Entropy-like functionals: conceptual background and some results

Miroslav Katětov

Displaying similar documents to “Entropy-like functionals: conceptual background and some results”

Convexity inequalities for estimating generalized conditional entropies from below

Alexey E. Rastegin (2012)

Kybernetika

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Generalized entropic functionals are in an active area of research. Hence lower and upper bounds on these functionals are of interest. Lower bounds for estimating Rényi conditional $α$ -entropy and two kinds of non-extensive conditional $α$ -entropy are obtained. These bounds are expressed in terms of error probability of the standard decision and extend the inequalities known for the regular conditional entropy. The presented inequalities are mainly based on the convexity of some functions....

Universally typical sets for ergodic sources of multidimensional data

Tyll Krüger, Guido F. Montúfar, Ruedi Seiler, Rainer Siegmund-Schultze (2013)

Kybernetika

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We lift important results about universally typical sets, typically sampled sets, and empirical entropy estimation in the theory of samplings of discrete ergodic information sources from the usual one-dimensional discrete-time setting to a multidimensional lattice setting. We use techniques of packings and coverings with multidimensional windows to construct sequences of multidimensional array sets which in the limit build the generated samples of any ergodic source of entropy rate below...

Entropy and growth of expanding periodic orbits for one-dimensional maps

A. Katok, A. Mezhirov (1998)

Fundamenta Mathematicae

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Let f be a continuous map of the circle $S^{1}$ or the interval I into itself, piecewise $C^{1}$ , piecewise monotone with finitely many intervals of monotonicity and having positive entropy h. For any ε > 0 we prove the existence of at least $e^{(h - ε) n_{k}}$ periodic points of period $n_{k}$ with large derivative along the period, $| {(f^{n_{k}})}^{'} | > e^{(h - ε) n_{k}}$ for some subsequence $n_{k}$ of natural numbers. For a strictly monotone map f without critical points we show the existence of at least $(1 - ε) e^{h n}$ such points.

The Dugundji extension property can fail in ωµ -metrizable spaces

Ian Stares, Jerry Vaughan (1996)

Fundamenta Mathematicae

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We show that there exist $ω_{μ}$ -metrizable spaces which do not have the Dugundji extension property ( $2^{ω_{1}}$ with the countable box topology is such a space). This answers a question posed by the second author in 1972, and shows that certain results of van Douwen and Borges are false.

Standardness of sequences of σ-fields given by certain endomorphisms

Jacob Feldman, Daniel Rudolph (1998)

Fundamenta Mathematicae

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Let E be an ergodic endomorphism of the Lebesgue probability space X, ℱ, μ. It gives rise to a decreasing sequence of σ-fields $ℱ, E^{- 1} ℱ, E^{- 2} ℱ, . . .$ A central example is the one-sided shift σ on $X = {0, 1}^{ℕ}$ with $\frac{1}{2}, \frac{1}{2}$ product measure. Now let T be an ergodic automorphism of zero entropy on (Y, ν). The [I|T] endomorphismis defined on (X× Y, μ× ν) by $(x, y) \to (σ (x), T^{x (1)} (y))$ . Here ℱ is the σ-field of μ× ν-measurable sets. Each field is a two-point extension of the one beneath it. Vershik has defined as “standard” any decreasing sequence of...

Each nowhere dense nonvoid closed set in Rn is a σ-limit set

Andrei Sivak (1996)

Fundamenta Mathematicae

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We discuss main properties of the dynamics on minimal attraction centers (σ-limit sets) of single trajectories for continuous maps of a compact metric space into itself. We prove that each nowhere dense nonvoid closed set in $ℝ^{n}$ , n ≥ 1, is a σ-limit set for some continuous map.

Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

Sergiĭ Kolyada, Michał Misiurewicz, L’ubomír Snoha (1999)

Fundamenta Mathematicae

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The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces ${(X_{i})}_{i = 1}^{\infty}$ and a sequence of continuous maps ${(f_{i})}_{i = 1}^{\infty}$ , $f_{i} : X_{i} \to X_{i + 1}$ , is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of $f_{n} ○ . . . ○ f_{2} ○ f_{1}$ . As an application we construct a large class of smooth triangular maps of the square of type $2^{\infty}$ and...

A classification of definable forcings on ω1

Jindřich Zapletal (1997)

Fundamenta Mathematicae

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Under the assumption of the existence of sharps for reals all simply definable posets on $ω_{1}$ are classified up to forcing equivalence.

Conspectus materiae tomorum LXXXI–XC (1997–1999)

(1999)

Acta Arithmetica

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