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

extending the bruhat order and the length function from the weyl group to the weyl monoid abstract for a symmetrizable kac moody algebra the category of admissible representations of the category o is an analogue of the category of finite dimensional representations of a semisimple lie algebra the monoid which is associated to this category and the category of restricted duals by a tannaka krein reconstruction contains the kac moody group as open dense unit group it has similar properties as a reductive algebraic monoid in particular there are bruhat and birkhoff decompositions the weyl group replaced by the weyl monoid c mokler an analogue of a reductive algebraic monoid whose unit group is a kac moody group arxiv math ag 0204246 mem amer math soc in press we determine the closure relations of the bruhat and birkhoff cells which give extensions of the bruhat order from the weyl group to the weyl monoid we show that the bruhat and birkhoff cells are irreducible and principal open in their closures we give product decompositions of the bruhat and birkhoff cells we define extended length functions which are compatible with the extended bruhat orders we show a generalization of some of the tits axioms for twin bn pairs

an inhomogeneous wave equation and non linear diophantine approximation abstract a non linear diophantine condition involving perfect squares and arising from an inhomogeneous wave equation on the torus guarantees the existence of a smooth solution the exceptional set associated with the failure of the diophantine condition and hence of the existence of a smooth solution is studied both the lebesgue and hausdorff measures of this set are obtained

conjugacy decomposition of reductive monoids we study the decomposition of a reductive monoid related to conjugacy classes and obtain a description of the associated finite poset open image in new window this poset is constructed via the gauss jordan elements of the renner monoid r for mn k the poset can be identified with the lattice of all partitions of m 0 m n where the order is an extension of the dominance order on partitions of n

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

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

rapidly convergent iteration method for simultaneous normal forms of commuting maps we propose a new rapidly convergent iteration scheme under weak simultaneous arithmetic conditions for overdetermined systems related to simultaneous conjugation of commuting maps the first application is for commuting holomorphic maps in mathcal c n fixing the origin not necessarily diffeomorphisms if n geq 2 the second one is for orientation preserving gevrey circle mappings by means of nonlinear superposition estimates

bowen topological entropy of subsets for amenable group actions abstract we extend the definition of bowen topological entropy of subsets to continuous action of amenable groups on a compact metrizable space we investigate how bowen topological entropy behaves for restricted actions of finite index subgroups and verify the validity of the subgroup formula we also show that the bowen topological entropy of subsets for amenable group actions can be determined via the local entropies of measures as applications we compute the bowen topological entropy of a subset for bernoulli shift over amenable groups

the hausdorff dimension of small divisors for lower dimensional kam tori given a bounded domain note see the image of page 359 for this formatted text omega subset r m and a lipschitz map note see the image of page 359 for this formatted text phi omega mapsto r n we determine the hausdorff dimension of sets of points omega in omega for which the inequality k cdot omega l cdot phi omega psi k l has infinitely many distinct integer solutions note see the image of page 359 for this formatted text k l in z m times z n satisfying l leq h where h is a fixed integer these sets interpolate between the cases h 0 and h infty which occur in the metric theory of diophantine approximation of independent and dependent quantities respectively they arise for example in the perturbation theories of lower dimensional tori in nearly integrable hamiltonian systems kam theory among others it turns out that their hausdorff dimension is independent of h and n it only depends on m and the lower order of psi at infinity part of this result even extends to the case n infty of infinite co dimension which is relevant in the kam theory of certain nonlinear partial differential equations

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

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

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

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

dynamical approximation entropies and topological entropy in operator algebras dynamical entropy invariants based on a general approximation approach are introduced for c and w algebra automorphisms this includes a noncommutative extension of topological entropy

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

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

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

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

entropy and mixing for amenable group actions for i a countable amenable group consider those actions of i as measurepreserving transformations of a standard probability space written asft g 2i acting on x f we sayft g 2i has completely positive entropy or simply cpe for short if for any flnite and nontrivial partition p of x the entropy h t p is not zero our goal is to demonstrate what is well known for actions of and even d that actions of completely positive entropy have very strong mixing properties let si be a list of flnite subsets of i we say the si spread if any particular 6 id belongs to at most flnitely many of the sets sis i1 i theorem 0 1 for ft g 2i an action of i of completely positive entropy and p any flnite partition for any sequence of flnite sets si i which spread we have

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

irreducible representations of semisimple algebraic groups and supersolvable lattices abstract let be the cross section lattice of an irreducible representation of a semisimple algebraic group certain combinatorial properties of are studied supersolvable s are determined in terms of dynkin diagrams the characteristic polynomial of a supersolvable is computed

finite uniform generators for ergodic finite entropy free actions of amenable groups let x be a lebesgue space we prove in the first part of this paper that any ergodic 2 action on x with finite entropy hlogk has generating partition p that is uniform and has k atoms in the second part we prove a similar result for any ergodic free g action with finite entropy hlog k 2 for any discrete amenable group g

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

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

strictly ergodic models for dynamical systems on etudie l action d un groupe g d homeomorphismes d un espace metrique compacte quand g est commutatif l action ergodique a un modele strictement ergodique

analog of the bruhat decomposition for algebraic monoids ii the length function and the trichotomy 0 introduction the purpose of this paper is to continue the study of b b orbits on a reductive monoidm this was first taken up systematically by the author in 9 where several basic properties analogous to the case of groups were established the b b orbits were identified with a finite monoid r defined in terms of the normalizer of a maximal torus and a direct analysis of these orbits led to an extension of tits axiom br brb b rb to the case of reductive monoids however there is more to this axiom when dealing with monoids obviously

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

a subadditive thermodynamic formalism for mixing repellers the thermodynamical description of fractals that has recently attracted much interest both experimentally and theoretically in the study of dynamical systems is in some ways limited being essentially an additive theory the author presents a subadditive thermodynamic formalism for which he derives a variational principle and shows how it may be used to study the dynamics of non conformal transformations in particular the author discusses an analogue of bowen s formula for the dimension of a mixing repeller

shellability in reductive monoids the purpose of this paper is to extend to monoids the work of bjorner wachs and proctor on the shellability of the bruhat chevalley order on weyl groups let m be a reductive monoid with unit group g borel subgroup b and weyl group w we study the partially ordered set of b x b orbits with respect to zariski closure inclusion within a g x g orbit of m this is the same as studying a w x w orbit in the renner monoid r such an orbit is the retract of a universal orbit which is shown to be lexicograhically shellable in the sense of bjorner and wachs

arithmetic cohen macaulayness and arithmetic normality for schubert varieties let g be a semisimple simply connected chevalley group over a field k let t be a maximal torus b a borel subgroup d t and w the weyl group of g relative to t let p be a maximal parabolic subgroup of classical type cf 7 8 i e a maximal parabolic subgroup such that if w is the associated fundamental weight then i w av i 1 2 w c a is 2 for all roots ae being a w invariant scalar product on x t q where x t character group of t for t e w wp let x t denote the schubert variety in g p associated to t namely x t is the zariski closure btp mod p in g p of the bruhat cell endowed with the canonical reduced structure for the canonical projective embedding g p in p h g p l let rt denote the homogeneous co ordinate ring of x t then the main result of this paper is given by

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

normal linear stability of quasi periodic tori abstract we consider families of dynamical systems having invariant tori that carry quasi periodic motions our interest is the persistence of such tori under small nearly integrable perturbations this persistence problem is studied in the dissipative the hamiltonian and the reversible setting as part of a more general kam theory for classes of structure preserving dynamical systems this concerns the parametrized kam theory as initiated by moser j k moser on the theory of quasiperiodic motions siam rev 8 2 1966 145 172 j k moser convergent series expansions for quasi periodic motions math ann 169 1967 136 176 and further developed in g b huitema unfoldings of quasi periodic tori phd thesis university of groningen 1988 h w broer g b huitema f takens unfoldings of quasi periodic tori mem amer math soc 83 421 1990 1 82 h w broer g b huitema unfoldings of quasi periodic tori in reversible systems j dynam differential equations 7 1 1995 191 212 the corresponding nondegeneracy condition involves certain trans versality conditions on the normal linear leading part at the invariant tori we show that as a consequence a cantor family of diophantine tori with positive hausdorff measure is persistent under nearly integrable perturbations this result extends the above references since presently the case of multiple floquet exponents is included our leading example is the normal 1 1 resonance which occurs a lot in applications both hamiltonian and reversible as an illustration of this we briefly describe the lagrange top coupled to an oscillator

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

analogue of the bruhat chevalley order for reductive monoids the purpose of this paper is to describe the adherence order of w x b b orbits on a reductive algebraic monoid by the results of 10 there is already a perfect analogue of the bruhat decomposition for reductive w x monoids indeed by 10 corollary 5 8 if m is reductive with unit group g and maximal torus t g with borel subgroup b t then the set of two sided b orbits b r mrb is canonically identified with a certain finite inverse monoid r in fact r s n t rt the orbit monoid of the zariski g closure of n t in m by definition for s t g r we define the adherg ence order by s f t if bs b bt b where bt b denotes the zariski closure in m of bt b

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