chaotic functions with zero topological entropy on caracterise la classe m c i i des applications chaotiques en ce sens on montre que m contient certaines appliquations qui ont a la fois une entropie topologique nulle et des attracteurs infinis on montre que le complement de m consiste en applications qui ont seulement des trajectoires approchables par des cycles

distributional chaos revisited in their famous paper schweizer and smital introduced the definition of a distributionally chaotic pair and proved that the existence of such a pair implies positive topological entropy for continuous mappings of a compact interval further their approach was extended to the general compact metric space case in this article we provide an example which shows that the definition of distributional chaos and as a result li yorke chaos may be fulfilled by a dynamical system with intuitively regular dynamics embedded in 3 next we state strengthened versions of distributional chaos which as we show are present in systems commonly considered to have complex dynamics we also prove that any interval map with positive topological entropy contains two invariant subsets x y c i such that f x has positive topological entropy and f y displays distributional chaos of type 1 but not conversely

grading refinements in the contractions of lie algebras and their invariants all mutually inequivalent toroidal gradings of the lie algebra are constructed each of them provides several different graded contractions to other eight dimensional lie algebras these can be either continuous or discrete the continuous ones are related to generalized wigner inonu contractions the behaviour of casimir operators under the graded contractions is studied and the casimir and generalized casimir operators of the contracted lie algebras are presented

classification of solvable lie algebras with a given nilradical by means of solvable extensions of its subalgebras abstract we construct all solvable lie algebras with a specific n dimensional nilradical n n 3 which contains the previously studied filiform n 2 dimensional nilpotent algebra n n 2 1 as a subalgebra but not as an ideal rather surprisingly it turns out that the classification of such solvable algebras can be deduced from the classification of solvable algebras with the nilradical n n 2 1 also the sets of invariants of coadjoint representation of n n 3 and its solvable extensions are deduced from this reduction in several cases they have polynomial bases i e the invariants of the respective solvable algebra can be chosen to be casimir invariants in its enveloping algebra

constant length substitution systems and countable distributionally scrambled sets abstract in this paper we study the compact dynamical systems which are on the edge of distributional chaos that is whose distributionally scrambled sets are finite or countable first we show that a constant length substitution system without eventually periodic substitution sequence has only finite distributionally scrambled sets then we give some constant length substitution systems whose distributionally scrambled sets may have any given finite cardinal number at last we provide a compact dynamical system generated by some constant length substitution systems whose distributionally scrambled set has at most countably many elements

distributional chaos for operators on banach spaces abstract four notions of distributional chaos namely dc1 dc2 dc 2 1 2 and dc3 are studied within the framework of operators on banach spaces it is known that for general dynamical systems dc1 dc2 dc 2 1 2 dc3 we show that dc1 and dc2 coincide in our context which answers a natural question in contrast there exist dc 2 1 2 operators which are not dc2 under the condition that there exists a dense set x 0 x such that t n x 0 for any x x 0 dc3 operators are shown to be dc1 moreover we prove that any upper frequently hypercyclic operator is dc 2 1 2 finally several examples are provided to distinguish between different notions of distributional chaos li yorke chaos and irregularity

distributional chaos for linear operators abstract we characterize distributional chaos for linear operators on frechet spaces in terms of a computable condition dcc and also as the existence of distributionally irregular vectors a sufficient condition for the existence of dense uniformly distributionally irregular manifolds is presented which is very general and can be applied to many classes of operators distributional chaos is also analyzed in connection with frequent hypercyclicity and the particular cases of weighted shifts and composition operators are given as an illustration of the previous results

a note on distributional chaos with respect to a sequence abstract the aim of this note is to use methods developed by kuratowski and mycielski to prove that some more common notions in topological dynamics imply distributional chaos with respect to a sequence in particular we show that the notion of distributional chaos with respect to a sequence is only slightly stronger than the definition of chaos due to li and yorke namely positive topological entropy and weak mixing both imply distributional chaos with respect to a sequence which is not the case for distributional chaos as introduced by schweizer and smital

a note on the map with the whole space being a scrambled set abstract in this paper we prove that there is no compact system with the whole space being a schweizer smital scrambled set and the huang ye homeomorphism w huang x d ye homeomorphisms with the whole compacta being scrambled sets ergod theory dynam systems 21 2001 77 91 has no schweizer smital pair moreover we present a simple noncompact system whose schweizer smital scrambled set can be the whole space

a differential operator realisation approach for constructing casimir operators of non semisimple lie algebras we introduce a search algorithm that utilises differential operator realisations to find polynomial casimir operators of lie algebras to demonstrate the algorithm we look at two classes of examples 1 the model filiform lie algebras and 2 the schrodinger lie algebras we find that an abstract form of dimensional analysis assists us in our algorithm and greatly reduces the complexity of the problem

distributionally chaotic systems of type 2 and rigidity abstract in this paper we deal with uniformly rigid systems obtained by a method introduced by katznelson and weiss and show that such systems never contain dc2 pairs on the other hand we introduce a modification of this technique that leads to a uniformly rigid system with dc2 pairs we also show that every dynamical system contains a pair of distinct points which is not dc2

chaos on hyperspaces abstract let f be a continuous self map of a compact metric space x the transformation f induces in a natural way a self map f defined on the hyperspace k x of all nonempty closed subsets of x we study which of the most usual notions of chaos for dynamical systems induced by f are inherited by f and vice versa we consider distributional chaos li yorke chaos chaos devaney chaos topological chaos positive topological entropy specification property and their variants this answers questions stated independently by roman flores and banks

the structure of the invariants of perfect lie algebras the structure of the invariants of perfect lie algebras with nontrivial centre is analysed it is shown that if the radical of a semidirect sum of a semisimple and a nilpotent lie algebra has a one dimensional centre then the defining representation r contains a copy of the trivial representation d0 of using this fact a criterion can be deduced to eliminate variables and to characterize the casimir operators of an algebra by means of certain subrepresentations r of r for rank 1 levi subalgebras all representations leading to perfect lie algebras with a radical isomorphic to a heisenberg lie algebra are determined we prove that the number of casimir operators of such an algebra is fixed for any dimension and moreover that they can be explicitly computed by means of determinantal formulae obtained from the brackets of the algebra for rank n 2 levi part a stabilization result of this nature is also given

isolating segments fixed point index and symbolic dynamics iii applications abstract we prove the existence of complicated dynamics in some class of periodic systems on the plane the proof is based on the continuation of the considered system to the model one

invariants of triangular lie algebras triangular lie algebras are the lie algebras which can be faithfully represented by triangular matrices of any finite size over the real complex number field in the paper invariants generalized casimir operators are found for three classes of lie algebras namely those which are either strictly or non strictly triangular and for so called special upper triangular lie algebras algebraic algorithm of boyko et al 2006 j phys a math gen 39 5749 preprint math ph 0602046 developed further in boyko et al 2007 j phys a math theor 40 113 preprint math ph 0606045 is used to determine the invariants a conjecture of tremblay and winternitz 2001 j phys a math gen 34 9085 concerning the number of independent invariants and their form is corroborated

li yorke and distributionally chaotic operators abstract we study li yorke chaos and distributional chaos for operators on banach spaces more precisely we characterize li yorke chaos in terms of the existence of irregular vectors sufficient computable criteria for distributional and li yorke chaos are given together with the existence of dense scrambled sets under some additional conditions we also obtain certain spectral properties finally we show that every infinite dimensional separable banach space admits a distributionally chaotic operator which is also hypercyclic

solvable lie algebras with heisenberg ideals all finite dimensional indecomposable solvable lie algebras l n f having the heisenberg algebra h n as the nilradical are constructed the number of non nilpotent elements f that can be added to h n satisfies f or n 1 the casimir and generalized casimir operators of the algebras l n f are obtained

chaotic sets of shift and weighted shift maps abstract in this paper li yorke chaotic sets generated by shift and weighted shift maps are studied the characterization of li yorke chaotic sets by p scrambled sets maximal scrambled sets and orbit invariants are proved for the general shift maps for weighted shift maps on infinite dimensional spaces the necessary and sufficient conditions for li yorke chaotic are proved both in the abstract sequence setting and in the eigenfunction setting besides a constructive proof is provided for the devaney chaos of weighted shift maps on the schwartz space

distributional chaos caused by snap back repeller in metric space abstract in this paper we extend the conception of snap back repeller of marotto s to general metric space and prove that a continuous map with snap back repeller is distributionally chaotic

dynamic parrondo s paradox abstract this paper is focused on studying parrondo s paradox in non linear dynamics specifically how the periodic combination of the individual maps f and g can give rise to chaos or order we construct dynamical systems exhibiting the paradox for several notions of chaos derived from topological dynamics the effect of altering the order of the combination i e considering f g or g f is analyzed as well as the robustness of the parrondo effect under small perturbations conditions for avoiding parrondian dynamics are also obtained placing special emphasis on the notion of chaos given by turbulence

stability of asymptotically stable sets for continuous functions let k be the class of nonempty compact subsets of i 0 1 and k consist of the nonempty closed subsets of k we study the map a c i i k defined so that a f is the closure of the collection of asymptotically stable sets of f while we find that a is not continuous we do get much more positive results when studying the semicontinuity of a in particular a is lower semicontinuous at a residual set of functions in c i i so that a is continuous if we restrict our map to a residual subset of c i i

isolating segments fixed point index and symbolic dynamics abstract an extension of the recently introduced srzednicki wojcik method for detecting chaotic dynamics in periodically forced ordinary differential equations is presented as an application of the method we construct a topological model for the planar equation z 1 e i t z 2 z z c 1 and we show by a continuation argument that the symbolic dynamics on three symbols for the topological model continues to eq 1 for 0

distributional chaos in a sequence abstract in this paper we prove a sufficient condition for the continuous map of a compact metric space for being distributively chaotic in a sequence as an application it is proved that a continuous map of an interval is chaotic in the li yorke sense if and only if it is distributively chaotic in a sequence

chaotic sets of continuous and discontinuous maps abstract this paper discusses li yorke chaotic sets of continuous and discontinuous maps with particular emphasis to shift and subshift maps scrambled sets and maximal scrambled sets are introduced to characterize li yorke chaotic sets the orbit invariant for a scrambled set is discussed some properties about maximality equivalence and uniqueness of maximal scrambled sets are also discussed it is shown that for shift maps the set of all scrambled pairs has full measure and chaotic sets of some discontinuous maps such as the gauss map interval exchange transformations and a class of planar piecewise isometries are studied finally some open problems on scrambled sets are listed and remarked

realizations of real low dimensional lie algebras using a new powerful technique based on the notion of megaideal we construct a complete set of inequivalent realizations of real lie algebras of dimension no greater than four in vector fields on a space of an arbitrary finite number of variables our classification amends and essentially generalizes earlier works on the subject

distributional chaos for backward shifts we provide sufficient conditions which give uniform distributional chaos for backward shift operators we also compare distributional chaos with other well studied notions of chaos for linear operators like devaney chaos and hypercyclicity and show that devaney chaos implies uniform distributional chaos for weighted backward shifts but there are examples of backward shifts which are uniformly distributionally chaotic and not hypercyclic

invariants of lie algebras with fixed structure of nilradicals an algebraic algorithm is developed for computation of invariants generalized casimir operators of general lie algebras over the real or complex number field its main tools are the cartan s method of moving frames and the knowledge of the group of inner automorphisms of each lie algebra unlike the first application of the algorithm in boyko et al 2006 j phys a math gen 39 5749 preprint math ph 0602046 which deals with low dimensional lie algebras here the effectiveness of the algorithm is demonstrated by its application to computation of invariants of solvable lie algebras of general dimension n restricted only by a required structure of the nilradical specifically invariants are calculated here for families of real complex solvable lie algebras these families contain with only a few exceptions all the solvable lie algebras of specific dimensions for whom the invariants are found in the literature

all solvable extensions of a class of nilpotent lie algebras of dimension n and degree of nilpotency n 1 we construct all solvable lie algebras with a specific n dimensional nilradical of degree of nilpotency n 1 and with an n 2 dimensional maximal abelian ideal we find that for given n such a solvable algebra is unique up to isomorphisms using the method of moving frames we construct a basis for the casimir invariants of the nilradical we also construct a basis for the generalized casimir invariants of its solvable extension consisting entirely of rational functions of the chosen invariants of the nilradical

distributional chaos for operators with full scrambled sets in this article we answer in the negative the question of whether hypercyclicity is sufficient for distributional chaos for a continuous linear operator we even prove that the mixing property does not suffice moreover we show that an extremal situation is possible there are hypercyclic and non hypercyclic operators such that the whole space consists except zero of distributionally irregular vectors

recurrence and topological entropy of translation operators abstract in this paper we provide several characterizations on weak recurrence of translation operators on weighted lebesgue spaces and on continuous function spaces it is shown that the translation operator admits a nonzero nonwandering point if and only if it has a nonzero recurrent point if and only if it is hypercyclic that the translation operator admits a nonzero almost periodic point if and only if it is devaney chaotic moreover every hypercyclic translation operator has infinite topological entropy but the converse is not true finally we remark that there exists a linear operator which has the specification property on the whole space

invariant measures for frequently hypercyclic operators abstract we investigate frequently hypercyclic and chaotic linear operators from a measure theoretic point of view among other things we show that any frequently hypercyclic operator t acting on a reflexive banach space admits an invariant probability measure with full support which may be required to vanish on the set of all periodic vectors for t that there exist frequently hypercyclic operators on the sequence space c 0 admitting no ergodic measure with full support and that if an operator admits an ergodic measure with full support then it has a comeager set of distributionally irregular vectors we also give some necessary and sufficient conditions which are satisfied by all the known chaotic operators for an operator t to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures finally we give a baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support

the fundamental invariants of inhomogeneous classical groups the casimir operators of the following groups are explicitly constructed rn gl n r rn sl n r rn o p q cn u p q r2n sp n r cn gl n c cn sl n c cn o n c c2n sp n c hn u 2n hn su 2n hn o 2n hn sp p q and r r2n sp n r the method is based on a particular fiber bundle structure of the generic orbits generated by the co adjoint representation of a semidirect product

computation of invariants of lie algebras by means of moving frames a new purely algebraic algorithm is presented for computation of invariants generalized casimir operators of lie algebras it uses the cartan method of moving frames and the knowledge of the group of inner automorphisms of each lie algebra the algorithm is applied in particular to computation of invariants of real low dimensional lie algebras a number of examples are calculated to illustrate its effectiveness and to make a comparison with the same cases in the literature bases of invariants of the real six dimensional solvable lie algebras with four dimensional nilradicals are newly calculated and listed in a table

linear operators with infinite entropy abstract we examine the chaotic behavior of certain continuous linear operators on infinite dimensional banach spaces and provide several equivalent characterizations of when these operators have infinite topological entropy for example it is shown that infinite topological entropy is equivalent to non zero topological entropy for translation operators on weighted lebesgue function spaces in particular finite non zero entropy is impossible for this class of operators which answers a question raised by yin and wei

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

distributional chaos for the forward and backward control traffic model the interest in car following models has increased in the last years due to its connection with vehicle to vehicle communications and the development of driverless cars some non linear models such as the gazes herman rothery model were already known to be chaotic we consider the linear forward and backward control traffic model for an infinite number of cars on a track we show the existence of solutions with a chaotic behaviour by using some results of linear dynamics of c0 semigroups in contrast we also analyse which initial configurations lead to stable solutions

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

solvable lie algebras with naturally graded nilradicals and their invariants the indecomposable solvable lie algebras with graded nilradical of maximal nilindex and a heisenberg subalgebra of codimension one are analysed and their generalized casimir invariants are calculated it is shown that rank one solvable algebras have a contact form which implies the existence of an associated dynamical system moreover due to the structure of the quadratic casimir operator of the nilradical these algebras contain a maximal non abelian quasi classical lie algebra of dimension 2n 1 indicating that gauge theories with ghosts are possible on these subalgebras

minimal presentations for irreducible sofic shifts we re cast a theorem of willems 1989 in terms of symbolic dynamics we then present a new result which characterizes when an irreducible sofic shift i e constrained system has a unique minimal irreducible presentation

distributionally scrambled invariant sets in a compact metric space abstract the paper solves a question posed by oprocha on the existence of invariant distributionally chaotic scrambled sets we show among other things that a continuous map f acting on compact metric space x d with a weak specification property fixed point and infinitely many mutually distinct periods has a dense mycielski i e c dense set of type f invariant distributionally scrambled set as a consequence we describe a class of maps with a distributionally scrambled invariant set of full lebesgue measure in the case when x is a k dimensional cube

the weak specification property and distributional chaos abstract in this paper we first discuss almost periodic points in a compact dynamical system with the weak specification property on the basis of this discussion we draw two conclusions i the weak specification property implies a dense mycielski uniform distributionally scrambled set ii the weak specification property and a fixed point imply a dense mycielski uniform invariant distributionally scrambled set these conclusions improve on some of the latest results concerning the specification property and give a final positive answer to an open problem posed in p oprocha invariant scrambled sets and distributional chaos dyn syst 24 2009 31 43

foundations of intrinsic habit formation we provide theoretical foundations for several common nested representations of intrinsic linear habit formation our axiomatization introduces an intertemporal theory of weaning a decision maker from her habits using the device of compensation we clarify differences across specifications of the model provide measures of habit forming tendencies and suggest methods for axiomatizing time nonseparable preferences

a function for which the limit points are not contained in the closure of the periodic points we develop a c 1 function f 1 6 1 1 6 1 for which f p f this answers a query from block and coppel 1992

some chaotic and mixing properties of fuzzified dynamical systems abstract let x be a compact metric space let be a continuous self map on x and let f x denote the space of fuzzy sets on x equipped with the levelwise topology in this paper we study relations between a given crisp dynamical system x and its g fuzzification f x g among other things we study various weak strong mild etc mixing properties and also several kinds of chaotic behaviors li yorke chaos chaos distributional chaos topological chaos etc we specified subspaces of fuzzy sets which makes sense to study further additionally we proved that numerous local characteristics of the original dynamical system can be found in its fuzzy extension while many global characteristics are preserved in the opposite direction thus among other things we obtained easy to check criteria for checking absence of complex behavior in induced fuzzy dynamical systems

sequentially continuous non monotonic choquet integrals schmeidler proc amer math soc 97 2 1986 255 261 established an integral representation theorem through the choquet integral for functionals satisfying monotonicity and a weaker condition than additivity namely comonotonic additivity murofushi sugeno machida fuzzy sets and systems 64 1994 73 86 generalize this representation to the case of bounded variation functionals omitting the monotonicity condition we give an alternative approach which is based on sequential continuity and tolerates non monotonicity this later condition is equivalent to s additivity in measure theory

solvable lie algebras with an mathbb n graded nilradical of maximal nilpotency degree and their invariants the class of solvable lie algebras with an graded nilradical of maximal nilpotency index is classified it is shown that such solvable extensions are unique up to isomorphism the generalized casimir invariants for the graded nilradicals and their associated solvable extensions are computed by the method of moving frames

solvable lie algebras with abelian nilradicals a procedure is presented for classifying solvable lie algebras with abelian nilradicals theorems on the structure of such algebras are proven and their centres are constructed many examples are analysed

measures of chaos and a spectral decomposition of dynamical systems on the interval let f 0 1 0 1 be continuous for x y e 0 1 the upper and lower distance distribution functions fx y and fxy are defined for any t 0 as the lim sup and lim inf as n oc of the average number of times that the distance ifi x fi y i between the trajectories of x and y is less than t during the first n iterations the spectrum of f is the system f of lower distribution functions which is characterized by the following properties 1 the elements of f are mutually incomparable 2 for any f e f there is a perfect set pf 8 0 such that fuv f and fuv 1 for any distinct u v e pf 3 if s is a scrambled set for f then there are f g in f and a decomposition s sf u sg sg may be empty such that fuv f if u v e sf and fuv g if u v e sg our principal results are 1 if f has positive topological entropy then f is nonempty and finite and any f e f is zero on an interval 0 e where e 0 and hence any pf is a scrambled set in the sense of li and yorke 2 if f has zero topological entropy then f f where f 1 it follows that the spectrum of f provides a measure of the degree of chaos of f in addition a useful numerical measure is the largest of the numbers fo 1 f t dt where f e f

a note on schweizer smital chaos abstract in this paper we consider a continuous map f x x where x is a compact metric space and prove that for any positive integer n f is schweizer smital chaotic if and only if f n is too

generalized casimir operators of solvable lie algebras with abelian nilradicals a solvable complex lie algebra l of dimension n with an abelian nilradical of dimension r is shown to have precisely 2r n generalized casimir invariants we always have r or n 2 they are constructed as invariants of the coadjoint representation of l and depend only on variables dual to elements of the nilradical their form in general involves logarithms of these variables in addition to rational and irrational functions they give rise to genuine casimir operators whenever they happen to be polynomials

a chaotic function with a distributively scrambled set of full lebesgue measure abstract misiurewicz proved that there exists a continuous map of the interval 0 1 onto itself for which there exists a scrambled set of full lebesgue measure in this paper we form a continuous interval map which has a distributively scrambled set of full lebesgue measure in which each point has dense orbit this contains misiurewicz s result since any distributively scrambled set must be scrambled but the converse is not generally true

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

invariants of solvable lie algebras of dimension six all invariant functions of the coadjoint representation are determined for real solvable lie algebras of dimension six having a nilradical of dimension four many questions related to the nature of these invariants are analysed in particular we derive the general type of functions in terms of which all the invariants can be expressed and give the characterization of all solvable lie algebras with abelian nilradical that have no non trivial invariants the existence of casimir operators among these invariants is also investigated

distributional chaos in dendritic and circular julia sets abstract if x and y belong to a metric space x we call x y a dc1 scrambled pair for f x x if the following conditions hold 1 for all t 0 lim sup n 1 n 0 i n d f i x f i y t 1 and 2 for some t 0 lim inf n 1 n 0 i n d f i x f i y t 0 if d x is an uncountable set such that every x y d form a dc1 scrambled pair for f we say f exhibits distributional chaos of type 1 if there exists t 0 such that condition 2 holds for any distinct points x y d then the chaos is said to be uniform a dendrite is a locally connected uniquely arcwise connected compact metric space in this paper we show that a certain family of quadratic julia sets one that contains all the quadratic julia sets which are dendrites and many others which contain circles has uniform dc1 chaos

a characterization of nonchaotic continuous maps of the interval stable under small perturbations recent results of the second author show that every continuous map of the interval to itself either has every trajectory approximable by cycles some times this is possible even in the case when the trajectory is not asymptotically periodic or is e chaotic for some e 0 in certain cases the first property is stable under small perturbations this means that a perturbed map can be chaotic but the chaos must be small whenever the perturbation is small in other words there are nonchaotic maps without chaos explosions in the paper we give a characterization of these maps along with some consequences using the main result it is possible to prove that generically the nonchaotic maps are stable asymptotic properties of trajectories have been intensively considered since the middle of the sixties when excellent papers by a n sarkovskii appeared the main reason is applicability of these results in mathematical modelling important are maps with trajectories having regular asymptotical behavior like asymptotical periodicity however this property can be easily destroyed by ar bitrarily small perturbations of the maps and the corresponding unstable models in connection with this there appeared some results guaranteeing the stability in 15 it is shown that every map having no periodic orbits of period dif ferent from 1 2 2 for some n with per nowhere dense is stable this result was generalized 16 to the maps with per nowhere dense which have all trajectories asymptotically periodic these maps can have cycles of period 2 for every n but no other cycles and are characterized e g by the property that per is closed 12 cf also 10

devaney s chaos implies distributional chaos in a sequence abstract let x be a complete metric space without isolated points and let f x x be a continuous map in this paper we prove that if f is transitive and has a periodic point of period p then f is distributionally chaotic in a sequence particularly chaos in the sense of devaney is stronger than distributional chaos in a sequence

operators with the specification property abstract we study a version of the specification property for linear dynamics operators having the specification property are investigated and relationships with other well known dynamical notions such as mixing devaney chaos and frequent hypercyclicity are obtained

dynamical systems over the space of upper semicontinuous fuzzy sets given a locally compact separable metric space e a perfect transformation defined on e induces a zadeh extension which is a transformation from the space of all upper semicontinuous fuzzy sets defined on e to itself here the latter space is equipped with the hit or miss topology in this setting each upper semicontinuous fuzzy set is identified with its hypograph a closed subset in the product space of e and 0 1 this approach does not require the relevant fuzzy sets to hold a compact support and it also overcomes the drawback of the traditional level set method further it is proved that the zadeh extension with this setting is continuous in the hit or miss topology and dynamical properties of zadeh extension regarding iteration have been explored

on open problems concerning distributional chaos for triangular maps abstract we show that in the class t of the triangular maps x y f x g x y of the square there is a map of type 2 with non minimal recurrent points which is not dc3 we also show that every dc1 continuous map of a compact metric space has a trajectory which cannot be weakly approximated by trajectories of compact periodic sets these two results make possible to answer some open questions concerning classification of maps in t with zero topological entropy and contribute to an old problem formulated by a n sharkovsky

mean li yorke chaos for random dynamical systems abstract in this paper we studied the complexity of some random dynamical systems with positive topological entropy the random dynamical systems are usually generated by stochastic partial differential equations spdes and contain randomness in many ways we proved that there exists mean li yorke chaotic phenomenon in some random dynamical systems with positive entropy

distributional chaos on compact metric spaces via specification properties abstract in this paper we show that a continuous function on a compact metric space exhibits distributional chaos as introduced in b schweizer and j smital trans amer math soc 344 1994 737 754 and elucidated in b schweizer a sklar and j smital to appear if the function has either a weaker form of the specification property see m denker c grillenberger and k sigmund springer lecture notes in mathematics vol 527 springer verlag new york heidelberg berlin 1976 or the generalized specification property introduced in f balibrea b schweizer a sklar and j smital to appear in particular any anosov diffeomorphism is distributionally chaotic regardless of the fact that in this case the trajectories of a e pair of points exhibit regular non chaotic behavior

invariants of solvable lie algebras with triangular nilradicals and diagonal nilindependent elements the invariants of solvable lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on cartan s method of moving frames

solvable lie algebras with triangular nilradicals all finite dimensional indecomposable solvable lie algebras having the triangular algebra t n as their nilradical are constructed the number of non nilpotent elements f in satisfies and the dimension of the lie algebra is

factor maps and invariant distributional chaos abstract the main aim of this article is to show that maps with the specification property have invariant distributionally scrambled sets and that this kind of scrambled set can be transferred from factor to extension under finite to one factor maps this solves some open questions in the literature of the topic we also show how our method can be applied in practice taking as example poincare map of time periodic nonautonomous planar differential equation z v t z 1 e i t z 2 z 3 n where and n are sufficiently small positive real numbers

on the invariants of some solvable rigid lie algebras we determine fundamental systems of invariants for complex solvable rigid lie algebras having nonsplit nilradicals of characteristic sequence 3 1 1 these algebras being the natural followers of solvable algebras having heisenberg nilradicals a special case of this allows us to obtain a criterion to determine the number of functionally independent invariants of rank one subalgebras of real or complex solvable lie algebras finally we give examples of the inverse procedure obtaining fundamental systems of an algebra starting from rank one subalgebras and a criterion for the nonexistence of nontrivial invariants

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

expectation and variation in multi period decisions multi period decisions are decisions which determine an individual s payoffs in several periods in the future this paper examines the theoretical foundations of the prevalent weighted average assumption more specifically we use a multi period interpretation of the famous ellsberg paradox in decision under uncertainty to show that in many cases of interest additively separable functionals in general and weighted average ones in particular do not seem appropriate for the representation of the decision maker s preferences we then suggest replacing the sure thing principle which may be used to axiomatize a weighted average functional by a weaker version of it using the weakened axiom in schmeidler s nonadditive measure model reinterpreted for the multi period context yields an axiomatization of a larger class of decision rules which are representable by a weighted average of the utility in each period und the utility variation between each two consecutive periods the weighted average assumption is a special case of the generalized model a case in which the decision maker is variation neutral similarly we define and characterize variation aversion and variation liking and show an example of the economic implications of these properties

chaotic semigroups from second order partial differential equations abstract we give general conditions on given parameters to ensure devaney and distributional chaos for the solution c 0 semigroup corresponding to a class of second order partial differential equations we also provide a critical parameter that led us to distinguish between stability and chaos for these semigroups in the case of chaos we prove that the c 0 semigroup admits a strongly mixing measure with full support we also give concrete examples of partial differential equations such as the telegraph equation whose solutions satisfy these properties

time continuity and nonadditive expected utility information consisting of probabilities of some but possibly not all events induces an integral with respect to a probability specified on a subalgebra a decision maker evaluates the alternatives using only the available information and completely ignoring unavailable information assume now that the decision maker assesses the worth of a different lottery at each point in a discrete time assume also that each such lottery is preferred to some fixed alternative lottery now consider the situation where the sequence of lotteries converges in some sense if the limiting lottery is preferred to the fixed alternative then the preference order is referred to as time continuous this paper studies time continuity for two preference functionals the choquet integral and the integral with respect to a probability specified on a subalgebra the integral with respect to probability specified on a subalgebra is determined by the structure of the available information by relating it to the choquet integral we characterize the structure of available information that would yield time continuity

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

the principal measure of a quantum harmonic oscillator at the end of his paper oprocha 2006 j phys a math gen 39 14 559 65 oprocha said that it is difficult and interesting to know the exact value of the principal measure of an annihilation operator of the unforced quantum harmonic oscillator motivated by this in this paper we prove that this exact value is equal to 1 besides we point out that exhibits distributional chaos for any 0 2

invariants of the nilpotent and solvable triangular lie algebras invariants of the coadjoint representation of two classes of lie algebras are calculated the first class consists of the nilpotent lie algebras t m isomorphic to the algebras of upper triangular m m matrices the lie algebra t m is shown to have m 2 functionally independent invariants they can all be chosen to be polynomials and they are presented explicitly the second class consists of the solvable lie algebras l m f with t m as their nilradical and f additional linearly nilindependent elements some general results on the invariants of l m f are given and the cases m 4 for all f and f 1 or m 1 for all m are treated in detail

invariants of solvable rigid lie algebras up to dimension 8 the invariants of all complex solvable rigid lie algebras up to dimension 8 are computed moreover we show for rank 1 solvable algebras some criteria to deduce the non existence of nontrivial invariants or the existence of fundamental sets of invariants formed by rational functions of the casimir invariants of the associated nilradical

a class of solvable lie algebras and their casimir invariants a nilpotent lie algebra with an n 1 dimensional abelian ideal is studied all indecomposable solvable lie algebras with as their nilradical are obtained their dimension is at most n 2 the generalized casimir invariants of and of its solvable extensions are calculated for n 4 these algebras figure in the petrov classification of einstein spaces for larger values of n they can be used in a more general classification of riemannian manifolds

invariants of triangular lie algebras with one nil independent diagonal element the invariants of solvable triangular lie algebras with one nil independent diagonal element are studied exhaustively bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on cartan s method of moving frames and the special technique developed for triangular and closed algebras in boyko et al j phys a math theor 2007 40 7557 the conjecture of tremblay and winternitz j phys a math gen 2001 34 9085 on the number and form of elements in the bases is completed and proved