Limit theorems and Markov approximations for chaotic dynamical systems

Summary We prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients. We also develop techniques of Markov approximations for dynamical systems. We apply our results to expanding interval maps, Axiom A diffeomorphisms, chaotic billiards and hyperbolic attractors.     

Generic continuity of restricted weak upper semi-continuous set-valued mappings

Set-valued mappings from a topological space into subsets of a Banach space which satisfy a restricted form of weak upper semi-continuity, have particularly noteworthy properties. We establish a selection theorem for certain set-valued mappings from a (-) unfavourable topological space into subsets of a Banach space and as a consequence derive the property that restricted weak upper semi-continuous set-valued mappings which satisfy a minimality condition, from a (-) unfavourable topological space into subsets of a Banach space are single-valued and norm upper semi-continuous at the points of a residual subset of their domain.     

Transcendency of local conjugacies in complex dynamics and transcendency of their values

Letp andq be polynomials of the same degree. A classical result of Böttcher says that there exists a functionf conformal in a neighborhood of infinity such thatf(p(z))=q(f(z)). We show thatf is transcendental and takes transcendental values at algebraic points unlessp andq are linearly conjugate to monomials or Chebychev polynomials. As an application, we show that the conformal map from the exterior of the Mandelbrot set onto the exterior of the unit disk takes transcendental values at algebraic points. A second application is the solution of a transcendency problem posed by Golomb.     

Some representations of midconvex set-valued functions

Summary In this note we establish conditions under which every midconvex set-valued function can be represented as sum of an additive function and a convex set-valued function. These results improve some theorems obtained in [8], [10] and [3]. Some results on local Jensen selections of midconvex set-valued functions are also given.     

A cohomological index of Fuller type for set-valued dynamical systems

In the paper we develop the theory of a cohomological index of the Fuller type detecting periodic orbits of a set-valued dynamical system generated by a differential inclusion or a differential equation without the uniqueness of solutions. The theory presented is applied to establish a general result on the existence of bifurcation of periodic orbits from an equilibrium point of a differential inclusion.     

Random attractors for second-order stochastic lattice dynamical systems

In this paper, the asymptotic behavior of second-order stochastic lattice dynamical systems is considered. We firstly show the existence of an absorbing set. Then an estimate on tails of the solutions is derived when the time is large enough, which ensures the asymptotic compactness of the random dynamical system. Finally, the existence of the random attractor is provided.     

Unimodular eigenvalues and invariant measures for linear operators

For a bounded linear operator in a complex separable Hilbert space we show that it accepts an invariant Borel probability measure of square integrable norm whose essential support spans the space, iff its eigenvectors with unimodular eigenvalues span the space.This research has been supported by the Economic Research Center (KOE) of the Athens University of Economics and Business.     

Random attractors for stochastic lattice dynamical systems in weighted spaces

In this paper, we first provide some sufficient conditions for the existence of global compact random attractors for general random dynamical systems in weighted space (p?1) of infinite sequences. Then we consider the existence of global compact random attractors in weighted space for stochastic lattice dynamical systems with random coupled coefficients and multiplicative/additive white noises. Our results recover many existing ones on the existence of global random attractors for stochastic lattice dynamical systems with multiplicative/additive white noises in regular l² space of infinite sequences.     

Metrizable extensions of dynamical systems from function algebras

Peters and Pennings introduced and studied in [9] and [11] a kind of extensions of dynamical systems induced by certain C^*-algebras, which they calltame extensions. In this paper we study the behaviour of metrizable tame extensions when some of the most important dynamical concepts related to the metric (such as sources, sinks, saddles, the shadowing property and distallity) are considered. We will confirm that these extensions can be troublesome in this context. We show, however, that sources and sinks are preserved under certain conditions. Entrata in Redazione il 24 febbraio 1997. Research partially supported by Generalitat Valenciana, (2223/94).     

Construction of invariant measures of Lagrangian maps: minimisation and relaxation

If F is an exact symplectic map on the{\it d}-dimensional cylinder , with a generating function h having superlinear growth and uniform bounds on the second derivative, we construct a strictly gradient semiflow on the space of shift-invariant probability measures on the space of configurations . Stationary points of are invariant measures of F, and the rotation vector and all spectral invariants are invariants of . Using and the minimisation technique, we construct minimising measures with an arbitrary rotation vector , and with an additional assumption that F is strongly monotone, we show that the support of every minimising measure is a graph of a Lipschitz function. Using and the relaxation technique, assuming a weak condition on (satisfied e.g. in Hedlund's counter-example, and in the anti-integrable limit) we show existence of double-recurrent orbits of F (and F-ergodic measures) with an arbitrary rotation vector , and action arbitrarily close to the minimal action . Received November 4, 1999; in final form July 29, 2000 / Published online April 12, 2001     

Poisson structures on Hilbert schemes of points¶of a surface and integrable systems

In this paper we prove that if S is a Poisson surface, i.e., a smooth algebraic surface with a Poisson structure, the Hilbert scheme of points of S has a natural Poisson structure, induced by the one of S. This generalizes previous results obtained by A. Beauville [B1] and S. Mukai [M2] in the symplectic case, i.e., when S is an abelian or K3 surface. Finally we apply our results to give some examples of integrable Hamiltonian systems naturally defined on these Hilbert schemes. In the simple case S=ℙ² we obtain by this construction a large class of integrable systems, which includes the ones studied by P. Vanhaecke in [V1] and, more generally, in [V2]. Received: 9 March 1998 / Revised version: 19 June 1998     

Schilder theorem for the Brownian motion on the diffeomorphism group of the circle

We prove a large deviation principle for flows associated to stochastic differential equations with non-Lipschitz coefficients. As an application we establish a Schilder Theorem for the Brownian motion on the group of diffeomorphisms of the circle.     

Riesz decompositions and subtractivity for excessive measures

The convex cone of excessive measures of a right Markov process is an example of a superharmonic semigroup in the abstract potential theory developed by Arsove and Leutwiler. We show that their theory of Riesz decompositions can be sharpened in the case of excessive measures. In particular there is always a Riesz decomposition relative to a given potential cone (resp. harmonic cone). An element of an ordered convex cone is subtractive if each majorant is a specific majorant. This notion of subtractivity features prominently in the theory of harmonic cones. We give a complete characterization of the subtractive elements in the cone of excessive measures.The research of both authors was supported in part by NSF Grant DMS 87-21347.     

Nonlinear versions of a vector maximal principle

Some nonlinear extensions of the vector maximality statement established by Goepfert et al. [A. Goepfert, C. Tammer, C. Z?linescu, On the vectorial Ekeland’s variational principle and minimal points in product spaces, Nonlinear Anal. 39 (2000) 909-922] are given. Basic instruments for these are the Brezis-Browder ordering principle [H. Brezis, F.E. Browder, A general principle on ordered sets in nonlinear functional analysis, Adv. Math. 21 (1976) 355-364] and its logical equivalent in Turinici [M. Turinici, Variational principles on semi-metric structures, Libertas Math. 20 (2000) 161-171].     

A process of runs and its convergence to the brownian motion

Let X₁,X₂,… be i.i.d. random variables with a continuous distribution function. Let R₀=0, R_k=min{j>R_k?1, such that X_j>X_j+1}, k?1. We prove that all finite-dimensional distributions of a process $\text{W}{}^{\mathrm{(n)}}\text{(t)=}\text{(R}{}_{\mathrm{[nt]}}\text{?2[nt])}\text{2}\text{3}\text{n}\text{, t ? [0,1]}$, converge to those of the standard Brownian motion.     

Selections for quasi-l.s.c. mappings with uniformly equi-LCn range

Every quasi-lower semi-continuous (q.l.s.c.) mapping admits a lower semi-continuous (l.s.c.) selection preserving all important (from the selection point of view) properties of the former mapping. Special-type extensions of l.s.c. mappings are established on this base.     

Homotopy Conley index for discrete multivalued dynamical systems

We define an index of Conley type for a certain class of upper semicontinuous multivalued dynamical systems. We use the Szymczak functor and apply techniques introduced by Reineck, Mrozek and Srzednicki for the index over the base. Moreover we introduce the notion of the homotopy partial functor for the usc maps. We show that the index possesses Wa?ewski and homotopy properties. We also give four examples that exhibit the benefits of our index over the cohomological index defined by Mrozek and Kaczyński.     

Mixing normal approximations of vectors of sums and maximum sums

Summary The conditioned central limit theorem for the vector of maximum partial sums based on independent identically distributed random vectors is investigated and the rate of convergence is discussed. The conditioning is that of Rényi (1958,Acta Math. Acad. Sci. Hungar.,9, 215–228). Analogous results for the vector of partial sums are obtained. University of Petroleum and Minerals