random walks of infinite moment on free semigroups we consider random walks on finitely or countably generated free semigroups and identify their poisson boundaries for classes of measures which fail to meet the classical entropy criteria namely measures with infinite entropy or infinite logarithmic moment

a multiplicative ergodic theorem and nonpositively curved spaces we study integrable cocycles u n x over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space y e g a cartan hadamard space or a uniformly convex banach space it is proved that for any y 2 y and almost all x there exist a 0 and a unique geodesic ray t x in y starting at y such that lim n 1 1 n d an x u n x y d 0 in the case where y is the symmetric space gln r on r and the cocycles take values in gln r this is equivalent to the multiplicative ergodic theorem of oseledec two applications are also described the first concerns the determination of poisson boundaries and the second concerns hilbert schmidt operators

the poisson boundary of teichmuller space abstract for a markov operator on teichmuller space commuting with the action of the mapping class group we prove convergence of sample paths of the associated markov chain in the thurston compactification and show that the poisson boundary of the markov operator can be identified with the space of projective measured foliations the approach consists in using the authors results on the poisson boundary of the mapping class group in combination with a discretization procedure based on a harnack inequality for markov operators on teichmuller space

uniform homeomorphisms of banach spaces and asymptotic structure we give a general result on the behavior of spreading models in banach spaces which coarse lipschitz embed into asymptotically uniformly convex spaces we use this result to study the uniqueness of the uniform structure in p sums of finite dimensional spaces for 1 p in particular we give some new examples of spaces with unique uniform structure

deviation inequalities for random walks we study random walks on groups with the feature that roughly speaking successive positions of the walk tend to be aligned we formalize and quantify this property by means of the notion of deviation inequalities we show that deviation inequalities have several consequences including central limit theorems the local lipschitz continuity of the rate of escape and entropy as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point in the second part of this article we show that the exponential deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups these include nonelementary relatively hyperbolic groups mapping class groups many groups acting on cat 0 spaces and small cancellation groups

harmonic operators the dual perspective the study of harmonic functions on a locally compactgroupghas recently been transferred to a non commutative setting in two different directions chu and lau replaced the algebra l g by the group von neumann algebra vn g and the convolution action of a probability measure on l g by the canonical action of a positive definite function on vn g on the other hand jaworski and the first author replaced l g by b l 2 g to which the convolution action by can be extended in a natural way we establish a link between both approaches the action of on vn g can be extended to b l 2 g we study the corresponding space h

sublinear deviation between geodesics and sample paths we give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic like properties as an application we prove sublinear tracking in teichmuller distance for random walks on mapping class groups and on cayley graphs of a large class of finitely generated groups

the poisson boundary of the mapping class group a theory of random walks on the mapping class group and its non elementary subgroups is developed we prove convergence of sample paths in the thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the poisson boundary of random walks the methods are based on an analysis of the asymptotic geometry of teichmuller space and of the contraction properties of the action of the mapping class group on the thurston boundary we prove in particular that teichmuller space is roughly isometric to a graph with uniformly bounded vertex degrees using our analysis of the mapping class group action on the thurston boundary we prove that no non elementary subgroup of the mapping class group can be a lattice in a higher rank semi simple lie group

the poisson boundary of text out f_n let mu be a probability measure on text out f n with finite first logarithmic moment with respect to the word metric finite entropy and whose support generates a nonelementary subgroup of text out f n we show that almost every sample path of the random walk on text out f n mu when realized in culler and vogtmann s outer space converges to the simplex of a free arational tree we then prove that the space mathcal fi of simplices of free and arational trees equipped with the hitting measure is the poisson boundary of text out f n mu using bestvina reynolds and hamenstadt s description of the gromov boundary of the complex mathcal ff n of free factors of f n this gives a new proof of the fact due to calegari and maher that the realization in mathcal ff n of almost every sample path of the random walk converges to a boundary point we get in addition that partial mathcal ff n equipped with the hitting measure is the poisson boundary of text out f n mu

a structure theorem for actions of semisimple lie groups we consider a connected semisimple lie group g with finite center an admissible probability measure on g and an ergodic g u space x v we first note lemma 0 1 that x v has a unique maximal projective factor of the form g q vo where q is a parabolic subgroup of g and then prove 1 theorem 1 if every noncompact simple factor of g has real rank at least two then the maximal projective factor is nontrivial unless v is a g

digitization scheme that assures faithful reconstruction of plane figures in the present extensive work we propose a digitization scheme for a broad class of plane figures it includes arbitrary polygons possibly non convex and with holes and plane sets whose boundary consists of smooth curves or straight segments our approach is based on an appropriate scaling of the original continuous real object so that the obtained magnified object and its appropriately constructed digitization feature analogous geometric properties as a bi product of the presented theory we prove the strong np hardness of the problem of obtaining an optimal i e with a minimal number of facets polyhedral reconstruction in which the facets are trapezoids or triangles this result implies the strong np hardness of the general polyhedral reconstruction problem which was a long standing open problem as another major result we show that the proposed digitization scheme allows faithful reconstruction of plane figures from the considered general class the reconstructed set features the basic properties of the original object such as location of curve and straight segments and inflection points of its boundary

efficient algorithms for geometric graph search problems in this paper we show that many graph search problems can be solved quite efficiently for a geometric intersection graph of horizontal and vertical line segments we first extract several basic operations for depth first search and breadth first search on a graph then we present data structures for the intersection graph in terms of which those operations can be implemented in an efficient manner the data structures enable us to solve various graph search problems besides depth first search and breadth first search specifying the results obtained in this paper for an intersection graph of n horizontal and vertical segments with m pairs of intersecting segments we obtain algorithms with the following complexity where n min m n log n i depth first search and breadth first search can be executed in o n log n time and o n space ii the biconnected components can be found in o n log n time and o n space iii a maximum matching and a maximum independent set can be found in o

contractive projections on banach algebras abstract in this paper we explore the properties of projections of norm one on general banach algebras in particular the relation with conditional expectations for algebras which arise in harmonic analysis

planar formulae and their uses we define the set of planar boolean formulae and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is np complete using these results we are able to provide simple and nearly uniform proofs of np completeness for planar node cover planar hamiltonian circuit and line geometric connected dominating set and of polynomial space completeness for planar generalized geography the np completeness of planar node cover and planar hamiltonian circuit and line were first proved elsewhere m r garey and d s johnson the rectilinear steiner tree is np complete siam j appl math 32 1977 pp 826 834 and m r garey d s johnson and r e tarjan the planar hamilton circuit problem is np complete siam j comp 5 1976 pp 704 714

on proximinality of subspaces and the lineability of the set of norm attaining functionals of banach spaces abstract we show that for every 1 n there exists a banach space x n containing proximinal subspaces of codimension n but no proximinal finite codimensional subspaces of higher codimension moreover the set of norm attaining functionals of x n contains n dimensional subspaces but no subspace of higher dimension this gives an n by n version of the solutions given by read and rmoutil to problems of singer and godefroy we also study the existence of strongly proximinal subspaces of finite codimension showing that for every 1 n and 1 k n there is a banach space x n k containing proximinal subspaces of finite codimension up to n but not higher and containing strongly proximinal subspaces of finite codimension up to k but not higher finally we deal with possible infinite dimensional versions of the previous results showing among other results that there are non separable banach spaces whose set of norm attaining functionals contains infinite dimensional separable linear subspaces but no non separable subspaces

the boundary of a square tiling of a graph coincides with the poisson boundary answering a question of benjamini and schramm ann probab 24 3 1219 1238 1996 we show that the poisson boundary of any planar uniquely absorbing e g one ended and transient graph with bounded degrees can be realised geometrically as a circle which arises from a discrete version of riemann s mapping theorem this implies a conjecture of northshield potential anal 2 4 299 314 1993 some of our technique apply to the non planar case and might have further applications when the graph is also hyperbolic then under mild conditions we prove the equivalence of several boundary constructions

a single exponential upper bound for finding shortest paths in three dimensions we derive a single exponential time upper bound for finding the shortest path between two points in 3 dimensional euclidean space with nonnecessarily convex polyhedral obstacles prior to this work the best known algorithm required double exponential time given that the problem is known to be pspace hard the bound we present is essentially the best in the worst case sense that can reasonably be expected

mathematical representation of clausius and kelvin s statements of the second law and irreversibility we provide a stochastic mathematical representation for clausius and kelvin planck s statements of the second law of thermodynamics in terms of the entropy productions of a finite compact driven markov system and its lift a surjective map is rigorously established through the lift when the state space is either a discrete graph or a continuous n dimensional torus mathbb t n the corresponding lifted processes have detailed balance thus a natural potential function but no stationary probability we show that in the long time limit the entropy production of the finite driven system precisely equals the potential energy decrease in the lifted system this theorem provides a dynamic foundation for the two equivalent statements of second law of thermodynamics a la kelvin s and clausius it suggests a modernized combined statement a mesoscopic engine that works in completing irreversible internal cycles statistically has necessarily an external effect that lowering a weight accompanied by passing heat from a warmer to a colder body

boundary and entropy of space homogeneous markov chains we study the poisson boundary equiv representation of bounded harmonic functions of markov operators on discrete state spaces that are invariant under the action of a transitive group of permutations this automorphism group is locally compact but not necessarily discrete or unimodular the main technical tool is the entropy theory which we develop along the same lines as in the case of random walks on countable groups while however the implementation is different and exploits discreteness of the state space on the one hand and the path space of the induced random walk on the nondiscrete group on the other various new examples are given as applications including a description of the poisson boundary for random walks on vertex transitive graphs with infinitely many ends and on the diestel leader graphs

regularity of the entropy for random walks on hyperbolic groups we consider non degenerate finitely supported random walks on a finitely generated gromov hyperbolic group we study how the entropy and the escape rate vary with the probability of constant support

jordan structures in harmonic functions and fourier algebras on homogeneous spaces we study the jordan structures and geometry of bounded matrix valued harmonic functions on a homogeneous space and their analogue the harmonic functionals in the setting of fourier algebras of homogeneous spaces

probability measures on groups and some related ideals in group algebras abstract let g be a locally compact group random walks on g some factorization problems in l 1 g and the significance for these of the amenability of g are studied these topics are linked in this paper via ideals in l 1 g of the form l 1 g e where is a probability measure on g cohen s factorization theorem and some ideas from ergodic theory play an important part

dimensional properties of the harmonic measure for a random walk on a hyperbolic group this paper deals with random walks on isometry groups of gromov hyperbolic spaces and more precisely with the dimension of the harmonic measure nu associated with such a random walk we first establish a link of the form dim nu leq h l between the dimension of the harmonic measure the asymptotic entropy h of the random walk and its rate of escape l then we use this inequality to show that the dimension of this measure can be made arbitrarily small and deduce a result on the type of the harmonic measure

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

spherical functions and harmonic analysis on free groups let f r 1 be a free group with r generators in this paper we study a principal series and a complementary series of irreducible unitary representations of f which are defined through the action of f on its poisson boundary relative to a simple random walk we show that the regular representation of f can be written as a direct integral of the representations of the principal series and that the resulting harmonic analysis on the free group bears a close resemblance with the harmonic analysis of sl 2 ir

on laws of large numbers for random walks we prove a general noncommutative law of large numbers this applies in particular to random walks on any locally finite homogeneous graph as well as to brownian motion on riemannian manifolds whi

norm attaining functionals need not contain 2 dimensional subspaces abstract g godefroy asked whether on any banach space the set of norm attaining functionals contains a 2 dimensional linear subspace we prove that a construction due to c j read provides an example of a space which does not have this property read found an equivalent norm on c 0 such that c 0 contains no proximinal subspaces of codimension 2 our result is obtained through a study of the relation between the following two sentences in which x is a banach space and y x is a closed subspace a y is proximinal in x and b y consists of norm attaining functionals we prove that these are equivalent if x is the space c 0 and our main theorem then follows as a corollary to read s result

ergodic properties of boundary actions and the nielsen schreier theory we study the basic ergodic properties ergodicity and conservativity of the action of a subgroup h of a free group f on the boundary pt f with respect to the uniform measure our approach is geometrical and combinatorial and it is based on choosing a system of nielsen schreier generators in h associated with a geodesic spanning tree in the schreier graph x h bs f we give several mod 0 equivalent descriptions of the hopf decomposition of the boundary into the conservative and the dissipative parts further we relate conservativity and dissipativity of the action with the growth of the schreier graph x and of the subgroup h equiv cogrowth of x respectively on the other hand our approach sheds a new light on entirely algebraic properties of subgroups of a free group we also construct numerous examples illustrating the connections between various relevant notions

amenability via random walks we use random walks to show that the basilica group is amenable answering an open question of grigorchuk and zuk our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits these classes are separated even within the realm of finitely presented groups

partitioning a polygonal region into trapezoids the problem of partitioning a polygonal region into a minimum number of trapezoids with two horizontal sides is discussed a triangle with a horizontal side is considered to be a trapezoid with two horizontal sides one of which is degenerate first a method of achieving a minimum partition is presented the number m of the trapezoids in the minimum partition of a polygonal region p is shown to be m n w h d 1 where n w and h are the number of vertices windows holes and horizontal edges of p respectively and d is the cardinality of a maximum independent set of the straight lines in the plane graph associated with p next this problem is shown to be polynomially equivalent to the problem of finding a maximum independent set of a straight lines in the plane graph and consequently it is shown to be np complete however for a polygonal region without windows an o n 2 time algorithm for partitioning it into a minimum number of trapezoids is presented finally an o n log n time approximation algorithm with the performance bound 3 is presented

entropy and drift in word hyperbolic groups the fundamental inequality of guivarc h relates the entropy and the drift of random walks on groups it is strict if and only if the random walk does not behave like the uniform measure on balls we prove that in any nonelementary hyperbolic group which is not virtually free endowed with a word distance the fundamental inequality is strict for symmetric measures with finite support uniformly for measures with a given support this answers a conjecture of s lalley for admissible measures this is proved using previous results of ancona and blachere haissinsky mathieu for non admissible measures this follows from a counting result interesting in its own right we show that in any infinite index subgroup the number of non distorted points is exponentially small the uniformity is obtained by studying the behavior of measures that degenerate towards a measure supported on an elementary subgroup

asymptotic structure lp estimates of sequences and compactness of multilinear mappings abstract we relate the moduli of asymptotic uniform smoothness and convexity of a banach space with the existence of upper and lower l p estimates of sequences in the space to this end we introduce two properties which are related to the m p property defined by kalton and werner in this way we obtain a connection between the moduli of asymptotic uniform smoothness and convexity and compactness or weak sequential continuity of multilinear mappings finally we give some applications to the existence of analytic and asymptotically flat norms on a banach space

asymptotic entropy and green speed for random walks on countable groups we study asymptotic properties of the green metric associated with transient random walks on countable groups we prove that the rate of escape of the random walk computed in the green metric equals its asymptotic entropy the proof relies on integral representations of both quantities with the extended martin kernel in the case of finitely generated groups where this result is known benjamini and peres probab theory related fields 98 1994 91 112 we give an alternative proof relying on a version of the so called fundamental inequality relating the rate of escape the entropy and the logarithmic volume growth extended to random walks with unbounded support

szlenk indices and uniform homeomorphisms we prove some rather precise renorming theorems for banach spaces with szlenk index wo we use these theorems to show the invariance of certain quantitative szlenk type indices under uniform homeomorphisms

boundary behavior for groups of subexponential growth in this paper we introduce a method for partial description of the poisson boundary for a certain class of groups acting on a segment as an application we find among the groups of subexponential growth those that admit nonconstant bounded harmonic functions with respect to some symmetric infinitely supported measure of finite entropy h this implies that the entropy h of the corresponding random walk is finite and positive as another application we exhibit certain discontinuity for the recurrence property of random walks finally as a corollary of our results we get new estimates from below for the growth function of a certain class of grigorchuk groups in particular we exhibit the first example of a group generated by a finite state automaton such that the growth function is subexponential but grows faster than exp n for any 1 we show that in some of our examples the growth function satisfies exp n ln

sur la dynamique unidimensionnelle en regularite intermediaire using probabilistic methods we prove new rigidity results for groups and pseudo groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class i e between c 1 and c 2 in particular we show some generalizations of denjoy theorem and the classical kopell lemma for abelian groups these techniques are also applied to the study of codimension 1 foliations for instance we obtain several generalized versions of sacksteder theorem in class c 1 we conclude with some remarks about the stationary measure

some geometric properties of read s space abstract we study geometric properties of the banach space r constructed recently by c read 8 which does not contain proximinal subspaces of finite codimension greater than or equal to two concretely we show that the bidual of r is strictly convex that the norm of the dual of r is rough and that r is weakly locally uniformly rotund but it is not locally uniformly rotund apart of the own interest of the results they provide a simplification of the proof by m rmoutil 9 that the set of norm attaining functionals over r does not contain any linear subspace of dimension greater than or equal to two note that if a banach space x contains proximinal subspaces of finite codimension at least two then the set of norm attaining functionals over x contain two dimensional linear subspaces of x our results also provides positive answer to the questions of whether the dual of r is smooth and of whether r is weakly locally uniformly rotund 9 finally we present a renorming of read s space which is smooth whose dual is smooth and such that its set of norm attaining functionals does not contain any linear subspace of dimension greater than or equal to two so the renormed space does not contain proximinal subspaces of finite codimension greater than or equal to two

the poisson formula for groups with hyperbolic properties the poisson boundary of a group g with a probability measure is the space of ergodic components of the time shift in the path space of the associated random walk via a generalization of the classical poisson formula it gives an integral representation of bounded harmonic functions on g in this paper we develop a new method of identifying the poisson boundary based on entropy estimates for conditional random walks it leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the poisson boundary with natural topological boundaries for several classes of groups word hyperbolic groups and discontinuous groups of isometries of gromov hyperbolic spaces groups with inflnitely many ends cocompact lattices in cartan hadamard manifolds discrete subgroups of semisimple lie groups