最大平均度量下的Bowen维数熵与测度下局部熵

熵是反映动力系统复杂性的一个非常重要的量.本文研究了平均意义下的动力系统的性质,对于最大平均度量,引入了Bowen维数熵以及测度下局部熵的概念.并研究了它们之间的关系,说明了在最大平均度量下,Bowen维数熵依然可以由测度下局部熵估计.     

On the Critical Dimension and AC Entropy for Markov Odometers

Markov odometers are natural models for non-homogeneous Markov chains, and are natural generalisations of infinite product measures. We show how to calculate the critical dimension of these measures: this is an invariant which describes the asymptotic growth rate of sums of Radon-Nikodym derivatives. This interesting invariant appears to give a kind of entropy for non-singular odometer actions. The techniques require a law of large numbers for inhomogeneous Markov chains.     

On the Critical Dimension and AC Entropy for Markov Odometers

Markov odometers are natural models for non-homogeneous Markov chains, and are natural generalisations of infinite product measures. We show how to calculate the critical dimension of these measures: this is an invariant which describes the asymptotic growth rate of sums of Radon-Nikodym derivatives. This interesting invariant appears to give a kind of entropy for non-singular odometer actions. The techniques require a law of large numbers for inhomogeneous Markov chains.     

Z~d-作用下的Bowen拓扑熵理论

在Z~d-作用下定义了Bowen维数熵,研究了Z~d-作用下Bowen维数熵的一些性质,证明了X中的任意子集的Bowen维数熵可以通过该子集中的点的测度下局部熵估计.     

一维单个守恒型方程的二阶熵耗散格式

本文考虑一维单个守恒律方程，对其设计了一种非线性守恒型差分格式，此格式为二阶Godunov型的，用的是分片线性重构，重构函数的斜率是根据熵耗散得到的，格式满足熵条件，且数值实验表明格式具有非线性稳定性，在此格式中一个所谓的熵耗散函数起了很重要的作用，它在每个网格的计算中耗散熵，在文中我们给出了熵耗散函数应满足的条件，并给出了一种具体的构造形式，最后给出了一些数值算例，从中可看出熵耗散函数是如何抑制非物理振荡的，及格式对计算的有效性。     

Delays, Recurrence and Ordinals

We apply set-theoretical ideas to an iteration problem of dynamicalsystems. Among other results, we prove that these iterationsnever stabilise later than the first uncountable ordinal; forevery countable ordinal we give examples in Baire space andin Cantor space of an iteration that stabilises exactly at thatordinal; we give an example of an iteration with recursive datawhich stabilises exactly at the first non-recursive ordinal;and we find new examples of complete analytic sets simply definablefrom concepts of recurrence. 2000 Mathematics Subject Classification:primary 03E15, 37B20, 54H05; secondary 37B10, 37E15.     

Fractal Percolation,Porosity, and Dimension

We study the porosity properties of fractal percolation sets \(E\subset \mathbb {R}^d\). Among other things, for all \(0<\varepsilon <\tfrac{1}{2}\), we obtain dimension bounds for the set of exceptional points where the upper porosity of E is less than \(\tfrac{1}{2}-\varepsilon \), or the lower porosity is larger than \(\varepsilon \). Our method works also for inhomogeneous fractal percolation and more general random sets whose offspring distribution gives rise to a Galton–Watson process.     

Dimension,Graph and Hypergraph Coloring

There is a natural way to associate with a poset P a hypergraph H _P, called the hypergraph of incomparable pairs, so that the dimension of P is the chromatic number of H _P. The ordinary graph G _P of incomparable pairs determined by the edges in H _P of size 2 can have chromatic number substantially less than H _P. We give a new proof of the fact that the dimension of P is 2 if and only if G _P is bipartite. We also show that for each t 2, there exists a poset P _t for which the chromatic number of the graph of incomparable pairs of P _t is at most 3 t – 4, but the dimension of P _t is at least (3 / 2) ^{t – 1}. However, it is not known whether there is a function f: NN so that if P is a poset and the graph of incomparable pairs has chromatic number at most t, then the dimension of P is at most f(t).