Normality and attracting fixed points

Let k 2 be a fixed integer. We construct a non-normal familyof meromorphic functions in the unit disc with kth iteratesthat have no non-attracting fixed points in the unit disc. Wealso prove that a family of meromorphic functions in a planedomain is normal if all fixed points in the domain of the k-iteratesof all functions in the family are uniformly attracting.     

On connectivity of Julia sets of transcendental meromorphic maps and weakly repelling fixed points I

It is known that the Julia set of the Newton method of a non-constantpolynomial is connected (Mitsuhiro Shishikura, Preprint, 1990,M/90/37, Inst. Hautes Études Sci.). This is, in fact,a consequence of a much more general result that establishesthe relationship between simple connectivity of Fatou componentsof rational maps and fixed points which are repelling or parabolicwith multiplier 1. In this paper we study Fatou components oftranscendental meromorphic functions; that is, we show the existenceof such fixed points, provided that immediate attractive basinsor preperiodic components are multiply connected.     

Weakly repelling fixed points and multiply-connected wandering domains of meromorphic functions

We consider the dynamics of a transcendental meromorphic function f(z) with only finitely many poles and prove that if / has only finitely many weakly repelling fixed points, then there is no multiply-connected wandering domain in its Fatou set.     

Every finitely generated group is weakly exact

We show that every finitely generated group admits weak analogues of an invariant expectation, whose existence characterizes exact groups. This fact has a number of applications. We show that Hopf G-modules are relatively injective, which implies that bounded cohomology groups with coefficients in all Hopf G-modules vanish in all positive degrees. We also prove a general fixed point theorem for actions of finitely generated groups on ?_∞-type spaces. Finally, we define the notion of weak exactness for certain Banach algebras.     

Random fixed points of pseudo-contractive random operators

The paper presents new random fixed point results for pseudo-contractive random operators.     

Common fixed points for weakly compatible maps

The purpose of this paper is to prove a common fixed point theorem, from the class of compatible continuous maps to a larger class of maps having weakly compatible maps without appeal to continuity, which generalizes the results of Jungck [3], Fisher [1], Kang and Kim [8], Jachymski [2], and Rhoades [9].     

Fixed points and random fixed points for weakly inward approximable maps

In this paper we obtain new fixed point and random fixed point theory for approximable maps.

     

Density of periodic points, invariant measures and almost equicontinuous points of cellular automata

Revisiting the notion of μ-almost equicontinuous cellular automata introduced by R. Gilman, we show that the sequence of image measures of a shift ergodic measure μ by iterations of such automata converges in Cesàro mean to an invariant measure μ_c. Moreover the dynamical system (cellular automaton F, invariant measure μ_c) has still the μ_c-almost equicontinuity property and the set of periodic points is dense in the topological support of the measure μ_c. We also show that the density of periodic points in the topological support of a measure μ occurs for each μ-almost equicontinuous cellular automaton when μ is an invariant and shift ergodic measure. Finally using most of these results we give a non-trivial example of a couple (μ-equicontinuous cellular automaton F, shift and F-invariant measure μ) such that the restriction of F to the topological support of μ has no equicontinuous points.     

Random fixed points of discontinuous random maps

We prove some random fixed-point theorems for random maps which are not necessarily continuous. This may lead to the discovery of some new results in random fixed-point theory for discontinuous maps.     

Random iterations, fixed points and invariant CRF-horospheres in complex Banach spaces

For holomorphic noncontractive maps on (not necessarily bounded) domains in complex Banach spaces, we establish the conditions guaranteeing locally uniform convergence of random iterations and study the existence of fixed points and boundary behaviour of iterations. In particular, we show that the problem, concerning the existence of the horospheres determined by Carathéodory-Reiffen-Finsler pseudometrics defined on unbounded domains, has the solution and we prove new results of type of Julia's lemma and Wolff's theorem.     

Stable manifolds associated to fixed points with linear part equal to identity

We consider maps defined on an open set of having a fixed point whose linear part is the identity. We provide sufficient conditions for the existence of a stable manifold in terms of the nonlinear part of the map.These maps arise naturally in some problems of Celestial Mechanics. We apply the results to prove the existence of parabolic orbits of the spatial elliptic three-body problem.     

Intermittency and deterministic diffusion in chaotic ratchets

We consider the problem of deterministic transport of particles in an asymmetric periodic ratchet potential of the rocking type. When the inertial term is taken into account, the dynamics can be chaotic and modify the transport properties. We calculate the bifurcation diagram as a function of the amplitude of forcing and analyze in detail the crisis bifurcation that leads to current reversals. Near this bifurcation we obtain intermittency and anomalous deterministic diffusion.     

Invariant measures for interval maps with critical points and singularities

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.     

Random fixed points of set-valued operators

Some random fixed point theorems for set-valued operators are obtained. The measurability of certain marginal maps is also studied. The underlying measurable space is not assumed to be a Suslin family.

     

Invariant Measures for a Random Evolution Equation with Small Perturbations

In this paper we consider a random evolution equation with small perturbations, and show how to construct coupled solutions to the equations. As applications, we prove the Feller continuity of the solutions and the existence and uniqueness of invariant measures. Furthermore, we establish a large deviations principle for the family of invariant measures as the perturbations tend to zero. Received March 20,1998, Accepted June 1, 2000     

带小扰动的随机发展方程的不变测度

本文考虑带小扰动的随机发展方程,证明如何建立此方程的耦合解.作为应用,我们证明解的Feller连续性和不变测度的存在唯一性.还进一步建立了当扰动趋于零时,关于这族不变测度的大偏差原理.     

Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition

In this work, we establish new coincidence and common fixed point theorems for hybrid strict contraction maps by dropping the assumption “f is T-weakly commuting” for a hybrid pair (f,T) of multivalued maps in Theorem 3.10 of T. Kamran (2004) [8]. As an application, an invariant approximation result is derived.