Ergodic group automorphisms are exponentially recurrent

We show that ergodic automorphisms of compact abelian groups have the property that for every nonempty open setU, the measure of the set first returning toU aftern iterates decays exponentially inn. This follows from a result about aperiodic automorphisms of countable abelian groups, whose proof employsp-adic analysis. Partially supported by NSF Grant MCS 7704915.     

Hall's universal group has ample generic automorphisms

We show that the automorphism group of Hall's universal locally finite group has ample generics, that is, it admits comeager diagonal conjugacy classes in all dimensions. Consequently, it has the small index property, is not the union of a countable chain of non‐open subgroups, and has the automatic continuity property. Also, we discuss some algebraic and topological properties of the automorphism group of Hall's universal group. For example, we show that every generic automorphism of Hall's universal group is conjugate to all of its powers, and hence has roots of all orders.     

On the group of conjugating automorphisms of a free group

The group of conjugating automorphisms of a free group and certain subgroups of this group, namely, the group of McCool basis-conjugating automorphisms and the Artin braid group are considered. The Birman theorem on the representation of a braid group by matrices is sharpened. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 92–108, July, 1996.     

Bowtie-free graphs and generic automorphisms

We show that the countable universal ω-categorical bowtie-free graph admits generic automorphisms. Moreover, we show that this graph is not finitely homogenisable.     

Cohomological dimension and symmetric automorphisms of a free group

The author gratefully acknowledges support from the Ruhr-Universit?t, Bochum and the Alexander von Humboldt-Foundation during the preparation of this paper.     

Linear Repetitivity, I. Uniform Subadditive Ergodic Theorems and Applications

This paper is concerned with the concept of linear repetitivity in the theory of tilings. We prove a general uniform subadditive ergodic theorem for linearly repetitive tilings. This theorem unifies and extends various known (sub)additive ergodic theorems on tilings. The results of this paper can be applied in the study of both random operators and lattice gas models on tilings. Received July 19, 2000, and in revised form January 30, 2001. Online publication July 25, 2001.     

超扩散过程的遍历定理

考虑初始测度为Ｌｅｂｅｓｇｕｅ测度μ 的一致椭圆超扩散过程，其分枝特征为ψ（狓，狕）＝犫（狓）狕＋γ（狓）狕２．该文研究这类超过程的占位时过程的极限性质．对系数犫（狓）及γ（狓）做必要的限制，得到了占位时过程在空间维数犱≤２的遍历定理，我们的结果是［６］的补充．     

The Generalized Entropy Ergodic Theorem for Nonhomogeneous Markov Chains

Let \((\xi _n)_{n=0}^\infty \) be a nonhomogeneous Markov chain taking values in a finite state-space \(\mathbf {X}=\{1,2,\ldots ,b\}\). In this paper, we will study the generalized entropy ergodic theorem with almost-everywhere and \(\mathcal {L}_1\) convergence for nonhomogeneous Markov chains; this generalizes the corresponding classical results for Markov chains.     

The converse of the Dominated Ergodic Theorem

The aim of the present paper is to derive a recursive formula of state estimation for a parabolic differential system with a “partially unknown input.” Here the word “partially” unknown input means that the input term of the system is a product of two functions: one is a known function of time variable and of spatial variable; the other is an unknown function of spatial variable. It is desired to perform the state estimation by identifying the latter function. The formula is derived applying the results of J. L. Lions for the decoupling of Riccati type.     

The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift

Research Supported in part by the Alfred P. Sloan Foundation, The National Science Foundation, and the PSC-CUNY Fellowship Award Program.     

A Uniform Ergodic Theorem for Commuting Semigroups

The existence of the limit, in the uniform topology, of theaverages of a finite family of commuting strongly continuoussemigroups of bounded linear operators in a Banach space ischaracterised in terms of infinitesimal generators.     

无穷迭代函数系统的遍历定理

度量空间的压缩映射的一个集合称为一个迭代函数系统．凝聚迭代函数系统可以被看成无穷迭代函数系统．研究了紧度量空间上的无穷迭代函数系统．利用Banach极限的特性和均匀压缩性,证明了紧度量空间上无穷迭代函数系统的随机迭代算法满足遍历性．于是,凝聚迭代函数系统的随机迭代算法也满足遍历性．     

The speed of convergence in the Ergodic Theorem

No general statement can be made about the speed with which convergence takes place in the Ergodic Theorem, in the sense that one can never be sure that convergence of the remainder to zero is fast enough to make, through term-by-term multiplication, any divergent series become convergent. As a corollary we obtain the nonexistence in general of the one-sided ergodic Hilbert transform; that is, there is no pointwise nonhomogeneous ergodic theorem.     

迭代函数系统的遍历性质—Elton定理的改进

本文改进Elton定理的结论,得到迭代函数系统的几个新的遍历定理。     

Determining whether a finite group of outer automorphisms of a free group is geometric

In this article we give necessary and sufficient conditions for a given finite group of outer automorphisms to be induced by the action of a group of orientation-preserving homeomorphisms on the fundamental group of a punctured surface. When the group is abelian, necessary and sufficient conditions can also be given in the absence of orientability assumptions. These properties are formulated in terms of the finite automorphism groups which project into the given outer automorphism group: each non-trivial automorphism in any such group can fix at most a cyclic subgroup of the fundamental group.