Measure-theoretic pressure and topological pressure in mean metrics

In this paper, using spanning sets in mean metrics we construct a new definition of measure-theoretic pressure of ergodic measures over a topological dynamical system. And we establish a pressure version of Katok's entropy formula in the case of mean metrics. Furthermore, we also introduce a new definition of topological pressure by replacing the Bowen metrics with the corresponding mean metrics, and prove that the new topological pressure is equivalent to the classical topological pressure. Then the variational principle for the newly defined topological pressure is obtained.     

Some results on the entropy of non-autonomous dynamical systems

In this paper, we advance the entropy theory of discrete non-autonomous dynamical systems that was initiated by Kolyada and Snoha in 1996. The first part of the paper is devoted to the measure-theoretic entropy theory of general topological systems. We derive several conditions guaranteeing that an initial probability measure, when pushed forward by the system, produces an invariant measure sequence whose entropy captures the dynamics on arbitrarily fine scales. In the second part of the paper, we apply the general theory to the non-stationary subshifts of finite type, introduced by Fisher and Arnoux. In particular, we give sufficient conditions for the variational principle, relating the topological and measure-theoretic entropy, to hold.     

关于齐次树指标马氏链的广义熵遍历定理

本文主要研究有限状态齐次树指标Markov链的强大数定律和广义熵遍历定理．熵遍历定理研究的是信息论中信源的渐近均分割性,树指标Markov链是近年来概率论的研究方向之一．首先,参照非齐次Markov链广义熵密度概念,本文给出了树指标Markov链的广义熵密度的定义．然后,通过构造一组期望值为1的随机变量,利用Markov不等式和Borel-Cantelli引理,证明得到了定义在树指标Markov链上一类随机变量的延迟平均的强极限定理．最后,利用上述定理的推论,我们证明得到了Cayley树上有限状态Markov链状态出现次数的延迟平均的强大数定律和广义熵遍历定理．本文的结果是对一些已有结果的推广．     

Families of ergodic and exact one-dimensional maps

We give a parametrized family of rational interval maps of degree two, each ergodic, exact and preserving a measure equivalent to a Lebesgue measure. The family includes the unique quadratic Chebyshev polynomial as its only polynomial map. We extend the family to other settings on the circle and real line. We also give numerical approximations to the entropy of the equivalent invariant measure and the Hausdorff dimension of the singular measure of maximal entropy.     

A self-dual induction for three-interval exchange transformations

We describe completely a new induction process on three-interval exchange transformations, under its additive and multiplicative forms; we describe the natural extension of the induction map, show that it is self-dual for the notion of duality which links the Rauzy/Zorich and da Rocha inductions, and compute a finite ergodic invariant measure (for the multiplicative form). Then we explain the connection with the published negative slope algorithm of Ferenczi, Holton and Zamboni [S. Ferenczi, C. Holton, and L. Zamboni, The structure of three-interval exchange transformations I: An arithmetic study, Ann. Inst. Fourier (Grenoble), 51 (2001), pp. 861–901], and use our measure-theoretic study to get new results on three-interval exchange transformations.     

Topological entropy dimension for noncompact sets

Similar with the fractal dimension, we introduce the concept of topological entropy dimension to classify the sets with entropy zero. We prove that the entropy dimension of the space in this article is not greater than that defined by De Carvalho, where he introduced the entropy dimension for the system, and give some examples indicating that such inequality is optimal. Some basic propositions of entropy dimension are discussed and it turns out that the entropy dimension is invariant under conjugacy. The property of the countable stability and a power rule for the entropy dimension of any set are obtained. It is shown that any set shares the same entropy dimension with its image set.     

Density of metric entropies for linear toral automorphisms

In this article we show that for every linear toral automorphism, including the non-hyperbolic ones, the entropies of ergodic measures form a dense set in the interval from zero to the topological entropy.     

A class of generalized Birnbaum-Saunders distributions for reliability modeling

In this paper, we generalize the Birnbaum-Saunders (BS) distribution by two ways. One is based on the mixture representation of BS distribution, and a flexible weight is adopted to describe the kurtosis of the distribution. The other way is based on the transformation property of BS distribution, and we incorporate a power parameter in the transformation to describe the skewness of the distribution. Then a four-parameter BS distribution including skewness and kurtosis parameters is induced by combining the two ways. The properties of these generalized BS distributions are investigated. Then, the expectation maximization (EM) algorithm is proposed to estimate the parameters. Real data analysis is performed to illustrate the superiority of the generalized BS distributions. Finally, some potential generalizations are discussed.     

Generator of quasi-periodic oscillations featuring two-dimensional torus doubling bifurcations

A new autonomous differential dynamical system with dimension N = 4 is introduced, which has solutions in the form of stable two-frequency oscillations and features a sequence of period-doubling bifurcations of two-dimensional ergodic tori. At the points of period-doubling bifurcations, no resonances are observed on a torus and only ergodic tori exhibit doubling.     

Relative rotation number for stochastic systems: dynamical and topological applications

We introduce a concept of relative rotation number to unify many different approaches of rotation number in non-linear dynamical systems. We present an ergodic result of existence a.s. for stochastic systems. In higher dimension, we show that the natural idea of projecting into a plane does work well a.s. for any plane (different from deterministic systems where projections may be degenerate). A number of further properties (invariance by homotopy and by conjugacy) and applications are presented.     

Topological pressure dimension for almost additive potentials

This paper is devoted to the study of the topological pressure dimension for almost additive sequences, which is an extension of topological entropy dimension. We investigate fundamental properties of the topological pressure dimension for almost additive sequences. In particular, we study the relationships among different types of topological pressure dimension and identify an inequality relating them. Also, we show that the topological pressure dimension is always equal to or greater than 1 for certain special almost additive sequence.     

Topological properties of invariant sets for two-dimensional hyperbolic toral automorphisms

We study topological properties of invariant sets of Anosov diffeomorphisms with holes. Results related to cardinality, local maximality, entropy and dimension are presented. The main result states that the Hausdorff dimension of the invariant set can be computed by the entropy of the invariant set together with the hyperbolicity constants.     

Random perturbations of codimension one homoclinic tangencies in dimension 3

Adding small random parametric noise to an arc of diffeomorphisms of a manifold of dimension 3, generically unfolding a codimension one quadratic homoclinic tangency q associated to a sectionally dissipative saddle fixed point p , we obtain not more than a finite number of 'physical' probability measures, whose ergodic basins cover the orbits which are recurrent to a neighbourhood of the tangency point q . This result is in contrast to the extension of Newhouse's phenomenon of coexistence of infinitely many sinks obtained by Palis and Viana in this setting. There is a similar result for the simpler bidimensional case whose proof relies on geometric arguments. We now extend the arguments to cover three-dimensional manifolds.     

Different forms of entropy dimension for zero entropy systems

The aim of this paper is to introduce the lower s-topological entropy to distinguish zero entropy systems. That this quantity is an invariant factor under topological conjugacy and a power rule is shown. Some examples are given to show that the lower entropy dimension can attain any value in (0, 1), and are different with the upper one and the entropy dimension in the sense of Bowen. A counterexample is used to indicate that the product rule does not hold, and the lower s-topological entropy of the subsystem for the non-wandering set can be strictly less than that of the system when 0 < s < 1. Finally, this study also constructs a dynamical system to show that the transitive system with zero entropy dimension may not be minimal.     

在共混物界面研究中分数维的物理意义

讨论了在共混物界面研究中分数维的物理意义,指出它是反映共混物界面特征的态函数。     

Entropy of partially polarized light and application to statistical processing techniques

We have analyzed entropy properties of coherent and partially polarized light in an arbitrary number of spatial dimensions. We show that for Gaussian fields, the Shannon entropy is a simple function of the intensity and of the Barakat degree of polarization. In particular, we provide a probabilistic interpretation of this definition of the degree of polarization. Using information theory results, we also deduce some physical properties of partially polarized light such as additivity of the entropy and depolarization effects induced by mixing partially polarized states of light. Finally, we demonstrate that entropy measures can play an important role in segmentation and detection tasks.     

On modeling and performance analysis of non-cooperative multi-antenna multi-user MIMO systems

This work presents modeling and performance analysis of beamforming in non-cooperative multi-user multiple-input multiple-output (MU-MIMO) cellular systems using stochastic geometry based abstraction models. Particularly, a downlink MU-MIMO system is investigated in which an array of antenna elements at the base stations (BS) serves several user devices. We derive the closed-form expression of the complimentary cumulative distribution function (CCDF) of signal-to-interference-plus-noise ratio (SINR) representing the coverage probability for orthogonal pilot sequences and thereby characterize a number of related special cases. Moreover, closed-form expressions are also obtained for the CCDF of signal-to-interference ratio (SIR) for non-orthogonal pilot sequences and we also derive useful special cases of it and identify the support region of our analysis. Lastly, we define the numerical expression of ergodic capacity and utilize a partial fraction expansion to solve a simple form. Our contribution in this work is twofold, we not only provide a pedagogical tutorial-like exposition of stochastic geometry technique to model terrestrial networks, but also augment the existing literature in the field with aspects of frequency reuse factor, an indicator function for pilot usage, and the effect of pilot contamination. Simulation results are presented to support the theoretical part.     

非线性科学及其在医学中的应用

简要介绍了非线性科学中的分形维方法和信息熵方法及其用于房颤信息的处理和分析，计算了未房颤时P波的及房颤时f波的格子维和信息熵，特别是，比较了未房颤时和发生房颤时的相关计算结果的显著差别，从而很容易判断是否发生了房颤。     

Fractal dimensions for fatigue fracture surfaces performed on micro- and meso-scale levels

A methodology was developed to use digital photograph fracture for calculating the integral fractal dimension and spectrum fractal dimensions for two perpendicular directions. The fractal dimension value, which was discovered by the variation of the pixel size up until the fractal dimension became unchangeable, i.e., where d _\min,d _\max are the borders of the scale level used. The effectiveness of the methodology performed showed the basis of the fracture surface analyses for Al-based alloys BS L65 and 2024 T351 (analogous to Al alloy D16T), specimens of which were tested by pull-push with imitation fretting corrosion damage of the sur- face and with four-point bending after the laser peening, respectively.