The Nonexistence of Expansive Z^d Actions on Graphs

It is well known that if X is an arc or a circle, then there is no expansive homeomorphism on X. In this paper we prove that there is no expansive Z^d action on X, which answers the two questions raised by us before, In 1979, Mané proved that there is no expansive homeomorphism on infinite dimensional spaces. Contrary to this result, we construct an expansive Z^2 action on an infinite dimensional space. We also construct an expansive Z^2 action on a zero dimensional space but no element in Z^2 is expansive.     

Dynamical properties of expansive one-sided cellular automata

To every one-sided one-dimensional cellular automatonF with neighbourhood radiusr we associate its canonical factor defined by considering only the firstr coordinates of all the images of points under the powers ofF. Whenever the cellular automaton is surjective, this factor is a subshift which plays a primary role in its dynamics. In this article we study the class of positively expansive one-sided cellular automata, i.e. those that are conjugate to their canonical factors. This class is a natural generalisation of the toggle or permutative cellular automata introduced in [He]. We prove that the canonical factors of positively expansive one-sided cellular automata are mixing subshifts of finite type that are shift equivalent to full shifts. Moreover, the uniform Bernoulli measure is the unique measure of maximal entropy forF. Consequently, their natural extensions are Bernoulli. We also describe a family of non-permutative positively expansive cellular automata. This research was partially supported by action E9314 of ECOS-CONICYT.     

On isomorphism problem for von Neumann flows with one discontinuity

A von Neumann flow is a special flow over an irrational rotation of the circle and under a piecewise C¹ roof function with a non-zero sum of jumps. We prove that the absolute value of the jump is a (measure theoretic) invariant in the class of von Neumann special flows with one discontinuity, i.e., two ergodic von Neumann flows with one discontinuity are not isomorphic if the jumps of the roof functions have different absolute values, regardless of the irrational rotation in the base.     

Expansive flows and the fundamental group

In this paper we construct stable and unstable foliations for expansive flows operating on 3-manifolds. We also prove that the fundamental group of the manifold has exponential growth.Partially supported by CNPq, Brasil and Centro de Matemática, Uruguay.     

Diophantine properties of measures invariant with respect to the Gauss map

Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss [8, 9], we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has finite Lyapunov exponent is extremal, i.e., gives zero measure to the set of very well approximable numbers. We show, on the other hand, that there exist examples where the Lyapunov exponent is infinite and the invariant measure is not extremal. Finally, we construct a family of Ahlfors regular measures and prove a Khinchine-type theorem for these measures. The series whose convergence or divergence is used to determine whether or not µ-almost every point is ψ-approximable is different from the series used for Lebesgue measure, so this theorem answers in the negative a question posed by Kleinbock, Lindenstrauss, and Weiss [8].     

Topological pressure of continuous flows without fixed points

In this paper, we give the equivalent definitions of topological pressure for flows by using spanning sets, weakly spanning sets, strongly separated sets and tracing sets, respectively. We get an inequality between the topological pressures of Lipschitz conjugate flows, and prove that the topological pressure of expansive flows with tracing property can be described by its periodic orbits.     

Embedding asymptotically expansive systems

A topological dynamical system is said asymptotically expansive when entropy and periodic points grow subexponentially at arbitrarily small scales. We prove a Krieger like embedding theorem for asymptotically expansive systems with the small boundary property. We show that such a system (X, T) embeds in the K-full shift if \( h_{top}(T)<\log K\) and \(\sharp Per_n(X,T)\le K^n\) for any integer n. The embedding is in general not continuous (unless the system is expansive and X is zero-dimensional) but the induced map on the set of invariant measures is a topological embedding. It is shown that this property implies asymptotical expansiveness. We prove also that the inverse of the embedding map may be continuously extended to a faithful principal symbolic extension.     

图上Z~2可扩作用的非存在性

文[9]中作者考虑连续统上可扩群作用的存在性问题,证明了单位闭区间上不存在自由交换群Z×Z的可扩作用,并且给出一个例子表明闭区间上存在自由积Z*Z的可扩作用.换句话说,由两个交换同胚生成的群是不能可扩作用在闭区间上的,但还是存在由两个非交换同胚生成的群能够可扩作用在闭区间上.本文证明了图上不存在Z×Z的可扩作用,解决了文[9]所提的一个问题.     

Existence d'une mesure de Sinai—Ruelle—Bowen pour des systèmes non uniformément hyperboliques

Let M be a smooth Riemannian manifold, and ƒ be a Csu1 + α-diffeomorphism of M onto itself Let Λ be a Pesin's set (see [5]). We prove that under some assumptions on Λ, and if the system (M, ƒ) is conservative for the Lebesgue measure, then there exists a SRB-measure, ƒ -invariant and σ-finite.     

Lyapunov Exponents versus Expansivity and Sensitivity in Cellular Automata

We establish a connection between the theory of Lyapunov exponents and the properties of expansivity and sensitivity to initial conditions for a particular class of discrete time dynamical systems; cellular automata (CA). The main contribution of this paper is the proof that all expansive cellular automata have positive Lyapunov exponents for almost all the phase space configurations. In addition, we provide an elementary proof of the non-existence of expansive CA in any dimension greater than 1. In the second part of this paper we prove that expansivity in dimension greater than 1 can be recovered by restricting the phase space to asuitablesubset of the whole space. To this extent we describe a 2-dimensional CA which is expansive over adense uncountablesubset of the whole phase space. Finally, we highlight the different behavior of expansive and sensitive CA for what concerns the speed at which perturbations propagate.     

Nonexistence of Bonatti-Langevin examples of Anosov flows on closed four manifolds

Bonatti and Langevin constructed an Anosov flow on a closed 3-manifold with a transverse torus intersecting all orbits except one [C. Bonatti, R. Langevin, Un exemple de flot d'Anosov transitif transverse à un tore et non conjugué à une suspension, Ergodic Theory Dynam. Systems 14 (4) (1994), 633-643]. We shall prove that these flows cannot be constructed on closed 4-manifolds. More precisely, there are no Anosov flows on closed 4-manifolds with a closed, incompressible, transverse submanifold intersecting all orbits except finitely many closed ones. The proof relies on the analysis of the trace of the weak invariant foliations of the flow on the transverse submanifold.     

Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups

We consider stochastic equations in Hilbert spaces with singular drift in the framework of [G. Da Prato, M. Röckner, Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Related Fields 124 (2) (2002) 261-303]. We prove a Harnack inequality (in the sense of [F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997) 417-424]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure μ (satisfying some mild integrability conditions). Finally, we prove existence of such a measure μ for noncontinuous drifts.     

The nonexistence of expansive commutative group actions on Peano continua having free dendrites

A dendrite D in a metric space X is said to be free if there exists a connected open set U in X such that . In this paper, we prove that there is no expansive commutative group action on any Peano continuum having a free dendrite. In particular, no 1-dimensional compact ANR admits an expansive commutative group action.     

Entropy of flows, revisited

We introduce a concept of measure-theoretic entropy for flows and study its invariance under measure-theoretic equivalences. Invariance properties of the corresponding topological entropy is studied too. We also answer a question posed by Bowen-Walters in [3] concerning the equality between the topological entropy of the time-one map of an expansive flow and the time-one map of its symbolic suspension.Partially supported by FAPESP-Brasil, Grant #96/11671-6.Partially supported by CNPq-Brasil, Grant #300557/89-2.     

On lévy measure and integrability of the norm in C[0, 1]

We prove that integrability of the norm is the best sufficient condition in terms of integrability of functions of the norm for a positive measure to be a Lévy Measure in C[0, 1].     

New approach to the incompressible Maxwell-Boussinesq approximation: Existence, uniqueness and shape sensitivity

The Boussinesq approximation to the Fourier-Navier-Stokes (F-N-S) flows under the electromagnetic field is considered. Such a model is the so-called Maxwell-Boussinesq approximation. We propose a new approach to the problem. We prove the existence and uniqueness of weak solutions to the variational formulation of the model. Some further regularity in W1,2+δ, δ>0, is obtained for the weak solutions. The shape sensitivity analysis by the boundary variations technique is performed for the weak solutions. As a result, the existence of the strong material derivatives for the weak solutions of the problem is shown. The result can be used to establish the shape differentiability for a broad class of shape functionals for the models of Fourier-Navier-Stokes flows under the electromagnetic field.     

Nonexistence of codimension one Anosov flows on compact manifolds with boundary

A flow is Anosov if it exhibits contracting and expanding directions forming with the flow a continuous tangent bundle decomposition. An Anosov flow is codimension one if its contracting or expanding direction is one-dimensional. Examples of codimension one Anosov flows on compact boundaryless manifolds can be exhibited in any dimension ?3. In this paper, we prove that there are no codimension one Anosov flows on compact manifolds with boundary. The proof uses an extension to flows of some results in Hirsch [On Invariant Subsets of Hyperbolic Sets, Essays on Topology and Related Topics, Memoires dédiés à Georges de Rham, 1970, pp. 126-135] related to Question 10(b) in Palis and Pugh [Fifty problems in dynamical systems, in: J. Palis, C.C. Pugh (Eds.), Dynamical Systems-Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E.C. Zeeman on his fiftieth birthday), Lecture Notes in Mathematics, vol. 468, Springer, Berlin, 1975, pp. 345-353].     

Quasi-invariant stochastic flows of SDEs with non-smooth drifts on compact manifolds

In this article we prove that stochastic differential equation (SDE) with Sobolev drift on a compact Riemannian manifold admits a unique ν-almost everywhere stochastic invertible flow, where ν is the Riemannian measure, which is quasi-invariant with respect to ν. In particular, we extend the well-known DiPerna-Lions flows of ODEs to SDEs on a Riemannian manifold.     

Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

We consider maps preserving a foliation which is uniformly contracting and a one-dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for suitable Poincaré maps of a large class of singular hyperbolic flows. From this we deduce a logarithm law for these flows.     

On the complexity of expansive measures

We prove that the existence of positively expansive measures for continuous maps on compact metric spaces implies the existence of e 0 and a sequence of(m, e)-separated sets whose cardinalities go to infinite as m →∞. We then prove that maps exhibiting such a constant e and the positively expansive maps share some properties.