Related papers: Square Entropy and Uniform n-to-1 Bernoulli Transf…

A {\it uniformly $p$-to-one endomorphism} is a measure-preserving map with entropy log $p$ which is almost everywhere $p$-to-one and for which the conditional expectation of each preimage is precisely $1/p$. The {\it standard} example of…

We show that expanding toral endomorphisms, together with their respective Lebesgue measure are isomorphic to 1-sided Bernoulli shifts. This result is then extended to smooth perturbations of expanding toral endomorphisms, together with…

This is the first part in a series in which sofic entropy theory is generalized to class-bijective extensions of sofic groupoids. Here we define topological and measure entropy and prove invariance. We also establish the variational…

In this paper we calculate the metric and folding entropies for a family of non-invertible symbolic dynamical systems $(\Sigma_{m_-,m_+}, \sigma_\phi)$ which generalizes the standard bilateral Bernoulli shifts. The space $\Sigma_{m_-,m_+}$…

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is AT$(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not…

We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This…

We prove that a shift ergodic measure on a topologically mixing sub-shift is isomorphic to a Bernoulli shift whenever it is quasi invariant under permutations of finite number of coordinates. We prove also that Gibbs measures on…

We define a notion of entropy for an infinite family $\mathcal{C}$ of measurable sets in a probability space. We show that the mean ergodic theorem holds uniformly for $\mathcal{C}$ under every ergodic transformation if and only if…

We prove that the entropy map for countable Markov shifts of finite entropy is upper semi-continuous at ergodic measures. Note that the phase space is non-compact. Applications to systems that can be coded by these shifts, such as positive…

We prove that if $G$ is a countably infinite group and $(L, \lambda)$ and $(K, \kappa)$ are probability spaces having equal Shannon entropy, then the Bernoulli shifts $G \curvearrowright (L^G, \lambda^G)$ and $G \curvearrowright (K^G,…

Using the entropic inequalities for Shannon and Tsallis entropies new inequalities for some classical polynomials are obtained. To this end, an invertible mapping for the irreducible unitary representation of groups $SU(2)$ and $SU(1,1)$…

Topological entropy or spatial entropy is a way to measure the complexity of shift spaces. This study investigates the relationships between the spatial entropy and the various periodic entropies which are computed by skew-coordinated…

The squarefree flow is a natural dynamical system whose topological and ergodic properties are closely linked to the behavior of squarefree numbers. We prove that the squarefree flow carries a unique measure of maximal entropy and express…

Permutation entropy quantifies the diversity of possible orderings of the values a random or deterministic system can take, as Shannon entropy quantifies the diversity of values. We show that the metric and permutation entropy…

We describe a general method of arithmetic coding of geodesics on the modular surface based on a two parameter family of continued fraction transformations studied previously by the authors. The finite rectangular structure of the…

In this paper, we extend the concept of generalized entropy to uniform spaces, allowing computations beyond metrizable settings. We apply this to parabolic dynamics - systems with a unique fixed point uniformly attracting all compact…

We show that the values of entropies of multidimensional shifts of finite type (SFTs) are characterized by a certain computation-theoretic property: a real number $h\geq 0$ is the entropy of such an SFT if and only if it is right…

We show that a class of robustly transitive diffeomorphisms originally described by Ma\~{n}\'{e} are intrinsically ergodic. More precisely we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic, but nevertheless have…

We introduce a new extension in symbolic dynamics on two sets of alphabets, called the zip shift space. In finite case, it represents a finite-to-1 local homeomorphism called zip shift map. Such extension, offers a conjugacy between some…

The Euler graph has vertices labelled (n,k) for n=0,1,2,... and k=0,1,...,n, with k+1 edges from (n,k) to (n+1,k) and n-k+1 edges from (n,k) to (n+1,k+1). The number of paths from (0,0) to (n,k) is the Eulerian number A(n,k), the number of…