Margulis Lemma, entropy and free products

Filippo Cerocchi

Displaying similar documents to “Margulis Lemma, entropy and free products”

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.

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...

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 free stochastic partial differential equation

Yoann Dabrowski (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We get stationary solutions of a free stochastic partial differential equation. As an application, we prove equality of non-microstate and microstate free entropy dimensions under a Lipschitz like condition on conjugate variables, assuming also the von Neumann algebra $R^{ω}$ embeddable. This includes an $N$ -tuple of $q$ -Gaussian random variables e.g. for $| q | N \leq 0.13$ .

ε-Entropy and moduli of smoothness in $L^{p}$ -spaces

A. Kamont (1992)

Studia Mathematica

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The asymptotic behaviour of ε-entropy of classes of Lipschitz functions in $L^{p} (^{d})$ is obtained. Moreover, the asymptotics of ε-entropy of classes of Lipschitz functions in $L^{p} (ℝ^{d})$ whose tail function decreases as $O (λ^{- γ})$ is obtained. In case p = 1 the relation between the ε-entropy of a given class of probability densities on $ℝ^{d}$ and the minimax risk for that class is discussed.

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)...

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.

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.

Topological disjointness from entropy zero systems

Wen Huang, Kyewon Koh Park, Xiangdong Ye (2007)

Bulletin de la Société Mathématique de France

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The properties of topological dynamical systems $(X, T)$ which are disjoint from all minimal systems of zero entropy, $ℳ_{0}$ , are investigated. Unlike the measurable case, it is known that topological $K$ -systems make up a proper subset of the systems which are disjoint from $ℳ_{0}$ . We show that $(X, T)$ has an invariant measure with full support, and if in addition $(X, T)$ is transitive, then $(X, T)$ is weakly mixing. A transitive diagonal system with only one minimal point is constructed. As a consequence, there exists...

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.

Orders of accumulation of entropy

David Burguet, Kevin McGoff (2012)

Fundamenta Mathematicae

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For a continuous map T of a compact metrizable space X with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the context of entropy structures and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if M is a metrizable Choquet...

On some nonlinear nonhomogeneous elliptic unilateral problems involving noncontrollable lower order terms with measure right hand side

C. Yazough, E. Azroul, H. Redwane (2013)

Applicationes Mathematicae

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We prove the existence of entropy solutions to unilateral problems associated to equations of the type $A u - d i v (ϕ (u)) = μ \in L ¹ (Ω) + W^{- 1, p^{'} (\cdot)} (Ω)$ , where A is a Leray-Lions operator acting from $W ₀^{1, p (\cdot)} (Ω)$ into its dual $W^{- 1, p (\cdot)} (Ω)$ and $ϕ \in C ⁰ (ℝ, ℝ^{N})$ .

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.

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.

Pattern avoidance in partial words over a ternary alphabet

Adam Gągol (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Blanched-Sadri and Woodhouse in 2013 have proven the conjecture of Cassaigne, stating that any pattern with $m$ distinct variables and of length at least $2^{m}$ is avoidable over a ternary alphabet and if the length is at least $3 \cdot 2^{m - 1}$ it is avoidable over a binary alphabet. They conjectured that similar theorems are true for partial words – sequences, in which some characters are left “blank”. Using method of entropy compression, we obtain the partial words version of the theorem for ternary words. ...

Some logarithmic function spaces, entropy numbers, applications to spectral theory

Haroske Dorothee

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AbstractIn [18] and [19] we have studied compact embeddings of weighted function spaces on ℝⁿ, $i d : H_{q}^{s} (w (x), ℝ ⁿ) \to L ₚ (ℝ ⁿ)$ , s>0, 1 < q ≤ p< ∞, s-n/q+n/p > 0, with, for example, $w (x) = ⟨ x ⟩^{α}$ , α > 0, or $w (x) = l o g^{β} ⟨ x ⟩$ , β > 0, and $⟨ x ⟩ = {(2 + | x | ²)}^{1 / 2}$ . We have determined the behaviour of their entropy numbers eₖ(id). Now we are interested in the limiting case 1/q = 1/p + s/n. Let $w (x) = l o g^{β} ⟨ x ⟩$ , β > 0. Our results in [18] imply that id cannot be compact for any β > 0, but after replacing the target space Lₚ(ℝⁿ) by a “slightly” larger one, $L ₚ {(l o g L)}_{- a} (ℝ ⁿ)$ , a...

On the directional entropy of ℤ²-actions generated by cellular automata

M. Courbage, B. Kamiński (2002)

Studia Mathematica

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We show that for any cellular automaton (CA) ℤ²-action Φ on the space of all doubly infinite sequences with values in a finite set A, determined by an automaton rule $F = F_{[l, r]}$ , l,r ∈ ℤ, l ≤ r, and any Φ-invariant Borel probability measure, the directional entropy $h_{v ⃗} (Φ)$ , v⃗= (x,y) ∈ ℝ², is bounded above by $m a x (| z_{l} |, | z_{r} |) l o g A$ if $z_{l} z_{r} \geq 0$ and by $| z_{r} - z_{l} |$ in the opposite case, where $z_{l} = x + l y$ , $z_{r} = x + r y$ . We also show that in the class of permutative CA-actions the bounds are attained if the measure considered is uniform Bernoulli.

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

B. Kamiński (1988)

Colloquium Mathematicae

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Measures of maximal entropy for random $β$ -expansions

Karma Dajani, Martijn de Vries (2005)

Journal of the European Mathematical Society

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Let $β > 1$ be a non-integer. We consider $β$ -expansions of the form $\sum_{i = 1}^{\infty} d_{i} / β^{i}$ , where the digits ${(d_{i})}_{i \geq 1}$ are generated by means of a Borel map $K_{β}$ defined on ${0, 1}^{ℕ} \times [0, ⌊ β ⌋ / (β - 1)]$ . We show that $K_{β}$ has a unique mixing measure $ν_{β}$ of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure $ν_{β}$ the digits ${(d_{i})}_{i \geq 1}$ form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness...

Local gradient estimates of $p$ -harmonic functions, $1 / H$ -flow, and an entropy formula

Brett Kotschwar, Lei Ni (2009)

Annales scientifiques de l'École Normale Supérieure

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In the first part of this paper, we prove local interior and boundary gradient estimates for $p$ -harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an existence theorem for weak solutions to the level set formulation of the $1 / H$ (inverse mean curvature) flow for hypersurfaces in ambient manifolds satisfying a sharp volume growth assumption. In the second part of this paper, we consider two parabolic analogues...

The Cauchy problem for a strongly degenerate quasilinear equation

F. Andreu, Vicent Caselles, J. M. Mazón (2005)

Journal of the European Mathematical Society

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We prove existence and uniqueness of entropy solutions for the Cauchy problem for the quasilinear parabolic equation $u_{t} = div 𝐚 (u, D u)$ , where $𝐚 (z, ξ) = \nabla_{ξ} f (z, ξ)$ , and $f$ is a convex function of $ξ$ with linear growth as $∥ ξ ∥ \to \infty$ , satisfying other additional assumptions. In particular, this class includes a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics.