On entropy and Hausdorff dimension of measures defined through a non-homogeneous Markov process

Athanasios Batakis

Displaying similar documents to “On entropy and Hausdorff dimension of measures defined through a non-homogeneous Markov process”

Homogeneity and non-coincidence of Hausdorff and box dimensions for subsets of ℝⁿ

Anders Nilsson, Peter Wingren (2007)

Studia Mathematica

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A class of subsets of ℝⁿ is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple (r,s,t) of numbers in the interval (0,n] with r < s < t, a compact set K is constructed so that for any non-empty subset U relatively open in K, we have $(d i m_{H} (U), {\underset{̲}{d i m}}_{B} (U), {\bar{d i m}}_{B} (U)) = (r, s, t)$ . Moreover, $2^{- n} \leq H^{r} (K) \leq 2 n^{r / 2}$ .

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),...

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...

A local approach to $g$ -entropy

Mehdi Rahimi (2015)

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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.

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)].

Dimensions of non-differentiability points of Cantor functions

Yuanyuan Yao, Yunxiu Zhang, Wenxia Li (2009)

Studia Mathematica

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For a probability vector (p₀,p₁) there exists a corresponding self-similar Borel probability measure μ supported on the Cantor set C (with the strong separation property) in ℝ generated by a contractive similitude $h_{i} (x) = a_{i} x + b_{i}$ , i = 0,1. Let S denote the set of points of C at which the probability distribution function F(x) of μ has no derivative, finite or infinite. The Hausdorff and packing dimensions of S have been found by several authors for the case that $p_{i} > a_{i}$ , i = 0,1. However, when p₀ < a₀...

Gradient flows of the entropy for jump processes

Matthias Erbar (2014)

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

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We introduce a new transport distance between probability measures on $ℝ^{d}$ that is built from a Lévy jump kernel. It is defined via a non-local variant of the Benamou–Brenier formula. We study geometric and topological properties of this distance, in particular we prove existence of geodesics. For translation invariant jump kernels we identify the semigroup generated by the associated non-local operator as the gradient flow of the relative entropy w.r.t. the new distance and show that the...

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...

Some dimensional results for a class of special homogeneous Moran sets

Xiaomei Hu (2016)

Czechoslovak Mathematical Journal

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We construct a class of special homogeneous Moran sets, called ${m_{k}}$ -quasi homogeneous Cantor sets, and discuss their Hausdorff dimensions. By adjusting the value of ${m_{k}}_{k \geq 1}$ , we constructively prove the intermediate value theorem for the homogeneous Moran set. Moreover, we obtain a sufficient condition for the Hausdorff dimension of homogeneous Moran sets to assume the minimum value, which expands earlier works.

The multifractal box dimensions of typical measures

Frédéric Bayart (2012)

Fundamenta Mathematicae

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We compute the typical (in the sense of Baire’s category theorem) multifractal box dimensions of measures on a compact subset of $ℝ^{d}$ . Our results are new even in the context of box dimensions of measures.

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.

Markov's property of the Cantor ternary set

Leokadia Białas, Alexander Volberg (1993)

Studia Mathematica

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We prove that the Cantor ternary set E satisfies the classical Markov inequality (see [Ma]): for each polynomial p of degree at most n (n = 0, 1, 2,...) (M) $| p^{'} (x) | \leq M n^{m} s u p_{E} | p |$ for x ∈ E, where M and m are positive constants depending only on E.

Semiconjugacy to a map of a constant slope

Jozef Bobok (2012)

Studia Mathematica

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It is well known that any continuous piecewise monotone interval map f with positive topological entropy $h_{t o p} (f)$ is semiconjugate to some piecewise affine map with constant slope $e^{h_{t o p} (f)}$ . We prove this result for a class of Markov countably piecewise monotone continuous interval maps.

Gibbs measures in a markovian context and dimension

L. Farhane, G. Michon (2001)

Colloquium Mathematicae

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The main goal is to use Gibbs measures in a markovian matrices context and in a more general context, to compute the Hausdorff dimension of subsets of [0, 1[ and [0, 1[². We introduce a parameter t which could be interpreted within thermodynamic framework as the variable conjugate to energy. In some particular cases we recover the Shannon-McMillan-Breiman and Eggleston theorems. Our proofs are deeply rooted in the properties of non-negative irreducible matrices and large deviations techniques...

Invariant measures related with randomly connected Poisson driven differential equations

Katarzyna Horbacz (2002)

Annales Polonici Mathematici

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We consider the stochastic differential equation (1) $d u (t) = a (u (t), ξ (t)) d t + \int_{Θ} σ {(u (t), θ)}_{p} (d t, d θ)$ for t ≥ 0 with the initial condition u(0) = x₀. We give sufficient conditions for the existence of an invariant measure for the semigroup ${P^{t}}_{t \geq 0}$ corresponding to (1). We show that the existence of an invariant measure for a Markov operator P corresponding to the change of measures from jump to jump implies the existence of an invariant measure for the semigroup ${P^{t}}_{t \geq 0}$ describing the evolution of measures along trajectories and vice versa. ...

Sets of $β$ -expansions and the Hausdorff measure of slices through fractals

Tom Kempton (2016)

Journal of the European Mathematical Society

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We study natural measures on sets of $β$ -expansions and on slices through self similar sets. In the setting of $β$ -expansions, these allow us to better understand the measure of maximal entropy for the random $β$ -transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing,...

Hausdorff dimension of scale-sparse Weierstrass-type functions

Abel Carvalho (2011)

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The aim of this paper is to calculate (deterministically) the Hausdorff dimension of the scale-sparse Weierstrass-type functions $W_{s} (x) : = \sum_{j \geq 1} ρ^{- γ^{j} s} g (ρ^{γ^{j}} x + θ_{j})$ , where ρ > 1, γ > 1 and 0 < s < 1, and g is a periodic Lipschitz function satisfying some additional appropriate conditions.

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...

Entropy of a doubly stochastic Markov operator and of its shift on the space of trajectories

Paulina Frej (2012)

Colloquium Mathematicae

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We define the space of trajectories of a doubly stochastic operator on L¹(X,μ) as a shift space $(X^{ℕ}, ν, σ)$ , where ν is a probability measure defined as in the Ionescu-Tulcea theorem and σ is the shift transformation. We study connections between the entropy of a doubly stochastic operator and the entropy of the shift on the space of trajectories of this operator.

Estimates of capacity of self-similar measures

Jozef Myjak, Tomasz Szarek (2002)

Annales Polonici Mathematici

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We give lower and upper estimates of the capacity of self-similar measures generated by iterated function systems $(S_{i}, p_{i}) : i = 1, . . ., N$ where $S_{i}$ are bi-lipschitzean transformations.

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.