on hausdorff dimension of unimodal attractors there exists a universal constant 1 such that every attractor of every c 4 unimodal map with a non degenerate critical point is an analytic manifold or its hausdorff dimension is equal to or less than

cat 1 spaces divergence groups and their commensurators a cat 1 space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane b briha gr the cat 1 property is one among many possible generalizations to singular spaces of the notion of negative curvature important examples of cat 1 spaces include riemannian manifolds of sectional curvature k 1 and their convex subsets b g s metric trees and piecewise hyperbolic cell complexes mou da hag be 1 be 2 b br in this paper we establish certain superrigidity results for isometric actions of a group on a cat 1 space in the following two settings a the group is a subgroup of a locally compact group g with comg where g is a sufficiently large discrete subgroup and comg g g g 1 g and share a subgroup of finite index is the commensurator of in g b the group is an irreducible lattice in g n 1g k where each g is a semisimple algebraic group defined over a local field k the issues addressed in this paper are motivated on one hand by earlier work of g a margulis ma dealing with the linear representation theory of where in case a g is a semisimple group and g a lattice and on the other hand by the results of lubotzky mozes and zimmer l m z concerning isometric actions of on trees where comg g is the group of automorphisms of a regular tree and g is a lattice our approach to establishing superrigidity results is based on ergodic theoretic methods developed by margulis ma zi 3 a c b in this context the following notion of boundary of a locally compact group will be useful let b be a standard borel space on which acts by borel automorphisms preserving a finite measure class

stable windings on hyperbolic surfaces let be a geometrically finite hyperbolic surface with infinite volume having at least one cusp we obtain the limit law under the patterson sullivan measure on t1 of the windings of the geodesics of around the cusps this limit law is stable with parameter 2 1 where is the hausdorff dimension of the limit set of the subgroup of mobius isometries associated with the normalization is t 1 2 1 for geodesics of length t our method relies on a precise comparison between geodesics and diffusion paths for which we need to approach the patterson sullivan measure mentioned above by measures that are regular along the stable leaves

surface subgroups of kleinian groups with torsion we prove that every finitely generated kleinian group that contains a finite non cyclic subgroup either is finite or virtually free or contains a surface subgroup hence every arithmetic kleinian group contains a surface subgroup

dynamical percolation on general trees haggstrom et al ann inst h poincare probab stat 33 4 497 528 1997 have introduced a dynamical version of percolation on a graph g when g is a tree they derived a necessary and sufficient condition for percolation to exist at some time t in the case that g is a spherically symmetric tree peres and steif in probab theory relat fields 111 1 141 165 1998 derived a necessary and sufficient condition for percolation to exist at some time t in a given target set d the main result of the present paper is a necessary and sufficient condition for the existence of percolation at some time t in d in the case that the underlying tree is not necessary spherically symmetric this answers a question of yuval peres personal communication we present also a formula for the hausdorff dimension of the set of exceptional times of percolation

restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems the restrictively preconditioned conjugate gradient rpcg method for solving large sparse system of linear equations of a symmetric positive definite and block two by two coefficient matrix is further studied in fact this rpcg method is essentially the classical preconditioned conjugate gradient pcg method with a specially structured preconditioner within this setting we present algorithmic descriptions of two restrictive preconditioners that respectively employ the block jacobi and the block symmetric gauss seidel matrix splitting matrices as approximations to certain matrices involved in them and give convergence analyses of the correspondingly induced two pcg methods numerical results show that these restrictive preconditioners can lead to practical and effective pcg methods for solving large sparse systems of linear equations of symmetric positive definite and block two by two coefficient matrices

on the calculation of quantum mechanical ground states from classical geodesic motion on certain spaces of constant negative curvature abstract we consider geodesic motion on three dimensional riemannian manifolds of constant negative curvature topologically equivalent to s 0 1 s a compact surface of genus two to those trajectories which are bounded and recurrent in both directions of the time evolution t t a fractal limit set is associated whose hausdorff dimension is intimately connected with the quantum mechanical energy ground state determined by the schrodinger operator on the manifold we give a rather detailed and pictorial description of the hyperbolic spaces we have in mind discuss various aspects of classical and quantum mechanical motion on them as far as they are needed to establish the connection between energy ground state and hausdorff dimension and give finally some examples of ground state calculations in terms of hausdorff dimensions of limit sets of classical trajectories

apollonian arrangements of spheres in d dimensional space the problem of arranging d dimensional spheres in such a way that each sphere touches exactly d 1 others is formulated by means of linear recursions involving d 1 different d 2 d 2 matrices the average curvatures are calculated exactly for any dimension special scaling limits for the powers of one matrix or of the products of two matrices show interesting behaviour the corresponding curvatures either diverge quadratically or form a six step cycle only for d 3 and one matrix or show ergodic behaviour sometimes with universal invariant measure the latter type of behaviour is also found if the results are formally extended to non integer values of d

generalized convexity in multiobjective programming abstract for the scalar programming problem some characterizations for optimal solutions are known in these characterizations convexity properties play a very important role in this work we study characterizations for multiobjective programming problem solutions when functions belonging to the problem are differentiable these characterizations need some conditions of convexity in differentiable scalar programming problems the concept of invexity is very important we prove that it is also necessary for the multiobjective programming problem and give some characterizations of multiobjective programming problem solutions under weaker conditions we define analogous concepts to those of stationary points and to the conditions of kuhn tucker and fritz john for the multiobjective programming problem

hausdorff dimension and conformal dynamics iii computation of dimension this paper presents an eigenvalue algorithm for accurately computing the hausdorff dimension of limit sets of kleinian groups and julia sets of rational maps the algorithm is applied to schottky groups quadratic polynomials and blaschke products yielding both numerical and theoretical results dimension graphs are presented for a the family of fuchsian groups generated by reflections in 3 symmetric geodesics b the family of polynomials f c z z 2 c c 1 1 2 and c the family of rational maps f t z z t 1 z t 0 1 we also calculate h dim 1 305688 for the apollonian gasket and h dim j f 1 3934 for douady s rabbit where f z z 2 c satisfies f 3 0 0

the asymptotic distribution of circles in the orbits of kleinian groups let mathcal p be a locally finite circle packing in the plane invariant under a non elementary kleinian group and with finitely many orbits when is geometrically finite we construct an explicit borel measure on which describes the asymptotic distribution of small circles in mathcal p assuming that either the critical exponent of is strictly bigger than 1 or mathcal p does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of our construction also works for mathcal p invariant under a geometrically infinite group provided admits a finite bowen margulis sullivan measure and the skinning size of mathcal p is finite some concrete circle packings to which our result applies include apollonian circle packings sierpinski curves schottky dances etc

plato apollonius and klein playing with spheres abstract we describe the group theoretical background and the numerical techniques which enable us to compute new families of apollonian circle packings we provide examples of these constructions and we motivate this research from the physical and the mathematical point of view

a divergence formula for randomness and dimension if s is an infinite sequence over a finite alphabet s and b is a probability measure on s then the dimension of s with respect to b written dim b s is a constructive version of billingsley dimension that coincides with the constructive hausdorff dimension dim s when b is the uniform probability measure this paper shows that dim b s and its dual dim b s the strong dimension of s with respect to b can be used in conjunction with randomness to measure the similarity of two probability measures a and b on s specifically we prove that the divergence formuladim b r dim b r h a h a d email protected email protected holds whenever a and b are computable positive probability measures on s and email protected email protected is random with respect to a in this formula h a is the shannon entropy of a and d email protected email protected is the kullback leibler divergence between a and b we also show that the above formula holds for all sequences r that are a normal in the sense of borel when dim b r and dim b r are replaced by the more effective finite state dimensions dim f s b r and dim f s b r in the course of proving this we also prove finite state compression characterizations of dim f s b s and dim f s b s

distribution of orbits in r 2 of a finitely generated group of sl 2 r in this work we study the asymptotic distribution of the non discrete orbits of a finitely generated group acting linearly on bbb r 2 to do this we establish new equidistribution results for the horocyclic flow on the unitary tangent bundle of the associated surface

the orthogonal spectrum of a hyperbolic manifold introduction in this paper we investigate the geometry of embedded hypersurfaces in hyperbolic manifolds of dimension greater than one our focus is on hypersurfaces that are either totally geodesic horospherical quotients of horospheres or the boundary of a hyperbolic ball these hypersurfaces have a natural decomposition into two pieces one of which corresponds to base points of normal rays that satisfy a specified crossing condition and the other piece being a disjoint union of embedded discs we show that these embedded discs are in one to one correspondence with a certain set of lengths of orthogonals the orthogonal spectrum emanating from the hypersurface in fact the area of each disc is a monotonic function of the corresponding length in the spectrum thus the area induced volume of the hypersurface is intimitely related to the orthogonal spectrum see theorem 1 1 for a precise statement in special cases corollary 1 2 the piece of the hypersurface corresponding to the base points of normal rays is related to the limit set of the discrete isometry group that represents the manifold as a consequence we show that the measure of the limit set of this group is zero if and only if the area of the piece consisting of base points of normal rays is zero the next theorem theorem 1 3 relates the orthogonal spectrum of a closed embedded totally geodesic hypersurface with the lengths of geodesic loops based at some point of the ambient manifold in particular by applying a result of sullivan we can show that the geodesic flow acts ergodically on the unit tangent bundle of the manifold if and only if the sum of the areas of the discs embedded in the hypersurface diverges corollary 1 4

free vs locally free kleinian groups we prove that kleinian groups whose limit sets are cantor sets of hausdorff dimension 0 examples of non free purely hyperbolic kleinian groups whose limit set is a cantor set of hausdorff dimension 1 epsilon

effective strong dimension in algorithmic information and computational complexity the two most important notions of fractal dimension are hausdorff dimension developed by hausdorff math ann 79 1919 pp 157 179 and packing dimension developed independently by tricot math proc cambridge philos soc 91 1982 pp 57 74 and sullivan acta math 153 1984 pp 259 277 both dimensions have the mathematical advantage of being defined from measures and both have yielded extensive applications in fractal geometry and dynamical systems lutz proceedings of the 15th ieee conference on computational complexity florence italy 2000 ieee computer society press piscataway nj 2000 pp 158 169 has recently proven a simple characterization of hausdorff dimension in terms of gales which are betting strategies that generalize martingales imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of hausdorff dimension including constructive computable polynomial space polynomial time and finite state dimensions work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science in this paper we show that packing dimension can also be characterized in terms of gales moreover even though the usual definition of packing dimension is considerably more complex than that of hausdorff dimension our gale characterization of packing dimension is an exact dual of and every bit as simple as the gale characterization of hausdorff dimension effectivizing our gale characterization of packing dimension produces a variety of effective strong dimensions which are exact duals of the effective dimensions mentioned above in general and in analogy with the classical fractal dimensions the effective strong dimension of a set or sequence is at least as great as its effective dimension with equality for sets or sequences that are sufficiently regular we develop the basic properties of effective strong dimensions and prove a number of results relating them to fundamental aspects of randomness kolmogorov complexity prediction boolean circuit size complexity polynomial time degrees and data compression aside from the above characterization of packing dimension our two main theorems are the following 1 if vec beta beta 0 beta 1 ldots is a computable sequence of biases that are bounded away from 0 and r is random with respect to vec beta then the dimension and strong dimension of r are the lower and upper average entropies respectively of vec beta 2 for each pair of delta 0 2 computable real numbers 0 alpha le beta le 1 there exists a in rm e such that the polynomial time many one degree of a has dimension alpha in e and strong dimension beta in e our proofs of these theorems use a new large deviation theorem for self information with respect to a bias sequence vec beta that need not be convergent