equations differentielles sur l espace de wiener et formules de cameron martin non lineaires proprietes algebriques des approximations sur r n une formule de perturbation des flots et convergence des approximations formule de changement de variable sur x et equation differentielle

maps close to identity and universal maps in the newhouse domain given an n dimensional c r diffeomorphism g its renormalized iteration is an iteration of g restricted to a certain n dimensional ball and taken in some c r coordinates in which the ball acquires radius 1 we show that for any r 1 the renormalized iterations of c r close to identity maps of an n dimensional unit ball b n n 2 form a residual set among all orientation preserving c r diffeomorphisms b n r n in other words any generic n dimensional dynamical phenomenon can be obtained by iterations of c r close to identity maps with the same dimension of the phase space as an application we show that any c r generic two dimensional map that belongs to the newhouse domain i e it has a so called wild hyperbolic set so it is not uniformly hyperbolic nor uniformly partially hyperbolic and that neither contracts nor expands areas is c r universal in the sense that its iterations after an appropriate coordinate transformation c r approximate every orientation preserving two dimensional diffeomorphism arbitrarily well in particular every such universal map has an infinite set of coexisting hyperbolic attractors and repellers

a poincare inequality on loop spaces we show that the laplacian on the loop space over a class of riemannian manifolds has a spectral gap the laplacian is defined using the levi civita connection the brownian bridge measure and the standard bismut tangent spaces

abundance of strange attractors in 1976 he h non performed a numerical study of the family of diffeomorphisms of the plane ha b x y 1 ax2 y bx and detected for parameter values a l 4 b 0 3 what seemed to be a non trivial attractor with a highly intricate geometric structure this family has since then been the subject of intense research both numerical and theoretical but its dynamics is still far from being completely understood in particular one could not exclude the possibility that the attractor observed by h6non were just a periodic orbit with a very high period recently in a remarkable paper bc2 benedicks and carleson were able to show that this is not the case at least for a positive lebesgue measure set of parameter values near a 2 b o more precisely they showed that if b 0 is small enough then for a positive measure set of a values near a 2 the corresponding diffeomorphism ha b exhibits a strange attractor their argument is a very creative extension of the techniques they had previously developed in bull for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated nonhyperbolic dynamics when acquainted in 1985 with the work by benedicks and carleson then in progress palls suggested that one should in this context think of the h6non family as a particular although important model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies he proposed that the correct setting for benedicks carleson s results is within this more general framework of homoclinic bifurcations and stated the following

de rham hodge kodaira operator on loop groups we consider a based loop group le g over a compact lie groupg endowed with its pinned wiener measure the law of the brownian bridge ong and we shall calculate the ricci curvature for differentialn forms over le g a type of bochner weitzenbock formula for general differentialn forms or shigekawa identity will be established

on the effect of invisibility of stable periodic orbits at homoclinic bifurcations a b s t r a c t we study bifurcations of a homoclinic tangency to a saddle focus periodic point we show that the stability domain for single round periodic orbits which bifurcate near the homoclinic tangency has a characteristic comb like structureanddependsstronglyonthesaddlevalue i e onthearea contracting properties of the map at the saddle focus in particular when the map contracts two dimensional areas we have a cascade of periodic sinks in any one parameter family transverse to the bifurcation surface that corresponds to the homoclinic tangency however when the area contraction property is broken while three dimensional volumes are still contracted the cascade of single round sinks appears with probability zero only thus if three dimensional volumes are contracted chaos associated with a homoclinictangencytoasaddle focusisalwaysaccompaniedbystabilitywindows howevertheviolation of the area contraction property can make the stability windows invisible in one parameter families

stable intersections of regular cantor sets with large hausdorff dimensions in this paper we prove a conjecture by j palis according to which the arithmetic difference of generic pairs of regular cantor sets on the line either has zero lebesgue measure or contains an interval more precisely we prove that if the sum of the hausdorff dimensions of two regular cantor sets is bigger than one then in almost all cases there are translations of them whose intersection persistently has hausdorff dimension

brownian pants and deligne cohomology we define stochastic pants and the stochastic parallel transport over them for the felder gawedzki kupiainen line bundle over the loop space it is related to the construction of stochastic integrals where we cannot use martingale theory in order to define them we get a stochastic fusion of the two line bundles when the two loops meet

wavelets bases adapted to a self similar quasicrystal given any self similar quasicrystal in rn with inflation 1 we construct bases of l2 rn having the following structure nj 2 jx j z where the mother wavelets are smooth and with exponential decay or compact support we also show that wavelets constitute a relatively compact set in some sobolev space and that they depend continuously on when is equipped with an appropriate topology

minimal distances in quasicrystals a general expression is derived for the minimal distance between points of a cut and project quasicrystal in with a convex acceptance window the study of minimal distances amounts to the study of one dimensional quasicrystals and their rescalings which occur in for an n dimensional ball as the exact value of is calculated for any radius for close to a ball a simple formula is given for all upper and lower bounds for are found the latter are easy to use even when is of a complicated shape

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3

on the existence of attractors on every compact 3 manifold we build a non empty open set u of diff m such that for every r 1 every cr generic diffeomorphism f u diffr m has no topological attractors on higher dimensional manifolds one may require that f has neither topological attractors nor topological repellers our examples have finitely many quasi attractors for flows we may require that these quasi attractors contain singular points finally we discuss alternative definitions of attractors which may be better adapted to generic dynamics

differential calculus on finite codimensional submanifolds of the wiener space the divergence operator abstract on finite codimensional submanifolds of the wiener space the ricci gaussian curvature is defined and used to prove a weitzenbock bochner identity from this identity sobolev estimates of the divergence of a vector field are obtained moreover using the method of the malliavin calculus covariant differentiation christoffel symbols and brackets of two vector fields are defined

logarithmic sobolev inequalities on loop groups let g be a compact lie group denote by m the brownian bridge measure on the loop group y g c 0 1 g g 0 g 1 e the finite energy subgroup of y determines in a natural way a gradient operation for functions on y the following logarithmic sobolev inequality is proven f2 log f dm gradf y 2 v y f y 2 dm f 2log f wherein f denotes the l2 m norm and v is a potential which is quadratic in the associated lie algebra valued brownian motion the inequality is derived by a method of inheritance from the known inequality for the g valued brownian motion

analysis on wiener spaces ii differential forms abstract the author gives the notions of p forms and p forms and shows some properties of them in particular a vanishing theorem of de rham cohomology of poincare type also he shows the hypoellipticity of certain elliptic operators on wiener space and a theorem of poinare inequality type

a cameron martin type quasi invariance theorem for pinned brownian motion on a compact riemannian manifold the results in driver 13 for quasi invariance of wiener measure on the path space of a compact riemannian manifold m are extended to the case of pinned wiener measure to be more explicit let h 0 1 tom be a c1 function where m is a compact riemannian manifold o e m is a base point and tom is the tangent space to m at o e m let w m be the space of continuous paths from 0 1 into m v be wiener measure on w m concentrated on paths starting at o e m and hs w denote the stochastic parallel translation operator along a path to e w m up to time s note hs w is only well defined up to v equivalence for c e w m let xh w denote the vector field along a given by xsh c hs cw h s for each s e 0 1 one should interpret xh as a vector field on w m the vector field xh induces a flow sh t w m w m which leaves wiener measure v quasi invariant see driver 13 it is shown in this paper that the same result is valid if h l 0 and the wiener measure v is replaced by a pinned wiener measure ive the measure ve is proportional to the measure v conditioned on the set of paths which start at 0 e m and end at a fixed end point e e m also as in 13 one gets an integration by parts formula for the vector fields xh defined above

tau wavelets in the plane first an iterative process is described for subdivision and for scaling up of the penrose robinson tiles then we build the wavelets of haar i e suitably normalized characteristic functions on the tiles so that the orthogonality and multiresolution conditions are satisfied here 12 1 5

integration by parts formulas and dilatation vector fields on elliptic probability spaces by coupling two arbitrary riemannian connections and on a riemannian manifold m we perform the stochastic calculus of variation on the path space p m of the manifold m the method uses direct calculations on ito s stochastic differential equations in this context we obtain intertwinning formulas with the ito map for first order operators on the path space p m of m by a judicious choice of the second connection in terms of the connection we can prolongate the intertwinning formulas to second order differential operators thus we obtain expressions of heat operators on the path space p m of a riemannian manifold m endowed with an arbitrary connection the integration by parts of the laplacians on p m leads us to the notion of dilatation vector field on the path space

dynamics of homoclinic tangles in periodically perturbed second order equations we obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second order ordinary differential equations with dissipation let be the size of perturbation and be the entire homoclinic tangle we prove in particular that i for infinitely many disjoint open sets of contains nothing else but a horseshoe of infinitely many branches ii for infinitely many disjoint open sets of contains nothing else but one sink and one horseshoe of infinitely many branches and iii there are positive measure sets of so that admits strange attractors with sinai ruelle bowen measure we also use the equation d2qdt2 q2 dqdt q q2 q2sin t to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

cycles and measure of bifurcation sets for two dimensional diffeomorphisms we describe a large class of one parameter families of two dimensional diffeomorphisms which arestable for 0 exhibit acycle for 0 and thereafter have a bifurcation set of positive but arbitrarily smallrelative measure for in small intervals 0 a main assumption is that the basic sets involved in the cycle havelimit capacities that are not too large

how often do simple dynamical processes have infinitely many coexisting sinks this work concerns the nature of chaotic dynamical processes sheldon newhouse wrote on dynamical processes depending on a parameter xx 1 t xn wherex is in the plane such as might arise when studying poincare return maps for autonomous differential equations in ir3 he proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane and that such parameter values would be dense in some interval the fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes we prove for an apparently typical situation that newhouse s construction yields only a set of parameter values of measure zero

toward discrete wavelets with irrational scaling factor scaling equations determining haar type wavelets are considered for three pisot numbers 1 5 2 1 2 2 3 appearing as dilation factors in scale invariance of observed diffraction patterns of quasicrystals simple summation formulas for exponentials of the corresponding integers are derived

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

tau wavelets of haar in this paper we construct a new type of haar wavelets called wavelets of haar using the arithmetics of the solutions and of the algebraic equation

integration on loop groups ii heat equation for the wiener measure abstract let g be a compact semi simple lie group and l g the loop group over g on l g is defined a family t l t 0 of quasi invariant wiener measures we construct on l g a laplace operator l such that t l are the measures generated by the process associated to l modified by a feynman kac density

integration by parts formula and logarithmic sobolev inequality on the path space over loop groups the geometric stochastic analysis on the riemannian path space developed recently gives rise to the concept of tangent processes roughly speaking it is the infinitesimal version of the girsanov theorem using this concept we shall establish a formula of integration by parts on the path space over a loop group following the martingale method developed in capitaine hsu and ledoux we shall prove that the logarithmic sobolev inequality holds on the full paths as a particular case of our result we obtain the driver lohrenz s heat kernel logarithmic sobolev inequalities over loop groups the stochastic parallel transport introduced by driver will play a crucial role

g model sets and their self similarities the model sets also called cut and project sets first defined by yves meyer in harmonic analysis play a central role in quasicrystal modelling each of them is defined by using a cut and project scheme containing two projectors and a lattice we present a method which can be used to study the self similarities of a model set based on the matrices of these projectors in a basis of the lattice this method also allows one to study the self similarities of the diffraction spectrum of a model set because generally the bragg peaks with intensity above a given threshold also form a model set the diffraction pattern corresponding to a quasicrystal is invariant under a finite group g and the local structure of the quasicrystal can be described by using a finite union of orbits of g called a g cluster the neighbours of each atom belong to some orbits of g and the quasicrystal can be regarded as a union of interpenetrating partially occupied translations of the corresponding g cluster we present a method to obtain a model set called the g model set by starting from a g cluster the experimental diffraction patterns allow one to determine the symmetry group g and high resolution electron microscopy images enable one to choose a g cluster describing the local structure the existing computer programs for the cut and project method allow one to pass directly from the local structure of the quasicrystal to a mathematical model to compute the theoretical diffraction spectrum and to compare it with the experimental data

integration on loop groups i quasi invariant measures on a riemannian manifold m the brownian motion defines a wiener measure on the loops over m in the case of r quasi invariance means the cameron martin theorem s which fully describes the quasi invariance by translation of the wiener measure in order to state a generalization of this theorem we suppose that m is an homogeneous space under the action of a compact lie group the noncommutativity of this situation introduces a completely new phenomenon the hypoellipticity associated to the lie bracket of the basic vector fields curiously enough hypoellipticity which helps to get regularity in finite dimensions works against quasi invariance in infinite dimensions 18 we denote by g a compact connected lie group by h a connected subgroup of g and by m the homogeneous space g h we denote by y and 2 the corresponding lie algebras we choose a euclidean metric on 9 which is ad g invariant this metric induces a riemannian metric on g and on m we denote p g the paths space on g that is the space of continuous maps y from 0 l into g such that y o e we denote by l g the space of loops on g that is

takens last problem and existence of non trivial wandering domains abstract in this paper we give an answer to a c r 2 r version of the open problem of takens in 42 which is related to historic behavior of dynamical systems to obtain the answer we show the existence of non trivial wandering domains near a homoclinic tangency which is conjectured by colli vargas 6 2 concretely speaking it is proved that any newhouse open set in the space of c r diffeomorphisms on a closed surface is contained in the closure of the set of diffeomorphisms which have non trivial wandering domains whose forward orbits have historic behavior moreover this result implies an answer in the c r category to one of the open problems of van strien 39 which is concerned with wandering domains for henon family

a cameron martin type quasi invariance theorem for brownian motion on a compact riemannian manifold abstract let m be a compact manifold without boundary o be a fixed base point in m g be a riemannian metric on m and be a g compatible covariant derivative on tm the tangent space to m assume the torsion t of satisfies the skew symmetry condition g t x y y 0 for all vector fields x and y on m for example take to be the levi civita covariant derivative on m g also let denote the wiener measure on w0 m c 0 1 m 0 o and let h s denote stochastic parallel translation relative to along the path w0 m up to time s 0 1 given a c1 function h 0 1 t0m it is shown that the differential equation t h t h with initial condition 0 id w m w m has a solution r maps w m w m the measurable maps from w m to w m this function is a flow on w m i e for all t r t t furthermore t has the quasi invariance property the law t of t with respect to the wiener measure is equivalent to for all t r this result is used to prove an integration by parts formula for the h derivative hf defined by h f d dt 0 f t where f is a c2 cylinder function on w m

bifurcation frequency for unimodal maps we consider some natural one parameter unfoldingsf of a unimodal mapf0 whose periodic points are hyperbolic and whose critical point is nondegenerate and eventually periodic among other facts it follows from our theorems that if the julia set off0 does not contain intervals the relative measure of the bifurcation set is zero at zero

dissipative homoclinic loops of two dimensional maps and strange attractors with one direction of instability we prove that when subjected to periodic forcing of the form p t d h x y c sin t certain two dimensional vector fields with dissipative homoclinic loops generate strange attractors with sinai ruelle bowen measures for a set of forcing parameters of positive lebesgue measure the proof extends ideas of afraimovich and shilnikov and applies the recent theory of rank 1 maps developed by wang and young we prove a general theorem and then apply this theorem to an explicit model a forced duffing equation of the form d 2 q

inflation centres of the cut and project quasicrystals a rather general form of the conventional cut and project scheme is used to define quasicrystals as point sets in real n dimensional euclidean space the inflation or equivalently the self similarity properties of such quasicrystals are studied here assuming only the convexity of the acceptance window our result is a description of inflation centres of all types in a quasicrystal and a proof that our description is complete there are no other inflation centres for any chosen quasicrystal point internal inflation centre u its inflation properties are given as a set of scaling factors it turns out that the scaling factors form a one dimensional quasicrystal with a u dependent acceptance window scaling window the intersection of the scaling windows associated with all points of a quasicrystal is the one dimensional quasicrystal of universal internal scaling symmetries its acceptance window is the interval external inflation centres of a cut and project quasicrystal are those which are not among quasicrystal points their complete description is given analogically to the description of the internal ones imposing some additional requirements on the scaling factors between any two adjacent quasicrystal points one finds a countable infinity of external inflation centres the scaling factors belonging to any such centre u form an infinite u dependent subset of points of the quasicrystal with acceptance window containing

on the equivalence of measures on loop space let k be a simply connected compact lie group equipped with an ad k invariant inner product on the lie algebra of k given this data there is a well known left invariant h 1 riemannian structure on l k the infinite dimensional group of continuous based loops in k as well as a heat kernel t k 0 associated with the laplace beltrami operator on l k here t 0 k 0 l k and t k 0 is a certain probability measure on l k in this paper we show that 1 e is equivalent to pinned wiener measure on k on s0 x t t 0 s 0 the algebra generated by truncated loops up to time s 0

poincare inequality for weighted first order sobolev spaces on loop spaces abstract let e be the loop space over a compact connected riemannian manifold with a torsion skew symmetric connection let l d be the ornstein uhlenbeck operator on a nonempty connected component d of the loop space e and let v d r be the restriction on d of the potential in the logarithmic sobolev inequality found by l gross on the loop group by s aida and by f z gong and z m ma on the loop space respectively we prove that the schrodinger operator l v l d v always has a spectral gap at the bottom 0 v of its spectrum and thus has its ground state transformed operator 1 l v 0 v where is the unique ground state of l v in particular our result proves l gross s conjecture about the existence of a spectral gap for the ground state transform of the schrodinger operator studied by him on the loop group in addition in all the above cases we identify the domain of the dirichlet forms associated with the ground state transforms as weighted first order sobolev spaces with weight given by 2 thus establishing a poincare inequality for them all these results are consequences from some new results in this paper on dirichlet forms characterizing certain classes with spectral gaps and from results by s aida and m hino

newhouse phenomenon for automorphisms of low degree in c3 abstract we show that there exists a polynomial automorphism f of c 3 of degree 2 such that for every automorphism g sufficiently close to f g admits a tangency between the stable and unstable laminations of some hyperbolic set as a consequence for each d 2 there exists an open set of polynomial automorphisms of degree at most d in which the automorphisms having infinitely many sinks are dense to prove these results we give a complex analogous to the notion of blender introduced by bonatti and diaz

dilatation vector fields on the loop group abstract on the loop space l g over a compact connected lie group g we explicitly determine the first order differential operator c t dilatation vector field such that the heat equation holds with respect to the wiener measure l t defined on the loop space

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

wavelet multiresolutions for the fibonacci chain most of the discrete wavelet families are periodic dyadic they live on the multiresolution sequence 2j j of lattice backgrounds at different scales we present here a construction of wavelet family based on a sequence of aperiodic discretizations of at a given scale this discretization is the point set of nodes of the fibonacci chain a well known stone inflation cut and project model for one dimensional quasicrystals corresponding multiresolution analysis and the elementary example of the haar system are presented

integration on loop group iii asymptotic peter weyl orthogonality using the wiener measure on the continuous loop group l g of a compact connected group g whose properties were studied in integration on loop group i we construct for g simply laced a stochastic central extension l g of l g despite the fact that this object has a multiplication defined almost everywhere we are able to define for it a convolution algebra

vanishing of one dimensional l 2 cohomologies of loop groups abstract let g be a simply connected compact lie group let l e g be the based loop group with the base point e which is the identity element let e be the pinned brownian motion measure on l e g and let l 2 1 t l e g e d p 1 t l e g e 1 p 2 be a closed 1 form on l e g using results in rough path analysis we prove that there exists a measurable function f on l e g such that d f moreover we prove that dim ker 0 for the hodge kodaira type operator acting on 1 forms on l e g

generic family with robustly infinitely many sinks we show for every r d ge 0 or r d ge 2 the existence of a baire generic set of c d families of c r maps f a a in mathbb r k of a manifold m of dimension ge 2 so that for every a small the map f a has infinitely many sinks when the dimension of the manifold is ge 3 the generic set is formed by families of diffeomorphisms when m is the annulus this generic set is formed by local diffeomorphisms this is a counter example to a conjecture of pugh and shub

solution of the basin problem for henon like attractors for a large class of non uniformly hyperbolic attractors of dissipative diffeomorphisms we prove that there are no holes in the basin of attraction stable manifolds of points in the attractor fill in a full lebesgue measure subset then lebesgue almost every point in the basin is generic for the srb sinai ruelle bowen measure of the attractor this solves a problem posed by sinai and by ruelle for this class of systems

hyperbolicity and the creation of homoclinic orbits we consider one parameter families sp t e r of diffeomorphisms on surfaces which display a homoclinic tangency for yt 0 and are hyperbolic for yt 0 i e pf has a hyperbolic nonwandering set the tangency unfolds into transversal homoclinic orbits for yt positive for many of these families we prove that p is also hyperbolic for most small positive values of yt which implies much regularity of the dynamical structure a main assumption concerns the limit capacities of the basic set corresponding to the homoclinic tangency

abundance of c 1 robust homoclinic tangencies a diffeomorphism f has a c 1 robust homoclinic tangency if there is a c 1 neighbourhood u of f such that every diffeomorphism in g u has a hyperbolic setg depending contin uously on g such that the stable and unstable manifolds ofg have some non transverse intersection for every manifold of dimension greater than or equal to three we exhibit a lo cal mechanism blender horseshoes generating diffeomorphisms with c 1 robust homoclinic tangencies using blender horseshoes we prove that homoclinic classes of c 1 generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings of appropriate dimensions display c 1 robust homoclinic tangencies

prevalent dynamics at the first bifurcation of henon like families we study the dynamics of strongly dissipative henon like maps around the first bifurcation parameter a at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set we prove that a is a full lebesgue density point of the set of parameters for which lebesgue almost every initial point diverges to infinity under positive iteration a key ingredient is that a corresponds to the non recurrence of every critical point reminiscent of misiurewicz parameters in one dimensional dynamics adapting on the one hand benedicks carleson s parameter exclusion argument we construct a set of good parameters having a as a full density point adapting benedicks viana s volume control argument on the other we analyze lebesgue typical dynamics corresponding to these good parameters