Orders of accumulation of entropy

David Burguet; Kevin McGoff

Displaying similar documents to “Orders of accumulation of entropy”

Operator entropy inequalities

M. S. Moslehian, F. Mirzapour, A. Morassaei (2013)

Colloquium Mathematicae

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We investigate a notion of relative operator entropy, which develops the theory started by J. I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341-348]. For two finite sequences A = (A₁,...,Aₙ) and B = (B₁,...,Bₙ) of positive operators acting on a Hilbert space, a real number q and an operator monotone function f we extend the concept of entropy by setting $S_{q}^{f} (A | B) : = \sum_{j = 1}^{n} A_{j}^{1 / 2} {(A_{j}^{- 1 / 2} B_{j} A_{j}^{- 1 / 2})}^{q} f (A_{j}^{- 1 / 2} B_{j} A_{j}^{- 1 / 2}) A_{j}^{1 / 2}$ , and then give upper and lower bounds for $S_{q}^{f} (A | B)$ as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004),...

A local approach to $g$ -entropy

Mehdi Rahimi (2015)

Kybernetika

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In this paper, a local approach to the concept of $g$ -entropy is presented. Applying the Choquet‘s representation Theorem, the introduced concept is stated in terms of $g$ -entropy.

The topological entropy versus level sets for interval maps (part II)

Jozef Bobok (2005)

Studia Mathematica

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Let f: [a,b] → [a,b] be a continuous function of the compact real interval such that (i) $c a r d f^{- 1} (y) \geq 2$ for every y ∈ [a,b]; (ii) for some m ∈ ∞,2,3,... there is a countable set L ⊂ [a,b] such that $c a r d f^{- 1} (y) \geq m$ for every y ∈ [a,b]∖L. We show that the topological entropy of f is greater than or equal to log m. This generalizes our previous result for m = 2.

On Pawlak's problem concerning entropy of almost continuous functions

Tomasz Natkaniec, Piotr Szuca (2010)

Colloquium Mathematicae

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We prove that if f: → is Darboux and has a point of prime period different from $2^{i}$ , i = 0,1,..., then the entropy of f is positive. On the other hand, for every set A ⊂ ℕ with 1 ∈ A there is an almost continuous (in the sense of Stallings) function f: → with positive entropy for which the set Per(f) of prime periods of all periodic points is equal to A.

Sequence entropy and rigid σ-algebras

Alvaro Coronel, Alejandro Maass, Song Shao (2009)

Studia Mathematica

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We study relationships between sequence entropy and the Kronecker and rigid algebras. Let (Y,,ν,T) be a factor of a measure-theoretical dynamical system (X,,μ,T) and S be a sequence of positive integers with positive upper density. We prove there exists a subsequence A ⊆ S such that $h_{μ}^{A} (T, ξ |) = H_{μ} (ξ | (X | Y))$ for all finite partitions ξ, where (X|Y) is the Kronecker algebra over . A similar result holds for rigid algebras over . As an application, we characterize compact, rigid and mixing extensions via relative...

Symbolic extensions for nonuniformly entropy expanding maps

David Burguet (2010)

Colloquium Mathematicae

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A nonuniformly entropy expanding map is any ¹ map defined on a compact manifold whose ergodic measures with positive entropy have only nonnegative Lyapunov exponents. We prove that a $^{r}$ nonuniformly entropy expanding map T with r > 1 has a symbolic extension and we give an explicit upper bound of the symbolic extension entropy in terms of the positive Lyapunov exponents by following the approach of T. Downarowicz and A. Maass [Invent. Math. 176 (2009)].

Further results on the generalized cumulative entropy

Antonio Di Crescenzo, Abdolsaeed Toomaj (2017)

Kybernetika

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Recently, a new concept of entropy called generalized cumulative entropy of order $n$ was introduced and studied in the literature. It is related to the lower record values of a sequence of independent and identically distributed random variables and with the concept of reversed relevation transform. In this paper, we provide some further results for the generalized cumulative entropy such as stochastic orders, bounds and characterization results. Moreover, some characterization results...

Entropy solutions for nonhomogeneous anisotropic $Δ_{p ⃗ (\cdot)}$ problems

Elhoussine Azroul, Abdelkrim Barbara, Mohamed Badr Benboubker, Hassane Hjiaj (2014)

Applicationes Mathematicae

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We study a class of anisotropic nonlinear elliptic equations with variable exponent p⃗(·) growth. We obtain the existence of entropy solutions by using the truncation technique and some a priori estimates.

Continuous dependence of the entropy solution of general parabolic equation

Mohamed Maliki (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

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We consider the general parabolic equation : $u_{t} - Δ b (u) + d i v F (u) = f$ in $Q =] 0, T [\times ℝ^{N}, T > 0$ with $u_{0} \in L^{\infty} (ℝ^{N}),$ $for a . e t \in] 0, T [, f (t) \in L^{\infty} (ℝ^{N})$ and $\int_{0}^{T} {∥f (t)∥}_{L^{\infty} (ℝ^{N})} d t < \infty .$ We prove the continuous dependence of the entropy solution with respect to $F,$ $b,$ $f$ and the initial data $u_{0}$ of the associated Cauchy problem. This type of solution was introduced and studied in [MT3]. We start by recalling the definition of weak solution and entropy solution. By applying an abstract result (Theorem...

Margulis Lemma, entropy and free products

Filippo Cerocchi (2014)

Annales de l’institut Fourier

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We prove a Margulis’ Lemma Besson-Courtois-Gallot, for manifolds whose fundamental group is a nontrivial free product $A * B$ , without 2-torsion. Moreover, if $A * B$ is torsion-free we give a lower bound for the homotopy systole in terms of upper bounds on the diameter and the volume-entropy. We also provide examples and counterexamples showing the optimality of our assumption. Finally we give two applications of this result: a finiteness theorem and a volume estimate for reducible manifolds. ...

Jumps of entropy for $C^{r}$ interval maps

David Burguet (2015)

Fundamenta Mathematicae

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We study the jumps of topological entropy for $C^{r}$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f \in C^{r} ([0, 1])$ with $h_{t o p} (f) > (l o g ⁺ | | f^{'} {| |}_{\infty}) / r$ . To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^{r}$ interval maps.

A representation theorem for perfect partitions of $Z^{2}$ -actions with finite entropy

B. Kamiński (1988)

Colloquium Mathematicae

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On the joint entropy of $d$ -wise-independent variables

Dmitry Gavinsky, Pavel Pudlák (2016)

Commentationes Mathematicae Universitatis Carolinae

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How low can the joint entropy of $n$ $d$ -wise independent (for $d \geq 2$ ) discrete random variables be, subject to given constraints on the individual distributions (say, no value may be taken by a variable with probability greater than $p$ , for $p < 1$ )? This question has been posed and partially answered in a recent work of Babai [Entropy versus pairwise independence (preliminary version), http://people.cs.uchicago.edu/ laci/papers/13augEntropy.pdf, 2013]. In this paper we improve some...

Symbolic extensions in intermediate smoothness on surfaces

David Burguet (2012)

Annales scientifiques de l'École Normale Supérieure

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We prove that $𝒞^{r}$ maps with $r > 1$ on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].

The entropy conjecture for diffeomorphisms away from tangencies

Gang Liao, Marcelo Viana, Jiagang Yang (2013)

Journal of the European Mathematical Society

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We prove that every $C^{1}$ diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub’s entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive. ...

Entropy and approximation numbers of embeddings between weighted Besov spaces

Iwona Piotrowska (2008)

Banach Center Publications

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The present paper is devoted to the study of the “quality” of the compactness of the trace operator. More precisely, we characterize the asymptotic behaviour of entropy numbers of the compact map $t r_{Γ} : B_{p ₁, q}^{s} (ℝ ⁿ, w_{ϰ}^{Γ}) \to L_{p ₂} (Γ)$ , where Γ is a d-set with 0 < d < n and $w_{ϰ}^{Γ}$ a weight of type $w_{ϰ}^{Γ} (x) d i s t {(x, Γ)}^{ϰ}$ near Γ with ϰ > -(n-d). There are parallel results for approximation numbers.

On lower bounds for the variance of functions of random variables

Faranak Goodarzi, Mohammad Amini, Gholam Reza Mohtashami Borzadaran (2021)

Applications of Mathematics

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In this paper, we obtain lower bounds for the variance of a function of random variables in terms of measures of reliability and entropy. Also based on the obtained characterization via the lower bounds for the variance of a function of random variable $X$ , we find a characterization of the weighted function corresponding to density function $f (x)$ , in terms of Chernoff-type inequalities. Subsequently, we obtain monotonic relationships between variance residual life and dynamic cumulative residual...

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Bernd Carl, David E. Edmunds (2003)

Studia Mathematica

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For a precompact subset K of a Hilbert space we prove the following inequalities: $n^{1 / 2} c ₙ (c o v (K)) \leq c_{K} (1 + \sum_{k = 1}^{ⁿ} k^{- 1 / 2} e_{k} (K))$ , n ∈ ℕ, and $k^{1 / 2} c_{k + n} (c o v (K)) \leq c [l o g^{1 / 2} (n + 1) ε ₙ (K) + \sum_{j = n + 1}^{\infty} ε_{j} (K) / (j l o g^{1 / 2} (j + 1))]$ , k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and $ε_{k} (K)$ and $e_{k} (K)$ denote the kth entropy and kth dyadic entropy number of K, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers cₙ(cov(K)) provided that the entropy numbers εₙ(K)...

Generalized Fokker-Planck equations and convergence to their equilibria

Piotr Biler, Grzegorz Karch (2003)

Banach Center Publications

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We consider extensions of the classical Fokker-Planck equation uₜ + ℒu = ∇·(u∇V(x)) on $ℝ^{d}$ with ℒ = -Δ and V(x) = 1/2|x|², where ℒ is a general operator describing the diffusion and V is a suitable potential.