The \"randomly generated graphs\", which include percolation graphs, random Erdős-Rényi graphs, and graphings of equivalence relations, are much easier to describe if they result as random objects in the context of unimodularity, with respect to either a vertex-transient \"host\"-graph or a probability measure. This volume tries to give an impression of the various fields in which the notion currently finds strong development and application: percolation theory, point processes, ergodic theory, and dynamical systems.