topological stability for conservative systems we prove that the c1 interior of the set of all topologically stable c1 incompressible flows is contained in the set of anosov incompressible flows moreover we obtain an analogous result for the discrete time case

on growth rates of subadditive functions for semiflows abstract let x t x be a semiflow on a compact metric space x a function f x t x is subadditive with respect to if f x t s f x t f x t nbsp s we define the maximal growth rate of f to be sup x x lim sup t 1 t f x t this growth rate is shown to equal the maximal growth rate of the subadditive function restricted to the minimal center of attraction of the semiflow applications to birkhoff sums characteristic exponents of linear skew product semiflows on banach bundles and average lyapunov functions are developed in particular a relationship between the dynamical spectrum and the measurable spectrum of a linear skew product flow established by r a johnson k j palmer and g r sell siam j math anal 18 1987 1 33 is extended to semiflows in an infinite dimensional setting

homoclinic tangencies and hyperbolicity for surface diffeomorphisms we prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms the ones exhibiting a homoclinic tangency are c 1 dense this represents a step towards the global understanding of dynamics of surface diffeomorphisms

singular robustly chain transitive sets are singular volume partial hyperbolic for diffeomorphisms or for non singular flows there are many results relating properties persistent under mathcal c 1 perturbations and a global structures for the dynamics such as hyperbolicity partial hyperbolicity dominated splitting however a difficulty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point in bonatti star flows and multisingular hyperbolicity arxiv 1705 05799 2017 with christan bonatti we propose a a general procedure for adapting the usual hyperbolic structures to the singularities using this tool we recover the results in bonatti et al ann math 158 2 355 418 2003 for flows showing that robustly chain transitive sets have a weak form of hyperbolicity allowing us to conclude as well the kind of hyperbolicity carried by the examples in bonatti et al j inst math jussieu 12 3 449 501 2013 a robust chain transitive singular attractor with periodic orbits of different indexes along with the results in 8 this shows that the way we propose to interpret the effect of singularities has the potential to adapt to other settings in which there is coexistence of singularities and regular orbits with the goal of re obtaining the results that we already know for diffeomorphisms

a new class of generalized set valued implicit variational inclusions in banach spaces with an application in this paper we introduce and study a new class of generalized set valued implicit variational inclusions in real banach spaces by using nadler s theorem and the resolvent operator technique for m accretive mapping in real banach spaces we construct some new iterative algorithms for solving this class of generalized set valued implicit variational inclusions we prove the existence of solution for this kind of generalized set valued implicit variational inclusions without compactness and the convergence of iterative sequences generated by the algorithms in banach spaces we also give an application to generalized set valued implicit variational inequalities in real hilbert spaces

robust permanence for interacting structured populations abstract the dynamics of interacting structured populations can be modeled by d x i d t a i x x i where x i r n i x x 1 x k and a i x are matrices with non negative off diagonal entries these models are permanent if there exists a positive global attractor and are robustly permanent if they remain permanent following perturbations of a i x necessary and sufficient conditions for robust permanence are derived using dominant lyapunov exponents i of the a i x with respect to invariant measures the necessary condition requires max i i 0 for all ergodic measures with support in the boundary of the non negative cone the sufficient condition requires that the boundary admits a morse decomposition such that max i i 0 for all invariant measures supported by a component of the morse decomposition when the morse components are axiom a uniquely ergodic or support all but one population the necessary and sufficient conditions are equivalent applications to spatial ecology epidemiology and gene networks are given

spectral decomposition and ω stability of flows with expanding measures abstract in this paper we present a measurable version of the classical spectral decomposition theorem for flows more precisely we prove that if a flow on a compact metric space x is invariantly measure expanding on its chain recurrent set c r and has the invariantly measure shadowing property on c r then has the spectral decomposition i e the nonwandering set is decomposed by a disjoint union of finitely many invariant and closed subsets on which is topologically transitive moreover we show that if is invariantly measure expanding on c r then it is invariantly measure expanding on x using this we characterize the measure expanding flows on a compact c manifold via the notion of stability

lipschitz shadowing and structural stability of flows abstract let be the flow generated by a smooth vector field x on a smooth closed manifold we show that the lipschitz shadowing property of is equivalent to the structural stability of x and that the lipschitz periodic shadowing property of is equivalent to the stability of x

direct sums of rickart modules abstract the notion of rickart modules was defined recently it has been shown that a direct sum of rickart modules is not a rickart module in general in this paper we investigate the question when are the direct sums of rickart modules also rickart we show that if m i is m j injective for all i j i 1 2 n then i 1 n m i is a rickart module if and only if m i is m j rickart for all i j i as a consequence we obtain that for a nonsingular extending module m e m m is always a rickart module other characterizations for direct sums to be rickart under certain assumptions are provided we also investigate when certain classes of free modules over a ring r are rickart it is shown that every finitely generated free r module is rickart precisely when r is a right semihereditary ring as an application we show that a commutative domain r is prufer if and only if the free r module r 2 is rickart we exhibit an example of a module m for which m 2 is rickart but m 3 is not so further von neumann regular rings are characterized in terms of rickart modules it is shown that the class of rings r for which every finitely cogenerated right r module is rickart is precisely that of right v rings examples which delineate the concepts and the results are provided

the star systems x and a proof of the c1ω stability conjecture for flows abstract in this paper we give a new proof for the c 1 stability conjecture for flows our approach is considerably different from that of hayashi or wen and a good deal of liao s obstruction sets theory is used

various expansive measures for flows abstract we discuss a characterization of countably expansive flows in measure theoretical terms as in the discrete case 2 more precisely we define the countably expansive flows and prove that a homeomorphism of a compact metric space is countable expansive just when its suspension flow is moreover we exhibit a measure expansive flow in the sense of 4 which is not countably expansive next we define the weak expansive measures for flows and prove that a flow of a compact metric space is countable expansive if and only if it is weak measure expansive i e every orbit vanishing measure is weak expansive furthermore unlike the measure expansive ones the weak measure expansive flows may exist on closed surfaces finally it is shown that the integrated flow of a c 1 vector field on a compact smooth manifold is c 1 stably expansive if and only if it is c 1 stably weak measure expansive

on the finiteness of uniform sinks abstract we study the finiteness of uniform sinks for flows precisely we prove that for 0 and t 0 if a vector field x has only hyperbolic singularities or sectionally dissipative singularities then x can have only finitely many t uniform sinks this is a generalized version of a theorem of liao 4

on direct sums of baer modules abstract the notion of baer modules was defined recently since a direct sum of baer modules is not a baer module in general an open question is to find necessary and sufficient conditions for such a direct sum to be baer in this paper we study rings for which every free module is baer it is shown that this is precisely the class of semiprimary hereditary rings we also prove that every finite rank free r module is baer if and only if r is right semihereditary left coherent necessary and sufficient conditions for finite direct sums of copies of a baer module to be baer are obtained for the case when m is retractable an example of a module m is exhibited for which m n is baer but m n 1 is not baer other results on direct sums of baer modules to be baer under some additional conditions are obtained some applications are also included

stability properties of one parameter flows this thesis is in two parts the first part consists of chapters 1 4 and the second part is chapter 5 in the first part we consider the idea of approximating pieces of orbits by a single orbit there are many examples of such properties approximation property a p see chapter one definitions 1 2 1 4 specification property s p 24 pseudo orbit tracing property p o t p 25 and definition 4 1 in chapter one we show that a p for a homeomorphism flow is equivalent to topological transitivity and density of periodic points and how this property a p is invariant under topological conjugacy in theorem 1 we prove that an expansive homeomorphism which is topologically mixing and has p o t p also has the s p in chapter two theorem 2 we prove that the p o t p for flows is invariant under topological conjugacy with preserved orientation velocity changes also we prove in theorem 3 that the suspension flow 2 for a homeomorphism t x x has p o t p if and only if t has the p o t p in chapter three we prove that an expansive flow which has the p o t p is topologically stable theorem 4 in chapter four the last theorem in this part is that every flow without fixed points on a compact manifold m which is topologically stable has the p o t p then some important corollaries are deduced in the second part of this thesis please see full thesis for rest of abstract

on c1 class local diffeomorphisms whose periodic points are nonuniformly expanding using a sifting shadowing combination we prove in this paper that an arbitrary c1 class local diffeomorphism f of a closed manifold mn is uniformly expanding on the closure mathrm cl m n mathrm per f of its periodic point set per f if it is nonuniformly expanding on per f

on the growth rate of periodic orbits for vector fields abstract we establish the relationship between the growth rate of periodic orbits and the topological entropy for c 1 generic vector fields this extends a classical result of katok for c 1 0 surface diffeomorphisms to c 1 generic vector fields of any dimension the main difficulty comes from the existence of singularities and the shear of the flow

orbital shadowing and stability for vector fields abstract the orbital shadowing property means that there exists a true orbit close to the approximate orbit in the sense of hausdroff distance in this paper we show that in any dimension the c 1 interior of the set of vector fields having the orbital shadowing property is contained in the set of stable vector fields we also discuss the structural stability of vector fields having the orbital shadowing property

a proof of stability conjecture for three dimensional flows we give a proof of the c1 stability conjecture for three dimensional flows i e prove that there exists a hyperbolic structure over the q set for the structurally stable three dimensional flows mafn s proof for the discrete case motivates our proof and we find his perturbation techniques crucial in proving this conjecture we have overcome several new difficulties e g the change of period after perturbation the ergodic closing lemma for flows the existence of dominated splitting over q y where 3 is the set of singularities for the flow the discontinuity of the contracting rate function on singularities etc based on these we finally succeed in separating the singularities from the other periodic orbits for the structurally stable systems i e we create unstable saddle connections if there are accumulations of periodic orbits on the singularities

characterization of dominated splittings for operator cocycles acting on banach spaces abstract versions of the oseledets multiplicative ergodic theorem for cocycles acting on infinite dimensional banach spaces have been investigated since the pioneering work of ruelle in 1982 and are a topic of continuing research interest for a cocycle to induce a continuous splitting in which the growth in one subbundle exponentially dominates the growth in a complementary subbundle requires additional assumptions a necessary and sufficient condition for the existence of such a dominated splitting also known as exponential separation was recently given by j bochi and n gourmelon for invertible finite dimensional cocycles in discrete time we extend this result to cocycles of injective bounded and possibly noncompact linear maps acting on banach spaces in both discrete and continuous time using an essentially geometric approach based on a notion of approximate singular value decomposition in banach spaces our method is constructive and in the finite dimensional case yields explicit growth estimates on the dominated splitting which may be of independent interest

vector fields with topological stability in this paper we give a characterization of the structurally stable vector fields by making use of the notion of topological stability more precisely it is proved that the c1 interior of the set of all topologically stable c1 vector fields coincides with the set of all vector fields satisfying axiom a and the strong transversality condition

measure topologically stable flows abstract topological stability is a kind of stability for given dynamical systems in which continuous perturbations are allowed very recently the concept of topological stability for borel measures with respect to a given homeomorphism was introduced by lee and morales in 7 in this paper we introduce a notion of measure topological stability for a continuous flow and prove that if every measure expansive flow has measure shadowing property then it is measure topologically stable

weak measure expansive flows abstract a notion of measure expansivity for flows was introduced by carrasco olivera and morales in 3 as a generalization of expansivity and they proved that there were no measure expansive flows on closed surfaces in this paper we introduce a concept of weak measure expansivity for flows which is really weaker than that of measure expansivity and show that there is a weak measure expansive flow on a closed surface moreover we show that any c 1 stably weak measure expansive flow on a c closed manifold m is stable and any c 1 stably measure expansive flow on m satisfies both axiom a and the quasi transversality condition

criteria for cr robust permanence abstract let xi xifi x i 1 n be a cr vector field that generates a dissipative flow on the positive cone of rn is called permanent if the boundary of the positive cone is repelling is called crrobustly permanent if remains permanent for sufficiently small cr perturbations of the vector field a necessary condition and a sufficient condition for cr robust permanence involving the average per capita growth rates fi d with respect to invariant measures are derived the necessary condition requires that inf maxi fi d 0 where the infimum is taken over ergodic measures with compact support in the boundary of the positive cone the sufficient condition requires that the boundary flow admit a morse decomposition m1 mk such that every mj satisfies min maxi fi d 0 where the minimum is taken over invariant measures with support in mj as applications we provide a sufficient condition for cr robust permanence of lotka volterra models and a topological characterization of cr robust permanence for food chain models

on thec1stability conjecture for flows we give a proof of thec1stability conjecture of palis and smale for flows which has reduced to proving thatc1structural stability implies axiom a the proof is based on the fundamental work of liao and mane and on the recent powerfulc1connecting lemma of hayashi

direct sums of quasi baer modules abstract while the research work on quasi baer rings has been quite extensive in existing literature the study of quasi baer modules introduced in 2004 remains quite limited with only little work available on the notion in this paper we extend the study of quasi baer modules a characterization of a quasi baer module in terms of its endomorphism ring is obtained we provide a complete characterization of arbitrary direct sums of quasi baer modules to be quasi baer we also show characterizations of some important properties of quasi baer modules examples which delineate the concepts and the results are provided

vector fields in the interior of kupka smale systems satisfy axiom a abstract in 1994 s hu conjectured in his paper 1994 trans amer math soc 342 753 772 that axiom a systems are dense in the interior of kupka smale systems we will show that if x is in the interior of kupka smale systems then x satisfies axiom a

the gluing orbit property uniform hyperbolicity and large deviations principles for semiflows abstract we introduce a gluing orbit property weaker than specification for both continuous maps and flows we prove that flows with the c 1 robust gluing orbit property are uniformly hyperbolic and that every uniformly hyperbolic flow satisfies the gluing orbit property we also prove a level 1 large deviations principle and a level 2 large deviations lower bound for semiflows with the gluing orbit property as a consequence we establish a level 1 large deviations principle for hyperbolic flows and every continuous observable and also a level 2 large deviations lower bound finally since many non uniformly hyperbolic flows can be modeled as suspension flows we also provide criteria for such flows to satisfy uniform and non uniform versions of the gluing orbit property

nonsingular star flows satisfy axiom a and the no cycle condition we give an affirmative answer to a problem of liao and mane which asks whether for a nonsingular flow to loose the stability it must go through a critical element bifurcation more precisely a vector field s on a compact boundaryless manifold is called a star system if s has a c 1 neighborhood mathcal u in the set of c 1 vector fields such that every singularity and every periodic orbit of every x in mathcal u is hyperbolic we prove that any nonsingular star flow satisfies axiom a and the no cycle condition

uniform persistence and chain recurrence soit 7b e d un espace metrique localement compact et e un sous ensemble clos de 7b e soit un systeme dynamique sur e on reconsidere la notion de persistance uniforme dans le cadre de la theorie de conley des ensembles invariants

the specification property for flows from the robust and generic viewpoint abstract we prove that if x has the weak specification property robustly where is an isolated set then is a hyperbolic topologically mixing set and as a consequence if x is a vector field that has the weak specification property robustly on a closed manifold m then the flow x t is a topologically mixing anosov flow also we prove that there exists a residual subset r x 1 m so that if x r and x has the weak specification property then x t is an anosov flow

shadowing in structurally stable flows 1 introductionit was shown by anosov 1 and bowen 3 that approximate trajec tories pseudotrajectories of a diffeomorphism in a neighborhood of ahyperbolic set are shadowed by real trajectories later different authors 10 11 proved that a structurally stable diffeomorphism on a closedmanifold m has the shadowing property on m see also appendix a in thebook 8 bowen 2 showed that pseudotrajectories of a flow near a hyperbolicset separated from rest points are shadowed by real trajectories see also 5 6 in this paper we study a flow generated by a system of ordinary differen tial equations on a closed manifold m it is assumed that the flow isstructurally stable we prove that pseudotrajectories of a structurally stable flow areshadowed by real trajectories and the shadowing is lipschitz with respectto the error theorem 2 1 the main difficulty of the considered problem is created by the followingfact specific for flows let p be a nonwandering point of our flow since astructurally stable flow satisfies axiom a 13 the trajectory of p is hyper bolic denote by s p and u p the corresponding stable and unstable subspaces of the hyperbolic structure if p

structural stability of vector fields with shadowing abstract let x be a c 1 vector field without singularities in this paper we show that x is in the c 1 interior of the set of vector fields with the shadowing property if and only if x satisfies both axiom a and the strong transversality condition that is x is structurally stable

lyapunov exponents periodic orbits and horseshoes for semiflows on hilbert spaces two settings are considered flows on finite dimensional riemann ian manifolds and semiflows on hilbert spaces with conditions consistent with those in systems defined by dissipative parabolic pdes under certain assump tions on lyapunov exponents and entropy we prove the existence of geometric structures called horseshoes this implies in particular the presence of infinitely many periodic solutions for diffeomorphisms of compact manifolds analogous results are due to a katok here we extend katok s results to i continuous time and ii infinite dimensions courant institute of mathematical sciences new york university 251 mercer street new york new york 10012 e mail address lian cims nyu edu courant institute of mathematical sciences new york university 251 mercer street new york new york 10012 e mail address lsy cims nyu edu

vector fields with the oriented shadowing property abstract we give a description of the c 1 interior int 1 orientsh of the set of smooth vector fields on a smooth closed manifold that have the oriented shadowing property a special class b of vector fields that are not structurally stable is introduced it is shown that the set int 1 orientsh b coincides with the set of structurally stable vector fields an example of a field of the class b belonging to int 1 orientsh is given bibliography 18 titles

c 1 stably expansive flows abstract in this paper the c 1 interior of the set of vector fields whose integrated flows are expansive is characterized as the set of vector fields without singularities satisfying both axiom a and the quasi transversality condition and it is proved that the above vector fields possessing the shadowing property must be structurally stable as a corollary there exists a non empty c 1 open set of vector fields whose integrated flows do not have the shadowing property

systems of nonempty intersection theorems with applications in this paper we study the systems of nonempty intersection theorems by the systems of nonempty intersection theorems we can study the existence theorem of systems of quasi kkm problems the existence theorems of systems of quasi variational inclusions problems the existence theorems of systems of quasi variational disclusions problems and fixed point theorem we also establish some of their equivalent relations

collective fixed points and maximal elements with applications to abstract economies abstract in this paper we first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is condensing as an application of our collective fixed points theorem we derive the coincidence theorem for two families of multivalued maps defined on product spaces then we give some existence results for maximal elements for a family of ls majorized multivalued maps whose product is condensing we also prove some existence results for maximal elements for a family of multivalued maps which are not ls majorized but their product is condensing as applications of our results some existence results for equilibria of abstract economies are also derived the results of this paper are more general than those given in the literature

robust transitive singular sets for 3 flows are partially hyperbolic attractors or repellers inspired by lorenz remarkable chaotic flow we describe in this paper the structure of all c 1 robust transitive sets with singularities for flows on closed 3 manifolds they are partially hyperbolic with volume expanding central direction and are either attractors or repellers in particular any c 1 robust attractor with singularities for flows on closed 3 manifolds always has an invariant foliation whose leaves are forward contracted by the flow and has positive lyapunov exponent at every orbit showing that any c 1 robust attractor resembles a geometric lorenz attractor

perturbation of the lyapunov spectra of periodic orbits we describe all lyapunov spectra that can be obtained by perturbing the derivatives along periodic orbits of a diffeomorphism the description is expressed in terms of the finest dominated splitting and lyapunov exponents that appear in the limit of a sequence of periodic orbits and involves the majorization partial order among the applications we give a simple criterion for the occurrence of universal dynamics

some results on systems of quasi variational inclusion problems and systems of generalized quasi variational inclusion problems abstract in this paper we study systems of quasi variational inclusion problem and systems of quasi variational disclusion problem from the existence theorems of solution for these two types of problems we study various types of systems of quasi variational inclusion problems systems of quasi equilibrium problems systems of quasi kkm theorem abstract economics and system of kkm theorem we also show their equivalent relations we study further existence theorems of solution for generalized quasi variational inclusion problem our results are different from any existence result in the literature

the solution existence of general variational inclusion problems we propose general variational inclusion problems which are slightly different from corresponding problems considered in several recent papers in the literature and show that they are advantageous sufficient conditions for the solution existence are established as applications we derive consequences for several special cases of variational inclusion problems quasioptimization problems equilibrium problems and implicit variational inequalities and show that they improve the results of some recent existing papers

partial hyperbolicity and robust transitivity throughout this paper m denotes a three dimensional boundaryless compact manifold and diff m the space of gl diffeomorphisms defined on m endowed with the usual cl topology a invariant set a is transitive if a w x for some xea here w x is the forward limit set of x the accumulation points of the positive orbit of x the maximal invariant set of in an open set u denoted by a u is the set of points whose whole orbit is contained in u i e a u i e z i u the set a u is robustly transitive if ar is transitive for every diffeomorphism r clclose to a diffeomorphism ediff m is transitive if m w x for some xem i e if a m m is transitive analogously is robustly transitive if every r glclose to also is transitive i e if a m m is robustly transitive in this paper we focus our attention on forms of hyperbolicity uniform partial and strong partial of a maximal invariant set a u derived from its robust transitivity observe that u can be equal to m and then is robustly transitive on one hand in dimension one there do not exist robustly transitive diffeomorphisms the diffeomorphisms with finitely many hyperbolic periodic points morse smale are open and dense in diff s1 on the other hand for two dimensional diffeomorphisms every robustly transitive set a u is a basic set i e a u is hyperbolic transitive and the periodic points of are dense in a u in particular every robustly transitive surface diffeomorphism is anosov and the unique surface which supports such diffeomorphisms is the torus t 2 these assertions follow from m3 and m4 in dimension bigger than or equal to three besides anosov hyperbolic diffeomorphisms there are robustly transitive diffeomorphisms of nonhyperbolic type as far as we know three types of such diffeomorphisms have been constructed skew products

quasi shadowing and limit quasi shadowing for quasi partially hyperbolic strings of flows abstract in this paper we study the quasi shadowing property for quasi partially hyperbolic pseudo orbit made by quasi partially hyperbolic strings we also investigate the corresponding l p limit and asymptotic quasi shadowing properties as an application we prove that the set of points with periodic center leaves is dense in the non wondering set for some partially hyperbolic flows

lyapunov stability and sectional hyperbolicity for higher dimensional flows we study c 1 generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices we prove that these flows exhibit finitely many sinks and sectional hyperbolic transitive lyapunov stable sets whose basins form a residual subset of the ambient manifold this represents a partial positive answer to conjectures in arbieto and morales proc am math soc 141 2817 2827 2013 the palis conjecture palis nonlinearity 21 t37 t43 2008 and gives a flow version of crovisier and pujals essential hyperbolicity and homoclinic bifurcations a dichotomy phenomenon mechanism for diffeomorphisms 2010

correction to connecting invariant manifolds and the solution of the c 1 stability and omega stability conjectures for flows stable vector elds satisfy axiom a in 1 the novel and essential part in the proof of the stability and stabilityconjectures for ows is the connecting lemma introduced in 1 a mistake inthe proof of the last conjecture was pointed out to me by toyoshiba 5 wholater also provided an independent proof of it again based on the connectinglemma and previous arguments by man e and palis the crucial step to the proof of theorem b is the separation of singu larities from periodic orbits 1 corollary iii by the c

hyperbolicity and stability for hamiltonian flows abstract we prove that a hamiltonian star system defined on a 2d dimensional symplectic manifold m d 2 is anosov as a consequence we obtain the proof of the stability conjecture for hamiltonians this generalizes the 4 dimensional results in bessa et al 2010 5

the studies of systems of variational inclusions problems and variational disclusions problems with applications abstract in this paper we study existence theorems of solutions for systems of variational inclusions problems and systems of variational disclusions problems from these existence results we establish existence theorems of solutions for systems of generalized vector quasiequilibrium problems and systems of quasioptimization problems

srb measures for partially hyperbolic systems whose central direction is mostly expanding we construct sinai ruelle bowen srb measures supported on partially hyperbolic sets of diffeomorphisms the tangent bundle splits into two invariant subbundles one of which is uniformly contracting under the assumption that the complementary subbundle is non uniformly expanding if the rate of expansion lyapunov exponents is bounded away from zero then there are only finitely many srb measures our techniques extend to other situations including certain maps with singularities or critical points as well as diffeomorphisms having only a dominated splitting and no uniformly hyperbolic subbundle

a spectral decomposition for flows on uniform spaces abstract we study the behavior of flows on uniform spaces and consider expansivity pseudo orbit tracing property and chain recurrence the main result of this article is a generalization of spectral decomposition theorem to flows that are expansive and have the pseudo orbit tracing property on uniform spaces

persistence in dynamical systems soit un espace metrique localement compact de metrique d et soit e un sous ensemble clos de a frontiere e et interieur e on suppose que l on a un flot continu 7b f defini sur e tel que e est invariant par 7b f on considere la notion de persistance et on donne un critere de persistance

a c 1 generic dichotomy for diffeomorphisms weak forms of hyperbolicity or infinitely many sinks or sources we show that for every compact n dimensional manifold n 1 there is a residual subset of diff m of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities either it is contained in the closure of an infinite set of sinks or sources newhouse phenomenon or it presents some weak form of hyperbolicity called dominated splitting this is a generalization of a bidimensional result of mafine ma3 in particular we show that any cl robustly transitive diffeomorphism admits a dominated splitting

the stability under perturbations of repulsive sets on considere le systeme suivant sur r n x f x ou f f 1 f n x x 1 x n on demontre qu un theoreme de fonction de lyapunov inverse est verifie garantissant l existence d une fonction de lyapunov u et que l on peut choisir u de lipschitz