We show that a transient graph can be extended to a recurrent graph of equal entropy which is either positive recurrent of null recurrent, and we give an example of each type. We extend the notion of local entropy to topological Markov chains and prove that a transitive Markov chain admits a measure of maximal entropy (or maximal measure) whenever its local entropy is less than its (global) ent...

It is well-known that maximum entropy distributions, subject to appropriate moment constraints, arise in physics and mathematics. In an attempt to find a physical reason for the appearance of maximum entropy distributions, the following theorem is offered. The conditional distribution of X, given the empirical observation (1 /n)X:, ,/I ( X,) = (Y, where X, , X2, . are independent identically di...

A labeling of a graph is a function from the vertex set of the graph to some finite set. Certain dynamical systems (such as topological Markov shifts) can be defined by directed graphs. In these instances, a labeling of the graph defines a continuous, shift-commuting factor of the dynamical system. We find sufficient conditions on the labeling to imply classification results for the factor dyna...

We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure μ on a countable group G, A. Nevo showed that a necessary condition for G to have property (T) is that the Furstenberg μ-entropy values of the ergodic, properly nonsingular G-actions are bounded away from zero. We show that this is also a sufficient condit...

In this note, we establish the first variation formula of the adjusted log entropy functional Ya introduced by Ye in [14]. As a direct consequence, we also obtain the monotonicity of Ya along the Ricci flow. Various entropy functionals play crucial role in the singularity analysis of Ricci flow. Let (M, g(t)) be a smooth family of Riemannian metrics on a closed manifold M and suppose g(t) is a ...

We provide a counterexample to show that the generic form of entropy ∑ = i i p g p S ) ( ) ( is not always stable against small variation of probability distribution (Lesche stability) even if g is concave function on [0,1] and analytic on ]0,1]. Our conclusion is that the stability of such a generic functional needs more hypotheses on the property of the function g, or in other words, the stab...

Abstract Let G be a countably infinite discrete amenable group. It should noted that -system $(X,G)$ naturally induces $(\mathcal {M}(X),G)$ , where $\mathcal {M}(X)$ denotes the space of Borel probability measures on compact metric X endowed with weak*-topology. A factor map $\pi : (X,G)\to (Y,G)$ between two -systems $\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$ . turns out }...

§0. Introduction Let X be a compact metric space and T : X → X is continuous transformation. Then the dynamics of T is a widely studied subject. In particular, h(T ) the entropy of T is a well understood object. Let Γ ⊂ X ×X be a closed set. Then Γ induces certain dynamics and entropy h(Γ). If X is a finite set then Γ can be naturally viewed as a directed graph. That is, if X = {1, ..., n} then...

For a measure-preserving transformation T which is a skew-product of a measure-preserving transformation S and a topological group endomorphism a, it is shown that the entropy h satisfies the following "addition theorem": h(T) = h(S) + h(o). Introduction. In a previous paper [4], conditions were given for a certain type of transformation of a G-space (G being a compact separable group) to have ...