On directional entropy functions

Given aZ ²-process, the measure theoretic directional entropy function,h( % MathType!End!2!1!), is defined on % MathType!End!2!1!. We relate the directional entropy of aZ ²-process to itsR ² suspension. We find a sufficient condition for the continuity of directional entropy function. In particular, this shows that the directional entropy is continuous for aZ ²-action generated by a cellular automaton; this finally answers a question of Milnor [Mil]. We show that the unit vectors whose directional entropy is zero form aG _δ subset ofS ¹. We study examples to investigate some properties of directional entropy functions. This research is supported in part by BSRI and KOSEF 95-0701-03-3.     

A note on entropy and inner functions

This paper constructs an inner function with infinite entropy.     

The Busemann-Petty problem on entropy of log-concave functions

Science China Mathematics - The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space ?nwith smaller central hyperplane sections necessarily have smaller volumes....     

Topological entropy of continuous functions on topological spaces

Adler, Konheim and McAndrew introduced the concept of topological entropy of a continuous mapping for compact dynamical systems. Bowen generalized the concept to non-compact metric spaces, but Walters indicated that Bowen’s entropy is metric-dependent. We propose a new definition of topological entropy for continuous mappings on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required), investigate fundamental properties of the new entropy, and compare the new entropy with the existing ones. The defined entropy generates that of Adler, Konheim and McAndrew and is metric-independent for metrizable spaces. Yet, it holds various basic properties of Adler, Konheim and McAndrew’s entropy, e.g., the entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have a same entropy, the entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new entropy coincides with Adler, Konheim and McAndrew’s entropy for compact systems.     

Rational functions with identical measure of maximal entropy

We discuss when two rational functions f and g   can have the same measure of maximal entropy. The polynomial case was completed by Beardon, Levin, Baker–Eremenko, Schmidt–Steinmetz, etc., 1980s–1990s, and we address the rational case following Levin and Przytycki (1997). We show: _μf=_μg${\mu }_f={\mu }_g$ implies that f and g   share an iterate (fⁿ=g^m$f^n=g^m$ for some n and m) for general f   with degree d≥3$d\ge 3$. And for generic f∈_Ratd≥3$f\in {\mathrm{Rat}}_{d\ge 3}$, _μf=_μg${\mu }_f={\mu }_g$ implies g=fⁿ$g=f^n$ for some n≥1$n\ge 1$. For generic f∈_Rat2$f\in {\mathrm{Rat}}_2$, _μf=_μg${\mu }_f={\mu }_g$ implies that g=fⁿ$g=f^n$ or _σf°fⁿ${\sigma }_f°f^n$ for some n≥1$n\ge 1$, where _σf∈_PSL2(C)${\sigma }_f\in {\mathit{PSL}}_2(\mathbb{C})$ permutes two points in each fiber of f. Finally, we construct examples of f and g   with _μf=_μg${\mu }_f={\mu }_g$ such that fⁿ≠σ°g^m$f^n\ne \sigma °g^m$ for any σ∈_PSL2(C)$\sigma \in {\mathit{PSL}}_2(\mathbb{C})$ and m,n≥1$m,n\ge 1$.     

Relative entropy functions for factor maps between subshifts

Let and be topological dynamical systems and a factor map. A function is a compensation function for if for all . For a factor code between subshifts of finite type, we analyze the associated relative entropy function and give a necessary condition for the existence of saturated compensation functions. Necessary and sufficient conditions for a map to be a saturated compensation function will be provided.

     

Zeta functions and analyticity of metric entropy for anosov systems

In this note we shall give an alternative proof, using generalized zeta functions, of a theorem of Contreras that the metric entropy of aC ^ω Anosov diffeomorphism or flow has a real analytic dependence on perturbations.     

On the topological entropy of continuous and almost continuous functions

We prove some results concerning the entropy of continuous and almost continuous functions. We first introduce the notions of bundle entropy and (strong) entropy points and then we study properties of these notions in connection with the theory of multifunctions. Based on these facts we give theorems about approximation of functions defined and assuming their values on compact manifold by functions having strong entropy points.     

Extension of quantum information by using entropy of deterministic functions

The concept of entropy of random variables first defined by Shannon has been generalized later in various ways by mathematicians who so obtained new measures of uncertainty, again for random variables. Recently, the author suggested another extension which provides a meaningful definition for the entropy of deterministic functions, both in the sense of Shannon and of Renyi. These measures of uncertainty are different from those which are utilized by physicists in the study of chaotic dynamics, like the Kolmogorov entropy for instance.

The aim of this paper is to go a step further, and to derive measures of uncertainty for operators, by using exactly the same rationale. After a short background on the entropies of deterministic functions, one obtains successively the entropy of a constant square matrix operator, the entropy of a varying square matrix operator, the entropy of the kernel of an integral transformation, and the entropy of differential operators defined by square matrices.

Then one carefully exhibits the relation which exists between these results and the quantum mechanical entropy first introduced by Von Neumann, and one so obtains a new generalized quantum mechanical entropy which applies to matrics which are not necessarily density matrices. Finally, some illustrative examples for future applications are outlined.     

On extractors and exposure‐resilient functions for sublogarithmic entropy

We study resilient functions and exposure‐resilient functions in the low‐entropy regime. A resilient function (a.k.a. deterministic extractor for oblivious bit‐fixing sources) maps any distribution on n ‐bit strings in which k bits are uniformly random and the rest are fixed into an output distribution that is close to uniform. With exposure‐resilient functions, all the input bits are random, but we ask that the output be close to uniform conditioned on any subset of n ‐ k input bits. In this paper, we focus on the case that k is sublogarithmic in n. We simplify and improve an explicit construction of resilient functions for k sublogarithmic in n due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in k rather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction (O(log k) output bits) is optimal for extractors computable by a large class of space‐bounded streaming algorithms. Next, we show that a random function is a resilient function with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure‐resilient function with high probability even if k is as small as a constant (resp. loglog n). No explicit exposure‐resilient functions achieving these parameters are known. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

基于自协调函数的信道容量和最大熵的计算

信道容量和最大熵的计算是信息论中的经典问题.讨论了利用自协调函数理论计算信道容量,尤其是带约束的信道容量的方法.将最大熵的计算作为信道容量计算的特殊情况.作为应用,在证明了单位成本信道容量函数的单峰性的基础上,提出了其相应的多项式时间算法.     

On a class of functions of the interval with zero topological entropy

We prove that functions defined on the Cantor set Q in such a way that each element of the transformed sequence depends only on the elements of the original one placed in the same or preceding positions, and extended by linearity on the whole unit interval I, have zero topological entropy.     

A note on the ε entropy of monotone functions in the levy norm

The ? entropy of the class $\text{F1}$ of real-valued monotone functions from [0, 1] to [0, 1] in the usual Chebyshev norm is infinite, due to the discontinuities in some of the $\text{f ? F1}$. One of the class of norms introduced by P. Lévy for analyzing the convergence of distribution functions gives finite ? entropy $\text{～(}\text{1}\text{?}\text{)}$. (There is an obvious extension to the class $\text{F1}{}_{\mathrm{AB}}$ of monotone functions from [0, A] to [0, B].     

Estimates for the entropy numbers of the Nikol'skii–Besov classes of functions with mixed smoothness in the space of quasi-continuous functions

We obtained order estimates for the entropy numbers of the Nikol'skii–Besov classes of functions $B_{p,\theta }^{\mathit{r}}({\mathbb{T}}^d)$ with mixed smoothness in the metric of the space of quasi-continuous functions $QC({\mathbb{T}}^d)$. We also showed that for $2\le p\le \infty$, , $r_1>\frac{1}{2}$, $d\ge 2$, the estimate of the corresponding asymptotic characteristic is exact in order.     

Computation of the ε entropy of the space of continuous functions with the Hausdorff metric

The number K_p,q, i.e., the number of (p, q) corridors of closed domains which are convex in the vertical direction, consist of elementary squares of the integral lattice, are situated within a rectangle of the size q × p, and completely cover the side of length p of this rectangle under projection is computed. The asymptotic (K_p,q/q²)^1/p → λ, as p, q → ∞, where λ = 0.3644255… is the maximum root of the equation₁F₁(-1/2 − 1/(16λ), 1/2, 1/(4λ)) = 0,₁F₁ being the confluence hypergeometric function, is established. These results allow us to compute the ε entropy of the space of continuous functions with the Hausdorff metric. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 39–50, January, 1977.     

Nonsymmetric entropy and maximum nonsymmetric entropy principle

Under the frame of a statistical model, the concept of nonsymmetric entropy which generalizes the concepts of Boltzmann’s entropy and Shannon’s entropy, is defined. Maximum nonsymmetric entropy principle is proved. Some important distribution laws such as power law, can be derived from this principle naturally. Especially, nonsymmetric entropy is more convenient than other entropy such as Tsallis’s entropy in deriving power laws.