noncommutative ricci flow in a matrix geometry we study noncommutative ricci flow in a finite dimensional representation of a noncommutative torus it is shown that the flow exists and converges to the flat metric we also consider the evolution of entropy and a definition of scalar curvature in terms of the ricci flow

extreme growth rate of periodic orbits for equivalent differentiable flows abstract we prove that the extreme growth rate of periodic orbits of differentiable flows is preserved by lipschitz equivalence when all the fixed points are hyperbolic and thus it is an invariant in an open dense subset x h r of the set of c r 1 r flows by contrast for any 1 r there exist two infinite dimensional connected subsets of equivalent c r flows disjoint from x h r such that the growth rate of periodic orbits attains infinity and zero respectively

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

parabolic monge ampere equations on riemannian manifolds on a compact riemannian manifold m g we consider the parabolic monge ampere equation t x t logdet g x hess x t detg x x t f x x 0 phiv 0 x here is a real parameter andf 0 m r are smooth functions we show existence of for all timestindependent of if 0 then t t converges exponentially towards a solution of the stationary problem ast in the special case 0 f 0 one has 0 and we determine the convergence rate t in thel2 norm more precisely

deforming three manifolds with positive scalar curvature in this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact three manifold is path connected the proof uses the ricci flow with surgery the conformal method and the connected sum construction of gromov and lawson the work of perelman on hamilton s ricci flow is fundamental as one of the applications we prove the path connectedness of the space of trace free asymptotically flat solutions to the vacuum einstein constraint equations on r 3

hopf bifurcation with cubic symmetry and instability of abc flow we examine the dynamics of generic hopf bifurcation in a system that is symmetric under the action of the rotational symmetries of the cube we classify the generic branches of periodic solutions at bifurcation there are generically 27 branches corresponding to maximal symmetries organized into five symmetry types there are also up to 22 periodic solution branches of two other symmetry types these results are found by examination of the normal form with s 1 normal form symmetry for the bifurcation truncated at the third order in addition to the periodic branches whose branching and stability we find there are several branches of tori homoclinic bifurcations and chaotic attractors in the dynamics of the third order normal form since many of these features are not amenable to analysis we give some numerical examples on breaking the normal form symmetry there may be breakup of the branches of tori but the predictions for the periodic solutions will be reliable for the navier stokes equations with a particular forcing an abc flow is a dynamically stable solution for small reynolds numbers r for the most symmetric case a b c 1 the first instability of this system is a hopf bifurcation at r 13 04 with rotational symmetry of the cube we use our normal form analysis to explain the observed behaviour of solutions at this primary instability numerical simulations show that there is supercritical branching to rotating waves that alternate between the three axes which undergo secondary hopf bifurcation to a 2 torus at approximately r 13 09 the eight symmetrically related tori break up and then merge to form a chaotic attractor with full symmetry we can explain all these features by use of the generic third order normal form and s 1 normal form symmetry breaking terms

the long time behavior of 3 dimensional ricci flow on certain topologies de gruyter 2017 in this paper we analyze the long time behavior of 3 dimensional ricci flow with surgery we prove that under the topological condition that the initial manifold only has non aspherical or hyperbolic components in its geometric decomposition there are only finitely many surgeries and the curvature is bounded by ct 1 for large t this proves a conjecture of perelman for this class of initial topologies the proof of this fact illustrates the fundamental ideas that are used in the subsequent papers of the author

perelman s entropy functional at type i singularities of the ricci flow we study blow ups around fixed points at type i singularities of the ricci flow on closed manifolds using perelman s w functional first we give an alternative proof of the result obtained by naber 18 and enders muller topping 8 that blow up limits are non flat gradient shrinking ricci solitons our second and main result relates a limit w density at a type i singular point to the entropy of the limit gradient shrinking soliton obtained by blowing up at this point in particular we show that no entropy is lost at infinity during the blow up process

collapsing irreducible 3 manifolds with nontrivial fundamental group let m be a closed orientable irreducible non simply connected 3 manifold we prove that if m admits a sequence of riemannian metrics which volume collapses and whose sectional curvature is locally controlled then m is a graph manifold this is the last step in perelman s proof of thurston s geometrisation conjecture

dynamical behavior of vibro impact machinery near a point of codimension two bifurcation a dual component system with vibro impact is considered local codimension two bifurcation of maps involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously is analyzed by using the center manifold and normal form method for maps the period n single impact motion and poincare map of the vibro impact system are derived analytically stability and local bifurcation of single impact periodic motion are analyzed by using the poincare map a center manifold theorem technique is applied to reduce the poincare map to a three dimensional one and the normal form map associated with the codimension two bifurcation is obtained local behavior of the dual component system with vibro impact near the point of codimension two bifurcation is analyzed it is found that near the point of codimension two bifurcation there exists not only hopf bifurcation of period one single impact motion but also hopf bifurcation of period two double impact motion period doubling bifurcation of period one single impact motion commonly exists near the point of codimension two bifurcation however no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of hopf bifurcation associated with period two fixed points the period two fixed points are symmetrical about the corresponding period one point the results from simulation show that the attracting invariant circles associated with period two points are symmetrical about the corresponding period one point near the value of hopf bifurcation of period two points two actual examples the impact forming machinery and inertial shaker are chosen to analyze further the phenomena of codimension two bifurcation of maps and local bifurcation analysis and numerical simulation are carried out to unfold dynamical behavior of these vibro impact systems near the points of codimension two bifurcations

a modified kahler ricci flow in this note we study a kahler ricci flow modified from the classic version in the non degenerate case strong convergence at infinite time is achieved the main focus should be on degenerate case where some partial results are presented

quasi genericity of bifurcations to high dimensional invariant tori for maps we consider a family of maps in a banach spacee near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle the other part of the spectrum being strictly inside the unit disc we focus our attention on the region of the parameter space where the truncated normal form of the maps shows a bifurcation of a family of invariantt 1 circles into a family of invariantt 2 tori we show that this problem needs a 3 dimensional parameter unfolding and that for the complete maps bifurcation occurs at points where is the rotation number on the non normally hyperbolict 1 circle ande 2i are the eigenvalues of the constant matrix conjugated to the non contracting part of the linearization on the normal fiber bundle overt 1 making some non resonance and diophantine assumptions on leading to a positive measure cantor set int 2 we show that in paraboloidal regions of the 3 dim parameter space we have clean bifurcations as for the truncated normal form the complement of these regions forms a set of bubbles such as the ones obtained by chenciner in chen for a codimension 2 problem for maps in 2 the main tool here is a generalization for a matrix function ont 1 close to a constant of the quasi conjugacy to a constant modulo a minimum of additional parameters moved quasi conjugacy for the infinite dimensional case we use ac decoupling result on the angular dependent linear parts into a contraction still angular dependent and another part quasi conjugated to a constant matrix this type of analysis applies for a wide range of problems where truncated normal forms of the maps give bifurcations fromt n tot n 1 tori and this needs a n 1 dimensional parameter unfolding

coupling of brownian motions and perelman s l functional abstract we show that on a manifold whose riemannian metric evolves under backwards ricci flow two brownian motions can be coupled in such a way that their normalized l distance is a supermartingale as a corollary we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non decreasing function and the normalized l distance in particular it provides a new proof of a recent result of topping p topping l optimal transportation for ricci flow j reine angew math 636 2009 93 122

geometrization of 3 dimensional orbifolds this paper is devoted to the proof of the orbifold theorem if o is a compact connected orientable irreducible and topologically atoroidal 3 orbifold with nonempty ramification locus then o is geometric i e has a metric of constant curvature or is seifert fibred as a corollary any smooth orientationpreserving nonfree finite group action on s3 is conjugate to an orthogonal action

curvature contraction of convex hypersurfaces by nonsmooth speeds the research of the second author was supported by an australian postgraduate award the research of the first third fourth and fifth authors was supported by discovery project grant dp120100097 of the australian research council

complete gradient shrinking ricci solitons with pinched curvature we prove that any n dimensional complete gradient shrinking ricci soliton with pinched weyl curvature is a finite quotient of mathbb r n mathbb r times mathbb s n 1 or mathbb s n in particular we do not need to assume the metric to be locally conformally flat

hopf saddle node bifurcation for fixed points of 3d diffeomorphisms analysis of a resonance bubble the dynamics near a hopf saddle node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study a two parameter model map g is constructed such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1 to investigate the effect of resonances the complex eigenvalues are selected to have a 1 5 resonance it is shown that near the origin of the parameter space the family g has two secondary hopf saddle node bifurcations of period five points a cone like structure exists in the neighbourhood formed by two surfaces of saddle node and a surface of hopf bifurcations quasi periodic bifurcations of an invariant circle forming a frayed boundary are numerically shown to occur in model g along such cantor like boundary an intricate bifurcation structure is detected near a 1 5 resonance gap subordinate quasi periodic bifurcations are found nearby suggesting the occurrence of a cascade of quasi periodic bifurcations c 2008 elsevier b v all rights reserved

the second twisted betti number and the convergence of collapsing riemannian manifolds let mi overset d gh longrightarrow x denote a sequence of n manifolds converging to a compact metric space x in the gromov hausdorff topology such that the sectional curvature is bounded in absolute value and dim x n we prove the following stability result if the fundamental groups of mi are torsion groups of uniformly bounded exponents and the second twisted betti numbers of mi vanish then there is a manifold m and a sequence of diffeomorphisms from m to a subsequence of mi such that the distance functions of the pullback metrics converge to a pseudo metric in c0 norm furthermore m admits a foliation with leaves diffeomorphic to flat manifolds not necessarily compact such that a vector is tangent to a leaf if and only if its norm converges to zero with respect to the pullback metrics these results lead to a few interesting applications

convergence of kahler ricci flow in this paper we prove a theorem on convergence of kahler ricci flow on a compact kahler manifold which admits a kahler ricci soliton

wave character of metrics and hyperbolic geometric flow in this paper we illustrate the wave character of the metrics and curvatures of manifolds and introduce a new understanding tool the hyperbolic geometric flow this kind of flow is new and very natural to understand certain wave phenomena in the nature as well as the geometry of manifolds it possesses many interesting properties from both mathematics and physics several applications of this method have been found

hyperbolic structures on 3 manifolds i deformation of acylindrical manifolds this is the first in a series of papers showing that haken manifolds have hyperbolic structures this first was published the second two have existed only in preprint form and later preprints were never completed this eprint is only an approximation to the published version which is the definitive form for part i and is provided for convenience only all references and quotations should be taken from the published version since the theorem numbering is different and not all corrections have been incorporated into the present version parts ii and iii will be made available as eprints shortly

conformal deformations of integral pinched 3 manifolds abstract in this paper we prove that under an explicit integral pinching assumption between the l 2 norm of the ricci curvature and the l 2 norm of the scalar curvature a closed 3 manifold with positive scalar curvature admits a conformal metric of positive ricci curvature in particular using a result of hamilton this implies that the manifold is diffeomorphic to a quotient of s 3 the proof of the main result of the paper is based on ideas developed in an article by m gursky and j viaclovsky

flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space abstract this paper concerns closed h convex hypersurfaces of dimension n 2 in the hyperbolic space h n 1 of constant sectional curvature evolving in direction of its normal vector where the speed equals a power 1 of a curvature function f which is monotone symmetric homogeneous of degree 1 it is shown that if the initial h convex hypersurface is pinched then this is maintained under the flow and the hypersurfaces shrink to a round point in h n 1 in finite time as a consequence when rescaling appropriately the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of r n 1

essential hyperbolicity and homoclinic bifurcations a dichotomy phenomenon mechanism for diffeomorphisms we prove that any diffeomorphism of a compact manifold can be approximated in the c 1 topology by another diffeomorphism exhibiting a homoclinic bifurcation a homoclinic tangency or a heterodimensional cycle or by one which is essentially hyperbolic has a finite number of transitive hyperbolic attractors with open and dense basin of attraction

ricci flow on quasi projective manifolds we consider the kahler ricci flow on complete finite volume metrics that live on the complement of a divisor in a compact kahler manifold x assuming certain spatial asymptotics on the initial metric we compute the singularity time in terms of cohomological data on x we also give a sufficient condition for the singularity if there is one to be type ii

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

smoothness of spectral subbundles and reducibility of quasi periodic linear differential systems abstract in this paper we study linear differential systems 1 x a t x where a is an n n matrix valued function defined on the k torus t k and t t is a given irrational twist flow on t k first we show that if a c n t k where n 0 1 2 then the spectral subbundles are of class c n on t k next we assume that a is sufficiently smooth on t k and satisfies a suitable small divisors inequality we show that if 1 satisfies the full spectrum assumption then there is a quasi periodic linear change of variables x p t y that transforms 1 to a constant coefficient system y by finally we study the case where the matrix a t in 1 is the jacobian matrix of a nonlinear vector field x evaluated along a quasi periodic solution x t of 2 x x we give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system z defined in the vicinity of h the hull of so that the linearized system 1 can be represented in the form z dz where d is a constant matrix our results represent substantial improvements over known methods because we do not require that a be close to a constant coefficient system

inverse mean curvature flow with forced term abstract in this paper we study the evolution of convex hypersurfaces by inverse mean curvature minus an external force field c we prove that the flow will preserve the convexity for any c if c 1 h on the initial surface we prove that the flow will expand the hypersurface for all time and after rescaling the hypersurface will converge to a sphere if c 1 h on the initial surface we show that the maximum existence time of the flow is finite and the hypersurface will contract to a point when approaching the final time

partial hyperbolicity far from homoclinic bifurcations abstract we prove that any diffeomorphism of a compact manifold can be c 1 approximated by a diffeomorphism which exhibits a homoclinic bifurcation a homoclinic tangency or a heterodimensional cycle or by a diffeomorphism which is partially hyperbolic its chain recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two we also study in a more systematic way the central models introduced in crovisier in press 10

three coupled oscillators mode locking global bifurcations and toroidal chaos abstract many systems exhibit behaviour which can be described by three coupled oscillators provided the coupling is not too strong such systems can be modelled by maps of the two torus to itself in this paper we describe and explain those aspects of the bifurcation diagram for two parameter families of torus maps that involve change of mode locking type we introduce the concepts of partial and full mode locking for the coupled oscillators these notions correspond to the presence of one or two rational relations between the frequencies respectively numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations in order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time 1 map of a flow on the torus then we analyse those condimension 1 and 2 bifurcations for flows on the torus which change the set of frequency ratios we find a large variety of bifurcation diagrams in particular many involving homotopically non trivial saddle connections next we ask how the picture changes when the time 1 maps of a family of flows are perturbed to a general family of diffeomorphisms we find toroidal chaos in the neighbourhood of all these bifurcations meaning that there exist orbits which perform a pseudo random sequence of rotations in different directions around the torus the corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk finally we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation

volume collapsed three manifolds with a lower curvature bound in this paper we determine the topology of three dimensional complete orientable riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small

real analyticity of solutions of hamilton s equation il existe des constantes positives cn et sn dependant seulement de la dimension n de la variete m telles que pour une metrique initiale g 0 avec r g 0 1 l equation de hamilton a une solution jusqu a t sn et l estimation suivante est vraie pour la metrique solution en t o t sn k 0 t k k 1 12 k r 2 c n

dynamical behavior of a class of vibratory systems with symmetrical rigid stops near the point of codimension two bifurcation a multi degree of freedom vibratory system having symmetrically placed rigid stops is considered the system consists of linear components but the maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops repeated impacts usually occur in the vibratory system due to the rigid amplitude constraints such models play an important role in the studies of mechanical systems with clearances or gaps local codimension two bifurcation of maps involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously is analyzed by using the center manifold theorem technique and normal form method for maps symmetrical double impact periodic motion and poincare map of the system are derived analytically a center manifold theorem technique is applied to reduce the poincare map to a three dimensional one and the normal form map associated with the codimension two bifurcation is obtained local behaviors of the vibratory systems with symmetrical rigid stops near the points of codimension two bifurcations are reported by the presentation of results for a two degree of freedom vibratory system with symmetrical stops the existence and stability of symmetrical double impact periodic motion are analyzed explicitly also local bifurcations at the points of change in stability are analyzed near the point of codimension two bifurcation there exists not only hopf bifurcation of period one double impact motion but also pitchfork bifurcation of the motion pitchfork bifurcation of period one double impact symmetrical motion results in the period one double impact unsymmetrical motion the unsymmetrical double impact motion is of two antisymmetrical forms due to different initial conditions and symmetrical stops with change of the forcing frequency the unsymmetrical double impact periodic motion will undergo hopf bifurcation moreover the period one double impact symmetrical motion will undergo hopf bifurcation directly as the forcing frequency is changed in the contrary direction the routes of quasi periodic impact motions to chaos are observed by results from simulation

routes to chaos in high dimensional dynamical systems a qualitative numerical study this paper examines the most probable route to chaos a high dimensional dynamical systems function space time delay neural networks endowed with a probability measure in a computational setting the most probable route to chaos relative to the measure we impose on the function space as the dimension is increased is observed to be a sequence of neimark sacker bifurcations into chaos the analysis is composed of the study of an example dynamical system followed by a probabilistic study of the ensemble of dynamical systems from which the example was drawn a scenario depicting the decoupling of the stable manifolds of the torus leading up to the onset of chaos in high dimensional dissipative dynamical systems is also presented c 2006 elsevier b v all rights reserved

on the ergodicity of partially hyperbolic systems pugh and shub have conjectured that essential accessibility implies ergodicity for a c partially hyperbolic volume preserving diffeomorphism we prove this conjecture under a mild center bunching assumption which is satisfied in particular by all partially hyperbolic systems with 1 dimensional center bundle we also obtain ergodicity results for c 1ci partially hyperbolic systems

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

the calabi flow on kahler surfaces with bounded sobolev constant i we consider the formation of singularities along the calabi flow by assuming the uniformly bounded sobolev constants on kahler surfaces we prove that if curvature tensor is not uniformly bounded then one can form a singular model called deepest bubble such deepest bubble has to be a scalar flat ale kahler metric in certain kahler classes on toric fano surfaces the sobolev constants are a priori bounded along the calabi flow with small calabi energy we can also show in certain cases no deepest bubble can form along the flow it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently to illustrate our results more clearly we focus on an example on mathbb cp 2 blown up three points at generic position our result also implies existence of constant scalar curvature metrics on mathbb cp 2 blown up three points at generic position in the kahler classes where the exceptional divisors have the same area

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

interaction of hopf and period doubling bifurcations of a vibro impact system abstract an inertial shaker as a vibratory system with impact is considered by means of differential equations periodicity and matching conditions the theoretical solution of periodic n 1 impacting motion can be obtained and the poincare map is established dynamics of the system are studied with special attention to interaction of hopf and period doubling bifurcations corresponding to a codimension 2 one when a pair of complex conjugate eigenvalues crosses the unit circle and the other eigenvalue crosses 1 simultaneously for the jacobi matrix the four dimensional map can be reduced to a three dimensional normal form by the center manifold theorem and the theory of normal forms the two parameter unfoldings of local dynamical behavior are put forward and the singularity is investigated it is proved that there exist curve doubling bifurcation a torus doubling bifurcation hopf bifurcation of 2 2 fixed points as well as period doubling bifurcation and hopf bifurcation of 1 1 fixed points near the critical point numerical results indicate that the vibro impact system presents complicated and interesting curve doubling bifurcation and hopf bifurcation as the two controlling parameters vary

on the positive mass theorem for manifolds with corners we study the positive mass theorem for certain non smooth metrics following p miao s work our approach is to smooth the metric using the ricci flow as well as improving some previous results on the behaviour of the adm mass under the ricci flow we extend the analysis of the zero mass case to higher dimensions

a c 2 generic trichotomy for diffeomorphisms hyperbolicity or zero lyapunov exponents or the c 1 creation of homoclinic bifurcations palis conjectured that densely in diff m r 1 diffeomorphisms are either hyperbolic or exhibit homoclinic bifurcations we prove a generic trichotomy for c2 diffeomorphisms an axiom a diffeomorphism with no cycles or kupka smale ones admitting zero lyapunov exponents or the c1 creation of homoclinic bifurcations i e the creation of homoclinic tangencies or heterodimensional cycles by some c1 small perturbations

homoclinic points in symplectic and volume preserving diffeomorphisms letmn be a compactn dimensional manifold and be a symplectic or volume form onmn let be ac1 diffeomorphism onmn that preserves and letp be a hyperbolic periodic point of we show that genericallyp has a homoclinic point and moreover the homoclinic points ofp is dense on both stable manifold and unstable manifold ofp takens 11 obtained the same result forn 2

a shadowing lemma for quasi hyperbolic strings of flows abstract in this paper we prove a shadowing lemma for pseudo orbits made by quasi hyperbolic strings we allow singularities in question and hence in particular the quasi hyperbolic strings are formulated by the rescaled linear poincare flow instead of the usual linear poincare flow we also introduce the sectional poincare map and rescaled sectional poincare map for lipschitz vector fields on banach spaces in the article

boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains

renormalization in the henon family ii the heteroclinic web we study highly dissipative henon maps f c b x y mapsto c x 2 by x with zero entropy they form a region in the parameter plane bounded on the left by the curve w of infinitely renormalizable maps we prove that morse smale maps are dense in but there exist infinitely many different topological types of such maps even away from w we also prove that in the infinitely renormalizable case the average jacobian bf on the attracting cantor set mathcal o f is a topological invariant these results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory along these lines we show that the unstable manifolds of the periodic points form a lamination outside mathcal o f if and only if there are no heteroclinic tangencies

lifespan theorem for simple constrained surface diffusion flows abstract we consider closed immersed hypersurfaces in r 3 and r 4 evolving by a special class of constrained surface diffusion flows this class of constrained flows includes the classical surface diffusion flow in this paper we present a lifespan theorem for these flows which gives a positive lower bound on the time for which a smooth solution exists and a small upper bound on the total curvature during this time the hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature this turns out to be a deep property of the initial manifold as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in l 2 for m 2 immersed in r 3 and additionally on the concentration in l 3 for m 3 immersed in r 4 this is stronger than a previous result on a different class of constrained surface diffusion flows as here we obtain an improved lower bound on maximal time a better estimate during this period and eliminate any assumption on the area of the evolving hypersurface

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

birth of homoclinic intersections a model for the central dynamics of partially hyperbolic systems creation d intersection homoclines un modele pour la dynamique centrale des systemes partiellement hyperboliques nous montrons une conjecture de j palis tout diffeomorphisme d une variete compacte peut etre approche en topologie c 1 par un diffeomorphisme morse smale ou par un diffeomorphisme ayant une intersection homocline transverse

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