Compact perturbations of a linear homeomorphism

Acta Mathematica Hungarica - Let X, Y be Banach spaces and let $$A,B :X rightarrow Y$$ be two operators, where A is a linear homeomorphism and B is a $$C^{1} $$ -compact operator. Sufficient...     

Approximation of entropy on hyperbolic sets for one-dimensional maps and their multidimensional perturbations

We consider piecewise monotone (not necessarily, strictly) piecewise C ² maps on the interval with positive topological entropy. For such a map f we prove that its topological entropy h _top(f) can be approximated (with any required accuracy) by restriction on a compact strictly f-invariant hyperbolic set disjoint from some neighborhood of prescribed set consisting of periodic attractors, nonhyperbolic intervals and endpoints of monotonicity intervals. By using this result we are able to generalize main theorem from [1] on chaotic behavior of multidimensional perturbations of solutions for difference equations which depend on two variables at nonperturbed value of parameter.     

About C 1-minimality of the hyperbolic Cantor sets

In this work we prove that a C ^1+α -hyperbolic Cantor set contained in S ¹ that is close to an affine Cantor set is not C ¹-minimal.     

On multiplicative perturbations of C 0 -groups and C 0 -cosine operator functions

Certain convolution operators of the form (K f) (t) = A∈t ₀ ^t L(t-s) f(s) ds , where A is the infinitesimal generator of either a C ₀ -group or a C ₀ -cosine family in a Banach space E , are considered. We obtain several lifting results guaranteeing that the continuity of K from L ^p to L ^q implies the continuity of K from L ^p to L ^∈ fty . These results are applied to the study of multiplicative perturbations of C ₀ -groups and C ₀ -cosine families in Banach spaces and to the study of the Maximal Regularly Property (MRP) in L ^p , 1 ≤ p ≤ +∈fty , for second-order Cauchy problem. It is proved that the MRP is equivalent to the boundedness of the infinitesimal generator. April 30, 1999     

On certain hyperbolic sets

Two invariant sets F of certain diffeomorphisms S that were described by A. Fathi, S. Crovisier, and T. Fisher as examples of hyperbolic sets with the property (unexpected at that time) that, in some neighborhood of such an F, there is no locally maximal set containing F are considered. It is proved that this property, although referring to the behavior of the orbits of S near F, is ultimately determined in the examples mentioned above by a combination of a certain explicitly stated intrinsic property of the action of S on F with the hyperbolicity of F. (This means that if a hyperbolic set F′ for a diffeomorphism S′ is equivariantly homeomorphic to a Fathi-Crovisier or a Fisher set, then F′ has a neighborhood in which S′ has no locally maximal set containing F′.)     

Local maximality of hyperbolic sets

Two properties of a hyperbolic set F are discussed: its local maximality and the property that, in any neighborhood U ⊃ F, there exists a locally maximal set F′ that contains F (we suggest calling the latter property local premaximality). Although both these properties of the set F are related to the behavior of trajectories outside F, it turns out that, in the class of hyperbolic sets, the presence or absence of these properties is determined by the interior dynamics on F.     

Limit sets of relatively hyperbolic groups

In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known results on limit sets of geometrically finite Kleinian groups are derived in general convergence groups. We also establish dynamical quasiconvexity of undistorted subgroups in finitely generated groups with nontrivial Floyd boundaries.     

Small tight sets of hyperbolic quadrics

An x-tight set of a hyperbolic quadric Q ⁺(2n + 1, q) can be described as a set M of points with the property that the number of points of M in the tangent hyperplanes of points of M is as big as possible. We show that such a set is necessarily the union of x mutually disjoint generators provided that x ≤ q and n ≤ 3, or that x < q, n ≥ 4 and q ≥ 71. This unifies and generalizes many results on x-tight sets that are presently known, see (J Comb Theory Ser A 114(7):1293–1314 [1], J Comb Des 16(4):342–349 [5], Des Codes Cryptogr 50:187–201 [4], Adv Geom 4(3):279–286 [8], Bull Lond Math Soc 42(6):991–996 [11]).     

General random perturbations of hyperbolic and expanding transformations

Small random perturbations of a general form of diffeomorphisms having hyperbolic invariant sets and expanding maps are considered. The convergence of invariant measures of perturbations to the Sinaî-Bowen-Ruelle measure in the case of a hyperbolic attractor and to the smooth invariant measure in the expanding case are proved. The convergence of corresponding entropy characteristics and the approximation of the topological pressure by means of perturbations is considered as well.     

Tangent measure distributions of hyperbolic Cantor sets

Tangent measure distributions were introduced byBandt [2] andGraf [8] as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by certain contractive mappings, which are not necessarily similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorff- or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models ofBedford andFisher [5].     

Random permutations of countable sets

Summary The main objective of this paper is a study of random decompositions of random point configurations onR ^d into finite clusters. This is achieved by constructing for each configurationZ a random permutation ofZ with finite cycles; these cycles then form the cluster decomposition ofZ. It is argued that a good candidate for a random permutation ofZ is a Gibbs measure for a certain specification, and conditions are given for the existence and uniqueness of such a Gibbs measure. These conditions are then verified for certain random configurationsZ.     

Random perturbations of transformations of an interval

Let μ^ε be invariant measures of the Markov chainsx _n ^F which are small random perturbations of an endomorphismf of the interval [0,1] satisfying the conditions of Misiurewicz [6]. It is shown here that in the ergodic case μ^ε converges as ε→0 to the smoothf-invariant measure which exists according to [6]. This result exhibits the first example of stability with respect to random perturbations while stability with respect to deterministic perturbations does not take place.