a note on self adjoint extensions of the laplacian on weighted graphs abstract we study the uniqueness of self adjoint and markovian extensions of the laplacian on weighted graphs we first show that for locally finite graphs and a certain family of metrics completeness of the graph implies uniqueness of these extensions moreover in the case when the graph is not metrically complete and the cauchy boundary has finite capacity we characterize the uniqueness of the markovian extensions

estimates for measures of lower dimensional sections of convex bodies abstract we present an alternative approach to some results of koldobsky on measures of sections of symmetric convex bodies which allows us to extend them to the not necessarily symmetric setting we prove that if k is a convex body in r n with 0 int k and is a measure on r n with a locally integrable non negative density g on r n then k c n k k max f g n n k k f k k n for every 1 k n 1 also if is even and log concave and if k is a symmetric convex body in r n and d is a compact subset of r n such that k f d f for all f g n n k then k c k l n k k d where l s is the maximal isotropic constant of a convex body in r s our method employs a generalized blaschke petkantschin formula and estimates for the dual affine quermassintegrals

the structure of tame minimal dynamical systems for general groups we use the structure theory of minimal dynamical systems to show that for a general group gamma a tame metric minimal dynamical system x gamma has the following structure open image in new window here i tilde x is a metric minimal and tame system ii eta is a strongly proximal extension iii y is a strongly proximal system iv pi is a point distal and rim extension with unique section v theta theta and iota are almost one to one extensions and vi sigma is an isometric extension when the map pi is also open this diagram reduces to open image in new window in general the presence of the strongly proximal extension eta is unavoidable if the system x gamma admits an invariant measure mu then y is trivial and x tilde x is an almost automorphic system i e x overset iota rightarrow z where iota is an almost one to one extension and z is equicontinuous moreover mu is unique and iota is a measure theoretical isomorphism iota x mu gamma rightarrow z lambda gamma with lambda the haar measure on z thus this is always the case when gamma is amenable

on partial shadowing of complete pseudo orbits abstract in recent years many new definitions of shadowing using a notion of ergodic or average pseudo orbit were introduced while under the assumption of chain mixing average pseudo orbit usually can be equivalently replaced by pseudo orbit in these definitions of shadowing it is not completely clear when these shadowing properties i e approximation of a pseudo orbit by a real orbit on a sufficiently large set of indices can or cannot occur in this paper we analyze necessary and sufficient conditions for shadowing over a set with positive density or syndetic while we do not provide full characterization a few relations to standard notions from topological dynamics are obtained

gaussian heat kernel upper bounds via the phragmen lindelof theorem we prove that in presence of l gaussian estimates so called davies gaffney estimates on diagonal upper bounds imply precise off diagonal gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces

on the lp spectrum of laplacians on graphs abstract we study the p independence of spectra of laplace operators on graphs arising from regular dirichlet forms on discrete spaces here a sufficient criterion is given solely by a uniform subexponential growth condition moreover under a mild assumption on the measure we show a one sided spectral inclusion without any further assumptions we study applications to normalized laplacians including symmetries of the spectrum and a characterization for positivity of the cheeger constant furthermore we consider laplacians on planar tessellations for which we relate the spectral p independence to assumptions on the curvature

quasi factors of zero entropy systems for minimal systems x t of zero topological entropy we demon strate the sharp difference between the behavior regarding entropy of the sys tems m x t and 2x t induced by t on the spaces m x of prob ability measures on x and 2x of closed subsets of x it is shown that the system m x t has itself zero topological entropy two proofs of this the orem are given the first uses ergodic theoretic ideas the second relies on the different behavior of the banach spaces ln and ln with respect to the existence of almost hilbertian central sections of the unit ball in contrast to this theorem we construct a minimal system x t of zero entropy with a minimal subsystem y t of 2 t whose entropy is positive school of mathematics tel aviv university ramat aviv israel e mail address glasner0math tau ac il institute of mathematics hebrew university jerusalem israel e mail address weissfmath huji ac il this content downloaded from 207 46 13 117 on sun 23 oct 2016 04 35 33 utc all use subject to http about jstor org terms

the mobius function and distal flows we prove that the mobius function is linearly disjoint from an analytic skew product on the 2 torus these flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points the previous cases for which such disjointness has been proved are all regular we also establish the linear disjointness of the mobius function from various distal homogeneous flows

chaos and indecomposability abstract we use recent developments in local entropy theory to prove that chaos in dynamical systems implies the existence of complicated structure in the underlying space earlier mouron proved that if x is an arc like continuum which admits a homeomorphism f with positive topological entropy then x contains an indecomposable subcontinuum barge and diamond proved that if g is a finite graph and f g g is any map with positive topological entropy then the inverse limit space lim g f contains an indecomposable continuum in this paper we show that if x is a g like continuum for some finite graph g and f x x is any map with positive topological entropy then lim x f contains an indecomposable continuum as a corollary we obtain that in the case that f is a homeomorphism x contains an indecomposable continuum moreover if f has uniformly positive upper entropy then x is an indecomposable continuum our results answer some questions raised by mouron and generalize the above mentioned work of mouron and also that of barge and diamond we also introduce a new concept called zigzag pair which attempts to capture the complexity of a dynamical systems from the continuum theoretic perspective and facilitates the proof of the main result

mobius disjointness for topological models of ergodic measure preserving systems with quasi discrete spectrum abstract by using a decomposition result for ergodic measure preserving system with quasi discrete spectrum we prove that a generic point of an ergodic quasi discrete spectrum measure in a topological dynamical system satisfies the required disjointness condition in sarnak s mobius disjointness conjecture as a direct application we have that sarnak s mobius disjointness conjecture holds for any topological model of an ergodic measure preserving system with quasi discrete spectrum

mobius orthogonality for generalized morse kakutani flows the prouhet thue morse sequence and its generalizations have occured in many settings morse kakutani flow refers to kakutani s 1967 generalization of the morse minimal flow 1922 these flows are bbb z 2 skew products of almost one to one extensions of the adding machine x rightarrow x 1 on the 2 adic completion of bbb z generalized morse kakutani flow is a k skew product of similar construct with base flow a factor of x rightarrow x 1 on the profinite completion of bbb z and k any compact group abelian or not of countable density the principal result is that sarnak s m o bius orthogonality conjecture holds for a restricted class of generalized morse kakutani flows

pointwise theorems for amenable groups in this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along folner sequences that obey some restrictions these restrictions are mild enough so that such sequences exist for all amenable groups we also prove a generalization of the shannon mcmillan breiman theorem to all discrete amenable groups

quantized arnold cat maps can be entropic k systems automorphisms on the irrational rotation algebra with respect to their ergodic properties are studied especially it is shown that for a dense set of the rotation parameter cat maps are entropic k systems

turbulence amalgamation and generic automorphisms of homogeneous structures we study topological properties of conjugacy classes in polish groups with emphasis on automorphism groups of homogeneous countable structures we first consider the existence of dense conjugacy classes the topological rokhlin property we then characterize when an automorphism group admits a comeager conjugacy class answering a question of truss and apply this to show that the homeomorphism group of the cantor space has a comeager conjugacy class answering a question of akin hurley and kennedy finally we study polish groups that admit comeager conjugacy classes in any dimension in which case the groups are said to admit ample generics we show that polish groups with ample generics have the small index property generalizing results of hodges hodkinson lascar and shelah and arbitrary homomorphisms from such groups into separable groups are automatically continuous moreover in the case of oligomorphic permutation groups they have uncountable cofinality and the bergman property these results in particular apply to automorphism groups of many stable 0 categorical structures and of the random graph in this connection we also show that the infinite symmetric group s has a unique non trivial separable group topology for several interesting groups we also establish serre s properties fh and fa

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

slicing inequalities for measures of convex bodies abstract we consider the following problem does there exist an absolute constant c so that for every n n every integer 1 k n every origin symmetric convex body l in r n and every measure with non negative even continuous density in r n 1 l c k max h gr n k l h l k n where gr n k is the grassmanian of n k dimensional subspaces of r n and l stands for volume this question is an extension to arbitrary measures in place of volume and to sections of arbitrary codimension k of the slicing problem a major open problem in convex geometry it was proved in 18 19 that 1 holds for arbitrary origin symmetric convex bodies all k and all with c o n in this article we prove inequality 1 with an absolute constant c for unconditional convex bodies and for duals of bodies with bounded volume ratio we also prove that for every 0 1 there exists a constant c c so that inequality 1 holds for every n n every origin symmetric convex body l in r n every measure with continuous density and the codimension of sections k n the proofs are based on a stability result for generalized intersection bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies in the last section we show that for some measures the behavior of minimal sections may be very different from the case of volume

automatic sequences fulfill the sarnak conjecture we present in this article a new method for dealing with automatic sequences this method allows us to prove a mobius randomness principle for automatic sequences from which we deduce the sarnak conjecture for this class of sequences furthermore we can show a prime number theorem for automatic sequences that are generated by strongly connected automata where the initial state is fixed by the transition corresponding to 0

davies gaffney grigor yan lemma on simplicial complexes we prove davies gaffney grigor yan lemma for heat kernels of bounded discrete hodge laplacians on simplicial complexes

local entropy theory for a countable discrete amenable group action in the paper we throw the first light on studying systematically the local entropy theory for a countable discrete amenable group action for such an action we introduce entropy tuples in both topological and measure theoretic settings and build the variational relation be tween these two kinds of entropy tuples by establishing a local variational principle for a given finite open cover moreover based the idea of topological entropy pairs we introduce and study two special classes of such an action uniformly positive entropy and completely positive entropy note that in the building of the local variational principle following romagnoli s ideas two kinds of measure theoretic entropy are introduced for finite borel covers these two kinds of entropy turn out to be the same where danilenko s orbital approach becomes an inevitable tool

an example related to the slicing inequality for general measures abstract for n n let s n be the smallest number s 0 satisfying the inequality k f s k 1 n max s n 1 k f for all centrally symmetric convex bodies k in r n and all even continuous probability densities f on k here k is the volume of k it was proved in 16 that s n 2 n and in analogy with bourgain s slicing problem it was asked whether s n is bounded from above by a universal constant in this note we construct an example showing that s n c n log log n where c 0 is an absolute constant additionally for any 0 2 we describe a related example that satisfies the so called condition

the rohlin property for shifts on uhf algebras and automorphisms of cuntz algebras abstract we show that the shift on the uhf algebra m n has the rohlin property for any odd n hence by brotteli et al the crossed product of a uhf algebra by a shift ergodic theory dynam systems 13 1993 615 626 for any n 2 3 and that any automorphism of the cuntz algebra o n n m is outer for any m 0

independence in topological and c dynamics we develop a systematic approach to the study of independence in topological dynamics with an emphasis on combinatorial methods one of our principal aims is to combinatorialize the local analysis of topological entropy and related mixing properties we also reframe our theory of dynamical independence in terms of tensor products and thereby expand its scope to c dynamics

positive topological entropy and δ weakly mixing sets abstract the notion of weakly mixing set is introduced and studied it is proved that weakly mixing sets share many properties with weakly mixing sets in particular if a dynamical system has positive topological entropy then the collection of weakly mixing sets is residual in the closure of the collection of entropy sets in the hyperspace the existence of weakly mixing sets in a topological dynamical system admitting an ergodic invariant measure which is not measurable distal is obtained these results generalize several well known results and also answer several open questions

entropy chaos and weak horseshoe for infinite dimensional random dynamical systems in this paper we study the complicated dynamics of infinite dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a polish space without assuming any hyperbolicity we prove if a continuous random map has a positive topological entropy then it contains a topological horseshoe we also show that the positive topological entropy implies the chaos in the sense of li yorke the complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system 2017 wiley periodicals inc

strong asymptotic abelianness for entropic k systems we prove that in entropick systems of type ii1 the automorphism is strongly asymptotically abelian

combinatorial lemmas and applications to dynamics abstract the well known combinatorial lemma of karpovsky milman and alon and a very recent one of kerr and li are extended the obtained lemmas are applied to study the maximal pattern entropy introduced in the paper it turns out that the maximal pattern entropy is equal to the supremum of sequence entropies over all sequences both in topological and measure theoretical settings moreover it is shown the maximal pattern entropy of any topological system is log k for some k n with k the maximal length of intrinsic sequence entropy tuples and a zero dimensional system has zero sequence entropy for any sequence if and only if the maximal pattern with respect to any open cover is of polynomial order

turbulence representations and trace preserving actions we establish criteria for turbulence in certain spaces of c algebra repre sentations and apply this to the problem of nonclassiability by countable structures for group actions on a standard atomless probability space x and on the hypernite ii1 factor r we also prove that the conjugacy action on the space of free actions of a countably innite amenable group on r is turbulent and that the conjugacy action on the space of ergodic measure preserving ows on x is generically turbulent

on li yorke pairs the li yorke definition of chaos proved its value for interval maps in this paper it is considered in the setting of general topological dynamics we adopt two opposite points of view on the one hand sufficient conditions for li yorke chaos in a topological dynamical system are given we solve a long standing open question by proving that positive entropy implies li yorke chaos on the other hand properties of dynamical systems without li yorke pairs are investigated in addition to having entropy 0 they are minimal when transitive and the property is stable under factor maps arbitrary products and inverse limits finally it is proved that minimal systems without li yorke pairs are disjoint from scattering systems

crossed products and entropy of automorphisms abstract let a be an exact c algebra let g be a locally compact group and let a g be a c dynamical system each automorphism g induces a spatial automorphism ad g on the reduced crossed product a g in this paper we examine the question first raised by e stormer of when the topological entropies of g and ad g coincide this had been answered by n brown for the particular case of discrete abelian groups using different methods we extend his result to preservation of entropy for g when the subgroup of aut g generated by the corresponding inner automorphism ad g has compact closure this property is satisfied by all elements of a wide class of groups called locally fia this class includes all abelian groups both discrete and continuous as well as all compact groups

intrinsic metrics for non local symmetric dirichlet forms and applications to spectral theory we present a study of what may be called an intrinsic metric for a general regular dirichlet form for such forms we then prove a rademacher type theorem for strongly local forms we show existence of a maximal intrinsic metric under a weak continuity condition and for dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals we then turn to applications on spectral theory and provide for measure perturbation of general regular dirichlet forms an allegretto piepenbrinck type theorem which is based on a ground state transform and a shnol type theorem our setting includes laplacian on manifolds on graphs and stable processes

on stochastic completeness of jump processes we prove the following sufficient condition for stochastic completeness of symmetric jump processes on metric measure spaces if the volume of the metric balls grows at most exponentially with radius and if the distance function is adapted in a certain sense to the jump kernel then the process is stochastically complete we use this theorem to prove the following criterion for stochastic completeness of a continuous time random walk on a graph with a counting measure if the volume growth with respect to the graph distance is at most cubic then the random walk is stochastically complete where the cubic volume growth is sharp

compact group automorphisms addition formulas and fuglede kadison determinants for a countable amenable group and an element f in the integral group ring z being invertible in the group von neumann algebra of we show that the entropy of the shift action of on the pontryagin dual of the quotient of z by its left ideal generated by f is the logarithm of the fuglede kadison determinant of f for the proof we establish an p version of rufus bowen s denition of topological entropy addition formulas for group extensions of countable amenable group actions and an approximation formula for the fuglede kadison determinant of f in terms of the determinants of perturbations of the compressions of f

dynamical entropy in banach spaces we introduce a version of voiculescu brown approximation entropy for isometric automorphisms of banach spaces and develop within this framework the connection between dynamics and the local theory of banach spaces as discovered by glasner and weiss our fundamental result concerning this contractive approximation entropy or ca entropy characterizes the occurrence of positive values both geometrically and topologically this leads to various applications for example we obtain a geometric description of the topological pinsker factor and show that a c algebra is type i if and only if every multiplier inner automorphism has zero ca entropy we also examine the behaviour of ca entropy under various product constructions and determine its value in many examples including isometric automorphisms of l p for 1 p and noncommutative tensor product shifts

stable sets and mean li yorke chaos in positive entropy systems abstract it is shown that in a topological dynamical system with positive entropy there is a measure theoretically rather big set such that a multivariant version of mean li yorke chaos happens on the closure of the stable or unstable set of any point from the set it is also proved that the intersections of the sets of asymptotic tuples and mean li yorke tuples with the set of topological entropy tuples are dense in the set of topological entropy tuples respectively

asymptotic pairs stable sets and chaos in positive entropy systems abstract we consider positive entropy g systems for certain countable discrete infinite left orderable amenable groups g we will use local analysis to study the connection between the existence of asymptotic pairs chaotic sets and stable sets examples are given for the case of integer lattice groups the heisenberg group and the groups of integral unipotent upper triangular matrices

generically there is but one self homeomorphism of the cantor set we describe a self homeomorphism r of the cantor set x and then show that its conjugacy class in the polish group h x of all homeomorphisms of x forms a dense g subset of h x we also provide an example of a locally compact second countable topological group which has a dense conjugacy class

combinatorial independence in measurable dynamics abstract we develop a fine scale local analysis of measure entropy and measure sequence entropy based on combinatorial independence the concepts of measure ie tuples and measure in tuples are introduced and studied in analogy with their counterparts in topological dynamics local characterizations of the pinsker von neumann algebra and its sequence entropy analogue are given in terms of combinatorial independence l 1 geometry and voiculescu s completely positive approximation entropy among the novel features of our local study is the treatment of general discrete acting groups with the structural assumption of amenability in the case of entropy

the mobius function is strongly orthogonal to nilsequences we show that the mobius function n is strongly asymptotically or thogonal to any polynomial nilsequence f g n n2n here g is a sim ply connected nilpotent lie group with a discrete and cocompact subgroup so g is a nilmanifold g z g is a polynomial sequence and f g r is a lipschitz function more precisely we show that j 1 n p n n 1 n f g n j f g a log a n for all a 0 in particu lar this implies the mobius and nilsequence conjecture mn s from our earlier paper for every positive integer s this is one of two major in gredients in our programme to establish a large number of cases of the generalised hardy littlewood conjecture which predicts how often a collec tion 1 t z d z of linear forms all take prime values the proof is a relatively quick application of the results in our recent companion paper we give some applications of our main theorem we show for exam ple that the mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c we also obtain a result about the distribution of nilse quences a n x n2n as n ranges only over the primes

measure complexity and mobius disjointness abstract in this paper the notion of measure complexity is introduced for a topological dynamical system and it is shown that sarnak s mobius disjointness conjecture holds for any system for which every invariant borel probability measure has sub polynomial measure complexity we then apply this result to a number of situations including certain systems whose invariant measure don t all have discrete spectrum

a n estimate for measures of hyperplane sections of convex bodies the hyperplane or slicing problem asks whether there exists an absolute constant c so that for any origin symmetric convex body k in rn k n 1n cmax sn 1 k where is the central hyperplane in rn perpendicular to and k stands for volume of proper dimension the problem is still open with the best to date estimate c n1 4 established by klartag who slightly improved the previous estimate of bourgain it is much easier to get a weaker estimate with c n in this note we show that the n estimate holds for arbitrary measure in place of volume namely if l is an origin symmetric convex body in rn and is a measure with non negative even continuous density on l then l nnn 1cnmax sn 1 l l 1 n where cn b2n n 1n b2n 1 1 and b2n is the unit euclidean ball in rn we deduce this inequality from a stability result for intersection bodies

spectra of combinatorial laplace operators on simplicial complexes abstract we first develop a general framework for laplace operators defined in terms of the combinatorial structure of a simplicial complex this includes among others the graph laplacian the combinatorial laplacian on simplicial complexes the weighted laplacian and the normalized graph laplacian this framework then allows us to define the normalized laplace operator i u p on simplicial complexes which we then systematically investigate we study the effects of a wedge sum a join and a duplication of a motif on the spectrum of the normalized laplace operator and identify some of the combinatorial features of a simplicial complex that are encoded in its spectrum

on distributionally chaotic and null systems abstract by a topological dynamical system we mean a pair x f where x is a compactum and f is a continuous self map on x a system is said to be null if its topological sequence entropies are zero along all strictly increasing sequences of natural numbers we show that there exists a null system which is distributionally chaotic this system admits open distributionally scrambled sets and its collection of all maximal distributionally scrambled sets has the same cardinality as the collection of all subsets of the phase space finally such system can even exist on continua

δ weakly mixing subset in positive entropy actions of a nilpotent group abstract the notion of weakly mixing subsets is introduced for countable torsion free discrete group actions it is shown that for a finitely generated torsion free discrete nilpotent group action positive topological entropy implies the existence of weakly mixing subsets and while there exists a finitely generated torsion free discrete solvable group action which has positive topological entropy but without any weakly mixing subsets

upper bounds for eigenvalues of the discrete and continuous laplace operators in this paper we are concerned with upper bounds of eigenvalues of laplace operator on compact riemannian manifolds and finite graphs while on the former the laplace operator is generated by the riemannian metric on the latter it reflects combinatorial structure of a graph respectively eigenvalues have many applications in geometry as well as in combinatorics and in other fields of mathematics we develop a universal approach to upper bounds on both continuous and discrete structures based upon certain properties of the corresponding heat kernel this approach is perhaps much more general than its realization here basically we start with the following entries 1 an underlying space m with a finite measure 2 a well defined laplace operator on functions on m so that is a self adjoint operator in l m with a discrete spectrum 3 if m has a boundary then the boundary condition should be chosen so that it does not disrupt self adjointness of and is of dissipative nature 4 a distance function dist x y on m so that dist 1 for an appropriate notion of gradient

on proximality with banach density one abstract let x t be a topological dynamical system a pair of points x y x 2 is called banach proximal if for any e 0 the set n z d t n x t n y e has banach density one we study the structure of the banach proximal relation a useful tool is the notion of the support of a topological dynamical system we show that a dynamical system is strongly proximal if and only if every pair in x 2 is banach proximal a subset s of x is banach scrambled if every two distinct points in s form a banach proximal pair but not asymptotic we construct a dynamical system with the whole space being a banach scrambled set even though the banach proximal relation of the full shift is of first category it has a dense mycielski invariant banach scrambled set we also show that for an interval map it is li yorke chaotic if and only if it has a cantor banach scrambled set

operational independence and tensor products of c algebras complete c independence of operator algebras is introduced equivalent characterization is given for c subalgebras to be completely independent in terms of maximal tensor product besides the independence of banach algebras is considered and we showed that hahn banach independence is a generalization of c independence and discussed hahn banach independence in mn a where a is a c algebra among others we characterize independence of operator algebras by projective and injective c tensor product in terms of simultaneous extensions of completely positive maps

asymptotic abelianness weak mixing and property t let g be a second countable locally compact group and h a closed subgroup we characterize the lack of kazhdan s property t for the pair g h by the genericity of g actions on the hyperfinite ii1 factor with a certain asymptotic abelianness property relative to h as well as by the genericity of measure preserving g actions on a nonatomic standard probability space that are weakly mixing for h the latter furnishes a definitive generalization of a classical theorem of halmos for single automorphisms and strengthens a recent result of glasner thouvenot and weiss on generic ergodicity we also establish a weak mixing version of glasner and weiss s characterization of property t for discrete g in terms of the invariant state space of a bernoulli shift and show that on the car algebra a type of norm asymptotic abelianness is generic for g actions when g is discrete and admits a nontorsion abelian quotient

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

intersection bodies positive definite distributions and the busemann petty problem the 1956 busemann petty problem asks whether symmetric convex bodies with larger central hyperplane sections also have greater volume in 1988 lutwak introduced the concept of an intersection body which is closely related to the busemann petty problem we prove that an origin symmetric star body k in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is an intersection body if and only if x 1 k is a positive definite distribution on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i where x k min a 0 x ak we use this result to show that for every dimension n there exist polytopes in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i which are intersection bodies for example the cross polytope the unit ball of every subspace of l p 0 p 2 is an intersection body the unit ball of the space l n q 2 q n 5 using lutwak s connection with the busemann petty problem we present new counterexamples to the problem for n 5 and confirm the conjecture of meyer that the answer to the problem is affirmative if the smaller body is a polar projection body

sharp davies gaffney grigor yan lemma on graphs in this note we prove the sharp davies gaffney grigor yan lemma for minimal heat kernels on graphs

topological pressure for sub additive potentials of amenable group actions abstract we introduce the topological pressure for any sub additive potentials of a countable discrete amenable group action and any given open cover and establish a local variational principle for it

homoclinic groups ie groups and expansive algebraic actions we give algebraic characterizations for expansiveness of algebraic actions of countable groups the notion of p expansiveness is introduced for algebraic actions and we show that for countable amenable groups a finitely presented algebraic action is 1 expansive exactly when it has finite entropy we also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms and show that the ie group determines the pinsker factor for such actions for an expansive algebraic action of a polycyclic by finite group on x it is shown that the entropy of the action is equal to the entropy of the induced action on the pontryagin dual of the homoclinic group the homoclinic group is a dense subgroup of the ie group the homoclinic group is nontrivial exactly when the action has positive entropy and the homoclinic group is dense in x exactly when the action has completely positive entropy

generalized specification property and distributionally scrambled sets abstract suppose the dynamical system x f has generalized specification property in this paper we research the existence of the invariant mycielski distributionally scrambled sets for dynamical system x f moreover some properties about measure center are discussed and a dichotomy theorem is proved

conditional entropy and fiber entropy for amenable group actions abstract in this paper we introduce the notions of topological conditional entropy and fiber entropy for a given factor map between two amenable group actions and prove three variational principles for conditional entropy and fiber entropy moreover as an application of our variational principle we prove that the countable to one extension and the distal extension have zero conditional topological entropy in particular the topological entropy of a distal system is zero

on the k property of quantized arnold cat maps we prove that some quantized arnold cat maps are entropic k systems this result was formulated by h narnhofer j math phys 33 1502 1510 1992 but the fact that the optimal decomposition for the multi channel entropy constructed there is not strictly local was not appropriately taken care of we propose a strictly local decomposition based on a construction of voiculescu

centroid bodies and the logarithmic laplace transform a unified approach abstract we unify and slightly improve several bounds on the isotropic constant of high dimensional convex bodies in particular a linear dependence on the body s 2 constant is obtained along the way we present some new bounds on the volume of l p centroid bodies and yet another equivalent formulation of bourgain s hyperplane conjecture our method is a combination of the l p centroid body technique of paouris and the logarithmic laplace transform technique of the first named author