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

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

semiparametric density estimation by local l2 fitting this article examines density estimation by combining a parametric approach with a nonparametric factor the plug in parametric estimator is seen as a crude estimator of the true density and is adjusted by a nonparametric factor the nonparametric factor is derived by a criterion called local l 2 fitting a class of estimators that have multiplicative adjustment is provided including estimators proposed by several authors as special cases and the asymptotic theories are developed theoretical comparison reveals that the estimators in this class are better than or at least competitive with the traditional kernel estimator in a broad class of densities the asymptotically best estimator in this class can be obtained from the elegant feature of the bias function

the symmetric rouche s form for robustness the objective of this work is to bring to control literature the strengthened and symmetric version of the classical rouche s form the author discusses the need for rouche s form as a tool for the analysis of linear time invariant control systems a robustness criterion is given and its corresponding geometrical interpretation is obtained a simplified implementation of the symmetric form is obtained and its relation to passivity is highlighted an application to time delay systems is given

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 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

a criterion for estimating robust time delays for closed loop stability a formula for estimating robust time delays for closed loop stability is presented continuous single input single output linear time invariant systems represented by their rational transfer functions which are not necessarily stable are considered the proposed formula is derived using the symmetric rouche form an illustrative example is included and some closing remarks concerning the merits of the formula are made

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

a semiparametric method for estimating nonlinear autoregressive model with dependent errors abstract the first order nonlinear autoregressive model is considered and a semiparametric method is proposed to estimate regression function in the presented model dependent errors are defined as first order autoregressive ar 1 the conditional least squares method is used for parametric estimation and the nonparametric kernel approach is applied to estimate regression adjustment in this case some asymptotic behaviors and simulated results for the semiparametric method are presented furthermore the method is applied for the financial data in iran s tejarat bank

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 topology and dynamics of the hyperspaces of normal fuzzy sets and their inverse limit spaces given a compact and connected metric space continuum x we study topological and dynamical properties of the hyperspace of normal fuzzy sets f1 x equipped with the hausdorff endograph or sendograph metric among the many results we show that it is contractible path connected locally contractible locally path connected locally simply connected and locally connected for the endograph metric the hyperspace f1 x is a continuum and then for a topological graph x we show how using the inverse limit approach of barge and martin the inverse limit of a fuzzy dynamical system on x can be realized as an attractor of a fuzzy dynamical system on a manifold

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

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

the dynamics of continuous maps of finite graphs through inverse limits suppose that f g g is a continuous piecewise monotone function on a finite graph g then the following are equivalent i f has positive topological entropy ii there are disjoint intervals i1 and i2 and a positive integer n with i1 u i2 c fn ji n fn i2 iii the inverse limit space constructed by using f on g as a single bonding map contains an indecomposable subcontinuum this result generalizes known results for the interval and circle

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

topological conjugacy and transitivity for a class of piecewise monotone maps of the interval on dit qu une application continue f d un intervalle compact dans lui meme est markov lineaire s il est lineaire par morceaux et l ensemble de tous les f k x ou k 0 et x une extremite d un morceau lineaire est fini on donne une classification effective pour des applications de markov lineaires et une procedure effective pour determiner quand une telle application est transitive

consistent model specification tests for time series econometric models in this paper we consider general hypothesis testing problems for nonparametric and semiparametric time series econometric models we apply the general methodology to construct a consistent test for omitted variables and a consistent test for a partially linear model the proposed tests are shown to have asymptotic normal distributions under their respective null hypotheses we also discuss the problems of testing portfolio conditional mean variance efficiency and testing a semiparametric single index model monte carlo simulations are conducted to examine the finite sample performances of the nonparametric omitted variable test and the test for a partially linear specification

chaos periodicity and snakelike continua the results of this paper relate the dynamics of a continuous map f of the interval and the topology of the inverse limit space with bonding map f these inverse limit spaces have been studied by many authors and are examples of what bing has called snakelike continua roughly speaking we show that when the dynamics of f are complicated the inverse limit space contains indecomposable subcontinua we also establish a partial converse

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

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

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

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

estimating the robust dead time for closed loop stability a method based on the small gain theorem is provided for estimating the robust dead time for closed loop stability the method lends itself to generalization for multivariable systems and also to graphical implementation some examples are provided to demonstrate the use of the method and the results are compared to the exact critical values of dead time

sensitivity of set valued discrete systems abstract consider the surjective continuous map f x x and the continuous map f of k x into itself induced by f where x is a compact metric space and k x is the space of all non empty compact subsets of x endowed with a hausdorff metric in this paper we give examples showing that sensitivity of f does not imply sensitivity of f furthermore we prove that if f is a surjective continuous interval map then f is sensitive if and only if f is sensitive

a transitive homeomorphism on the pseudoarc which is semiconjugate to the tent map a powerful theorem and construction of wayne lewis are used to build two homeomorphisms on the pseudoarc each of which is semiconjugate to the tent map on the unit interval the first homeomorphism is transitive thus answering a question of marcy barge as to whether such homeomorphisms exist the second homeomorphism admits wandering points also it is proven that any homeomorphism on the pseudoarc that is semiconjugate to the tent map and is irreducible with respect to the semiconjugacy must either be transitive or admit wandering points a continuum is a compact connected metric space a continuum x is indecomposable if every proper subcontinuum of x is nowhere dense in x and x is hereditarily indecomposable if each subcontinuum of x is indecomposable a continuum x is chainable if for each e 0 there is a chain c c0 cn of open sets of diameter less than e that covers x c is a chain means that ci n ej 0 iff ii ii 2 or in the hilbert cube are pseudoarcs here we will show that this continuum admits a transitive homeomorphism thus answering a question of marcy barge in addition this homeomorphism has the property that it is semiconjugate to the tent map on the unit interval for more information on pseudoarcs and indecomposable continua see bil bi4 l1 l2 k km kr and ot exotic continua have been making their appearance in dynamical systems ever since the early part of this century with the work of c caratheodory c received by the editors may 20 1988 and in revised form july 25 1989 presented at the 1988 spring topology conference university of florida gainesville florida 32611 1980 mathematics subject classification 1985 revision primary 54f20 secondary 54h20

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

basins of wada abstract we describe situations in which there are several regions more than two with the wada property namely that each point that is on the boundary of one region is on the boundary of all we argue that such situations arise even in studies of the forced damped pendulum where it is possible to have three attractor regions coexisting and the three basins of attraction have the wada property

locally parametric nonparametric density estimation this paper develops a nonparametric density estimator with parametric overtones suppose f x is some family of densities indexed by a vector of parameters we define a local kernel smoothed likelihood function which for each x can be used to estimate the best local parametric approximant to the true density this leads to a new density estimator of the form f x o x thus inserting the best local parameter estimate for each new value of x when the bandwidth used is large this amounts to ordinary full likelihood parametric density estimation while for moderate and small bandwidths the method is essentially nonparametric using only local properties of data and the model alternative ways more general than via the local likelihood are also described the methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class properties of this new semiparametric estimator are investigated our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias the new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework the methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression

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

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

nonlinear stochastic trends abstract a nonlinear growth process has its change equal to a positive function of the lagged series plus an innovation whose variance depends on the lagged series if the innovations were identically zero the resulting deterministic trends take a wide variety of shapes the balance between growth in mean and variance to guarantee eventual growth of the process is determined the relationship between stochastic and deterministic trends found parametric and nonparametric estimates of the functions considered and applications analysed the natural bivariate generalization is found to produce a form of cointegration and a nonlinear error correction model

local polynomial estimators of the volatility function in nonparametric autoregression abstract in this paper we consider a class of dynamic models in which both the conditional mean and the conditional variance volatility are unknown functions of the past we first derive probabilistic conditions under which nonparametric estimation of these functions is possible we then construct an estimator based on local polynomial fitting we examine the rates of convergence of these estimators and give a result on their asymptotic normality the local polynomial fitting of the volatility function is applied to different foreign exchange rate series we find an asymmetric u shaped smiling face form of the volatility function

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

nonparametric regression with errors in variables the effect of errors in variables in nonparametric regression estimation is examined to account for errors in covariates deconvolution is involved in the construction of a new class of kernel estimators it is shown that optimal local and global rates of convergence of these kernel estimators can be characterized by the tail behavior of the characteristic function of the error distribution in fact there are two types of rates of convergence according to whether the error is ordinary smooth or super smooth it is also shown that these results hold uniformly over a class of joint distributions of the response and the covariate which is rich enough for many practical applications furthermore to achieve optimality we show that the convergence rates of all possible estimators have a lower bound possessed by the kernel estimators

on extended partially linear single index models aiming to explore the relation between the response y and the stochastic explanatory vector variable x beyond the linear approximation we consider the single index model which is a well known approach in multidimensional cases specifically we extend the partially linear single index model to take the form y 0 t x o 0 0 t x e where e is a random variable with ee 0 and var e 2 unknown 0 and o are unknown parametric vectors and o is an unknown real function the model is also applicable to nonlinear time series analysis in this paper we propose a procedure to estimate the model and prove some related asymptotic results both simulated and real data are used to illustrate the results

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 splitting theorem for transitive maps in this paper we prove a splitting theorem for continuous transitive maps on locally connected compact metric spaces which generalizes the results of barge and martin 4 for transitive maps on the interval and of blokh 7 for transitive graph maps as a consequence lower bounds for the topological entropy of transitive graph maps in terms of the cardinality of their splittings are obtained also we show that for every connected graph which is not a tree the infimum of the topological entropy of the transitive maps having a periodic point is zero since the topological entropy of transitive maps without periodic points is zero 7 and positive lower bounds of the entropy were given for the transitive maps on trees 2 this paper completes the problem of obtaining lower bounds of the topological entropy for transitive maps of any connected graph

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

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