Topological pressure via saddle points

Let be a compact locally maximal invariant set of a - diffeomorphism on a smooth Riemannian manifold . In this paper we study the topological pressure (with respect to the dynamical system ) for a wide class of Hölder continuous potentials and analyze its relation to dynamical, as well as geometrical, properties of the system. We show that under a mild non-uniform hyperbolicity assumption the topological pressure of is entirely determined by the values of on the saddle points of in . Moreover, it is enough to consider saddle points with ``large' Lyapunov exponents. We also introduce a version of the pressure for certain non-continuous potentials and establish several variational inequalities for it. Finally, we deduce relations between expansion and escape rates and the dimension of . Our results generalize several well-known results to certain non-uniformly hyperbolic systems.

     

Entropy-expansiveness of geodesic flows on closed manifolds without conjugate points

In this article, we consider the entropy-expansiveness of geodesic flows on closed Riemannian manifolds without conjugate points. We prove that, if the manifold has no focal points, or if the manifold is bounded asymptote, then the geodesic flow is entropy-expansive. Moreover, for the compact oriented surfaces without conjugate points, we prove that the geodesic flows are entropy-expansive. We also give an estimation of distance between two positively asymptotic geodesics of an uniform visibility manifold.     

Topological convexities, selections and fixed points

A convexity on a set X is a family of subsets of X which contains the whole space and the empty set as well as the singletons and which is closed under arbitrary intersections and updirected unions. A uniform convex space is a uniform topological space endowed with a convexity for which the convex hull operator is uniformly continuous. Uniform convex spaces with homotopically trivial polytopes (convex hulls of finite sets) are absolute extensors for the class of metric spaces; if they are completely metrizable then a continuous selection theorem à la Michael holds. Upper semicontinuous maps have approximate selections and fixed points, under the usual assumptions.     

连续自映射拓扑压的两个注记

令T:X→ X是紧度量空间(X,d)上的连续映射.该文给出了T的拓扑压和T在非游荡集上的限制的拓扑压相等的不依赖于变分原理的一个直接证明.同时,还讨论了半共轭的两个系统的拓扑压之间的关系,证明了拓扑压在一致有限对一条件下是半共轭不变量.     

Monotone flows and fixed points for dynamic systems on time scales in a banach space

Utilizing the dynamic systems on time scales in a Banach space, the theory of monotone flows and fixed points for discrete as well as continuous dynamical systems are unified.     

Topological modules of continuous homomorphisms

In this paper, we consider the notion of module homomorphisms in the general topological module setting and establish their linearity and continuity under some suitable conditions. We also introduce the strict and uniform topologies on the modules of continuous linear homomorphisms and study their various properties.     

Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces

Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudo-contraction on C. Consider Mann's iteration algorithm given by

     

Surjectivity and fixed points of relaxed dissipative multifunctions. Differential inclusions approach

One from the most important properties of accretive and monotone operators is the existence of zeros and surjectivity. In the paper we introduce relaxed variants of dissipative, accretive and monotone operators. Using essentially the properties of the solution set of appropriate differential inclusions we study the existence of zeros of such operators. As corollaries the existence of fixed points of relaxed contractive and relaxed nonexpansive multifunctions are obtained.     

Limiting points of stochastic flows

It is known that if the flow of a stochastic differential equation on a compact manifold has only negative Lyapunov exponents, then its limiting behavior can be described by a moving random set σof n points. We study the properties of σ and the associated domains of attraction. For example, we will show that it is supported by a compact set on which the induced flow has only one limiting point. On a d-dimensional sphere, if the support of the stochastic flow contains all the isometries, then n=1 or 2, and in the latter case, the two random points and their domains of attractions are antipodal symmetric     

Topological degree and number of Nash equilibrium points of bimatrix games

In this paper, using the topological degree, we give a new proof of a well-known result: the number of Nash equilibrium points of a nondegenerate bimatrix game is odd. The calculation of the topological degree allows the localization of the whole set of non-degenerate equilibrium points.     

Unstable manifold,Conley index and fixed points of flows

We study dynamical and topological properties of the unstable manifold of isolated invariant compacta of flows. We show that some parts of the unstable manifold admit sections carrying a considerable amount of information. These sections enable the construction of parallelizable structures which facilitate the study of the flow. From this fact, many nice consequences are derived, specially in the case of plane continua. For instance, we give an easy method of calculation of the Conley index provided we have some knowledge of the unstable manifold and, as a consequence, a relation between the Brouwer degree and the unstable manifold is established for smooth vector fields. We study the dynamics of non-saddle sets, properties of existence or non-existence of fixed points of flows and conditions under which attractors are fixed points, Morse decompositions, preservation of topological properties by continuation and classify the bifurcations taking place at a critical point.     

Limit sets of typical continuous functions

Given a metrizable compact topological n-manifold X with boundary and a finite positive Borel measure μ on X, we prove that for the typical continuous function , it is true that for every point x in a full μ-measure subset of X the limit set ω(f,x) is a Cantor set of Hausdorff dimension zero, f maps ω(f,x) homeomorphically onto itself, each point of ω(f,x) has a dense orbit in ω(f,x) and f is non-sensitive at each point of ω(f,x); moreover, the function x→ω(f,x) is continuous μ-almost everywhere.     

紧拓扑半群上概率测度的简单半群及其支撑集

本文讨论一类拓扑半群上概率测度的极限性质．首先在紧半群上研究测度的简单半群和它的支撑集的相依关系;然后讨论测度的卷积幂ｕｎ收敛到Ｈａａｒｒ测度的充要条件．     

Topological degree of a family of convex functions and localization of nash equilibrium points

We define the degree of a family of convex functionsf _x, wherex , and prove that, if this degree is different from zero, one has at least one Nash equilibrium point in . This is a criterion of localization of such equilibrium points.     

Topological Properties of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

We establish a continuous mapping of the space of the matrix elements of an arbitrary nonempty set of quasi solutions of Lipschitzian quantum stochastic differential inclusion (QSDI) into the space of the matrix elements of its solutions. As a corollary, we furnish a generalization of a previous selection result. In particular, when the coefficients of the inclusion are integrably bounded, we show that the space of the matrix elements of solutions is an absolute retract, contractible, locally and integrally connected in an arbitrary dimension. Permanent address: E.O. Ayoola, Department of Mathematics, University of Ibadan, Ibadan, Nigeria.     

Pressure regularity criteria of the three‐dimensional micropolar fluid flows

In the study of the regularity criteria of weak solutions of the three‐dimensional (3D) micropolar fluid flows, the regularity of solutions are examined by imposing some critical growth conditions only on the pressure field in the Lebesgue space, Morrey space, Multiplier space, BMO space and Besov space, respectively. Copyright © 2010 John Wiley & Sons, Ltd.     

On the topological pressure of random bundle transformations in sub-additive case

In this paper, we define the topological pressure for sub-additive potential via separated sets in random dynamical systems and give a proof of the relativized variational principle for the topological pressure.     

INVARIANT OF TOPOLOGICAL PRESSURE UNDER SOME SEMI-CONJUGATES

In this paper, the topological pressure is preserved under some semi-conjugates, and a formula of computing topological pressure by use of periodic points for positively expansive continuous map with specification is given.     

Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption

In the present paper, the following result is shown: Let be a real Banach space with a uniformly convex dual , and let be a nonempty closed convex and bounded subset of . Assume that is a continuous strong pseudocontraction. Let and be two real sequences satisfying (i) for all ; (ii) ; and (iii) as Then the Ishikawa iterative sequence generated by

converges strongly to the unique fixed point of .