Expansive Subdynamics

This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let be a continuous action of on an infinite compact metric space. For each subspace of we introduce a notion of expansiveness for along , and show that there are nonexpansive subspaces in every dimension . For each the set of expansive -dimensional subspaces is open in the Grassmann manifold of all -dimensional subspaces of . Various dynamical properties of are constant, or vary nicely, within a connected component of , but change abruptly when passing from one expansive component to another. We give several examples of this sort of ``phase transition,' including the topological and measure-theoretic directional entropies studied by Milnor, zeta functions, and dimension groups. For we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an . The unresolved case is that of the complement of a single irrational direction. Algebraic examples using commuting automorphisms of compact abelian groups are an important source of phenomena, and we study several instances in detail. We conclude with a set of problems and research directions suggested by our analysis.

     

The fixed subgroups of homeomorphisms of Seifert manifolds

Let M be a compact connected orientable Seifert manifold with hyperbolic orbifold B M,and fπ : π1(M) →π1(M) be an automorphism induced by an orientation-reversing homeomorphism f of M. We give a bound on the rank of the fixed subgroup of fπ, namely, rank Fix(fπ) 2rankπ1(M),which is an analogue of inequalities on surface groups and hyperbolic 3-manifold groups.     

The Godbillon-Vey cyclic cocycle for PL-foliations

We define a cyclic cocycle which corresponds to the piecewise linear Godbillon-Vey class of Ghys and Sergiescu [E. Ghys, V. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv. 62 (1987) 185-239]. Using Connes's pairing [A. Connes, Non-commutative differential geometry. Part II: De Rham homology and noncommutative algebra, Publ. Math. Inst. Hautes Études Sci. 62 (1985) 257-360; A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, in: H. Araki, G. Effros (Eds.), Geometric Methods in Operator Algebras, Pitman Res. Notes Math. Ser., vol. 123, Longman, Harlow, 1986, pp. 52-144] between cyclic cohomology and K-theory, we then evaluate this cocycle on a suitable K-theory class and obtain a nontrivial result, for foliations of the 3-torus by slope components.     

膏溶角砾岩侧限膨胀特性试验研究

研究了膏溶角砾岩的侧限膨胀特性及其影响因素。这种岩石受到扰动后可吸水发生膨胀变形使其结构发生变形甚至破坏。通过GDG高压固结仪采用平衡加压法和充分膨胀法研究了侧限情况下岩石的膨胀力指标, 解释了充分膨胀法膨胀力指标大于平衡加压法膨胀力指标的原因。通过试验研究了供水充足条件下膏溶角砾岩侧限自由膨胀变形演化规律, 阐述了膏溶角砾岩侧限膨胀变形的影响因素。岩石试样采自山西石太客运专线太行山隧道, 试验结果表明, 膏溶角砾岩的膨胀特性不同于常规的膨胀岩, 其轴向膨胀变形随含水率增加而增加, 随密度增大而减小。     

The group of asynchronous automata and rational homeomorphisms of the cantor set

The asynchronous automaton group is defined; its fundamental properties are established. Embeddability of many familiar groups in this group is proved. The isomorphism type of this group is shown to be independent of the alphabet, which makes it possible to define the group of rational homeomorphisms of the Cantor set. Applications of the asynchronous automaton group to dynamical systems are given. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 680–685, May, 2000.     

膨胀土掺石灰对土的工程特性影响机制探讨

结合工程实践,通过对南阳膨胀土特性研究,尤其是对2∶8 灰土垫层和原状膨胀土做了对比试验,并探讨了膨胀土掺石灰对土的工程特性的影响机制。     

Stochastic flows of diffeomorphisms on an open set in rn

In this paper, we will give sufficient conditions for the solution to a stochastic differential equation (SDE) on an open set D in R" to define a stochastic flow of diffeomorphisms of D onto itself. Since a necessary and sufficient condition for the solution to determine a stochastic flow of diffeomorphisms is that the original SDE and its adjoint SDE are both strictly conservative, we will concentrate our attention on finding sufficient conditions for the SDE to be strictly conservative. It will be etablished that the strict conservativeness follows if the vector fields governing the SDE decay suitably near the boundary dD in the direction transversal to 3D and some additional assumptions are satisfied.     

Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2, ℝ) cocycles

We consider the linear cocycle (T, A) induced by a measure preserving dynamical system T : X → X and a map A: X → SL(2, ℝ). We address the dependence of the upper Lyapunov exponent of (T, A) on the dynamics T when the map A is kept fixed. We introduce explicit conditions on the cocycle that allow to perturb the dynamics, in the weak and uniform topologies, to make the exponent drop arbitrarily close to zero. In the weak topology we deduce that if X is a compact connected manifold, then for a C^r (r ≥ 1) open and dense set of maps A, either (T, A) is uniformly hyperbolic for every T, or the Lyapunov exponents of (T, A) vanish for the generic measurable T. For the continuous case, we obtain that if X is of dimension greater than 2, then for a C^r (r ≥ 1) generic map A, there is a residual set of volume-preserving homeomorphisms T for which either (T, A) is uniformly hyperbolic or the Lyapunov exponents of (T, A) vanish. *Partially supported by CNPq-Profix and Franco-Brazilian cooperation program in Mathematics.     

Solving stochastic differential equations on Homeo(S)

The Brownian motion with respect to the metric H^3/2 on Diff(S¹) has been constructed. It is realized on the group of homeomorphisms Homeo(S¹). In this work, we shall resolve the stochastic differential equations on Homeo(S¹) for a given drift Z.