axiom a maps are dense in the space of unimodal maps in the ck topology in this paper we prove c k structural stability conjecture for unimodal maps in other words we shall prove that axiom a maps are dense in the space of c k unimodal maps in the c k topology here k can be 1 2 1 1 the structural stability conjecture the structural stability conjecture was and remains one of the most interesting and important open problems in the theory of dynamical systems this conjecture states that a dynamical system is structurally stable if and only if it satisfies axiom a and the transversality condition in this paper we prove this conjecture in the simplest nontrivial case in the case of smooth unimodal maps these are maps of an interval with just one critical turning point to be more specific let us recall the definition of axiom a maps definition 1 1 let x be an interval we say that a c k map f x satisfies the axiom a conditions if f has finitely many hyperbolic periodic attractors the set f x f i sh yperbolic where f i sa union of the

universal teichmuller space and diff s 1 s 1 we point out that the coset space diffs1 s1 is a dense complex submanifold of the universal teichmuller spaces of compact riemann spaces of genus g 1 a holomorphic map ofs into the inifinite dimensional segal diskd1 is constructed this is the universal analogue of the map of teichmuller spaces into the siegel disk provided by the period matrix the kahler potential for the general homogenous metric on diffs1 s1 is computed explicitly using the map intod1 some applications to string theory are discussed

getting rid of the negative schwarzian derivative condition in this paper we will show that the assumption on the negative schwarzian derivative is redundant in the case of c 3 unimodal maps with a non at critical point the following theorem will be proved for any c 3 unimodal map of an interval with a non at critical point there exists an interval around the critical value such that the flrst entry map to this interval has negative schwarzian derivative another theorem proved in the paper provides useful cross ratio estimates thus all theorems proved only for unimodal maps with negative schwarzian derivative can be easily generalized

regular or stochastic dynamics in real analytic families of unimodal maps in this paper we prove that in any non trivial real analytic family of quasiquadratic maps almost any map is either regular i e it has an attracting cycle or stochastic i e it has an absolutely continuous invariant measure to this end we show that the space of analytic maps is foliated by codimension one analytic submanifolds hybrid classes this allows us to transfer the regular or stochastic property of the quadratic family to any non trivial real analytic family

integration of brownian vector fields summary using the wiener chaos decomposition we show that strong solutions of non lipschitzian s d e s are given by random markovian kernels the example of sobolev flows is studied in some detail exhibiting interesting phase transitions

real bounds ergodicity and negative schwarzian for multimodal maps over the last 20 years many of the most spectacular results in the field of dynamical systems dealt specifically with interval and circle maps or perturbations and complex extensions of such maps primarily this is because in the one dimensional case much better distortion control can be obtained than for general dynamical systems however many of these spectacular results were obtained so far only for unimodal maps the aim of this paper is to provide all the tools for studying general multimodal maps of an interval or a circle by obtaining real bounds controlling the geometry of domains of certain first return maps and providing a new and we believe much simpler proof of absense of wandering intervals provided certain combinatorial conditions are satisfied large real bounds implying that certain first return maps are almost linear koebe distortion controlling the distortion of high iterates of the map and negative schwarzian derivative for certain return maps showing that the usual assumption of negative schwarzian derivative is unnecessary control of distortion of certain first return maps ergodic properties such as sharp bounds for the number of ergodic components

modulus of continuity of the canonic brownian motion on the group of diffeomorphisms of the circle abstract an explicit modulus of holder continuity is given for the flow associated to the canonic brownian motion on the diffeomorphism group of the circle

the fibonacci unimodal map the fibonacci recurrence of the critical orbit appeared in the work of branner and hubbard on complex cubic polynomials bh 12 and in yoccoz s work yl y2 on quadratic ones as the worst pattern of recurrence on the other hand a real quadratic fibonacci map was suggested by hofbauer and keller hk as a possible candidate for a map having a wild attractor that is a set a which is the w limit set for lebesgue almost every orbit but is strictly smaller than the w limit set for a generic orbit the w limit set of the critical orbit in hk possesses all known topological properties of wild attractors compare bl2 in fact we will see below that the quadratic fibonacci map does not have a wild attractor however the corresponding question for a map with a degenerate critical point remains open actually the first indication of the fibonacci map appeared in the numerical work of tsuda t related to the belousova zhabotinskii reaction and also in numerical work of shibayama sh more precisely they studied the sequence of fibonacci bifurcations creating the fibonacci map this paper will study topological geometrical and measure theoretical properties of the real fibonacci map our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by sullivan s it turns out that the situation can be understood completely and is of quite regular nature in particular any fibonacci map with negative schwarzian and nondegenerate critical point has an absolutely continuous invariant measure so we deal with a regular type of chaotic dynamics it turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the feigenbaum map its hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter branner and hubbard introduce the concept of a tableau in order to describe recurrence of critical orbits their fibonacci tableau is a basic example which corresponds to one particularly close and regular pattern of recurrence if a complex quadratic map z 1 z2 c realizes this fibonacci tableau then the

some remarks on quasi invariant actions of loop groups and the group of diffeomorphisms of the circle we construct the series of quasi invariant actions of the group diff of diffeomorphisms of the circle and loop groups on the functional spaces provided by non wiener gauss measures we construct some measures which can be considered as analogues of haar measure for loop groups and the group diff these constructions allow us to construct series of representations of these groups including all known types of representations higest weight representations energy representations almost invariant structures etc

brownian motion on volume preserving diffeomorphisms group and existence of global solutions of 2d stochastic euler equation abstract a geometric brownian motion performs a continuous time infinitesimal perturbation of the state of the system this perturbation conserves the energy exact expression for the energy transfer towards high modes is obtained ensuring existence for all time of the solution

bi lipschitz parameterization of surfaces we consider surfaces z homeomorphic to the plane with complete possibly singular riemannian metrics if we have z k 0 and c 0 this result implies a conjecture by j fu

integrated harnack inequalities on lie groups we show that the logarithmic derivatives of the convolution heat kernels on a uni modular lie group are exponentially integrable this result is then used to prove an integrated harnack inequality for these heat kernels it is shown that this integrated harnack inequality is equivalent to a version of wang s harnack inequality a key feature of all of these inequalities is that they are dimension independent finally we show these inequalities imply quasi invariance properties of heat kernel measures for two classes of infinite dimensional lie groups

julia fatou sullivan theory for real one dimensional dynamics our aim is to show that the julia fatou sullivan structure theory for the dynamics of rational maps is also valid for smooth endomorphisms of the circle and of the interval under extremely mild smoothness and non flatness conditions in order to stress the similarity between real and complex one dimensional dynamics let us recall the main results from the fatou julia sullivan theory if f is a rational map then there is a dynamical decomposition of the riemann sphere into the disjoint union of two totally invariant i e both forward and backward invariant sets j f f f here f f is the domain of normality of the family of iterates of f and is called the fatou set its complement which is called the julia set off is a compact set which contains all the complications of the dynamics off the connected components of the open set f f are mapped onto each other by f hence the orbit of a component of f f is the union of some components of f f julia proved in the beginning of the century that if a component of f f is periodic and contains an attracting periodic point then the orbit of this component must contain a critical point sullivan in the remarkable paper su showed via quasi conformal deformations that the components are eventually periodic and fall into finitely many orbits let n be either the circle s j or a compact interval of the real line and f n n be a smooth endomorphism a critical point of f is a point where the derivative vanishes a critical point is non flat if some higher derivative is non zero a critical point is an inflection point if it has a neighbourhood where f is monotone otherwise it is called a turning point assume f is not a homeomorphism i f f is a homeomorphism then these maps correspond to degree 1 rational maps of the riemann sphere and the situation is

entropic measure and wasserstein diffusion we construct a new random probability measure on the circle and on the unit interval which in both cases has a gibbs structure with the relative entropy functional as hamiltonian it satisfies a quasi invariance formula with respect to the action of smooth diffeomorphism of the sphere and the interval respectively the associated integration by parts formula is used to construct two classes of diffusion processes on probability measures on the sphere or the unit interval by dirichlet form methods the first one is closely related to malliavin s brownian motion on the homeomorphism group the second one is a probability valued stochastic perturbation of the heat flow whose intrinsic metric is the quadratic wasserstein distance it may be regarded as the canonical diffusion process on the wasserstein space

measure and dimension of solenoidal attractors of one dimensional dynamical systems letf m m be ac map of the interval or the circle with non flat critical points a closed invariant subseta m is called a solenoidal attractor off if it has the following structure a mathop cap limits n 1 infty mathop cup limits k 0 p n 1 l k n where ik n is the cycle of intervals of periodpn we prove that the lebesgue measure ofa is equal to zero and if sup pn 1 pn then the hausdorff dimension ofa is strictly less than 1

the ricci curvature of diff s1 sl 2 r previous calculations of the ricci curvature for the manifold diff diff s1 s1 are extended to diff s1 sl 2r these manifolds are distinguished by being coadjoint orbits of diff s1 which admit compatible symphectic and complex structures making them kahler manifolds

compactification of a class of conformally flat 4 manifold in this paper we generalize huber s result on complete surfaces of finite total curvature for complete locally conformally flat 4 manifolds of positive scalar curvature with q curvature integrable where q is a variant of the chern gauss bonnet integrand we first derive the cohn vossen inequality we then establish finiteness of the topology this allows us to provide conformal compactification of such manifolds

canonical brownian motion on the diffeomorphism group of the circle for infinitesimal data given on the group of diffeomorphism of the circle with respect to the metric h3 2 the associated brownian motion has been constructed by malliavin c r acad sci parist 329 1999 325 329 in this work we shall give another approach and prove the invariance of heat measures under the adjoint action of s1

on the chern gauss bonnet integral for conformal metrics on r4 where m is the euler number ofm later huber hu extended this inequality to metrics with much weaker regularity more importantly he proved that such a surface m is conformally equivalent to a closed surface with finitely many punctures the deficit in formula 1 1 has an interpretation as an isoperimetric constant on a complete and open surface with gaussian curvature absolutely integrable one may represent each end conformally as r2 k for some compact set k we consider the isoperimetric ratio

heat kernel analysis on semi infinite lie groups this paper studies brownian motion and heat kernel measure on a class of infinite dimensional lie groups we prove a cameron martin type quasi invariance theorem for the heat kernel measure and give estimates on the lp norms of the radon nikodym derivatives we also prove that a logarithmic sobolev inequality holds in this setting

nonergodicity of euler fluid dynamics on tori versus positivity of the arnold ricci tensor abstract brownian motions above the group g of volume preserving diffeomorphisms of the torus t d d 2 are constructed the asymptotic behaviour for large time of those processes shows the nonexistence of a probability measure invariant under the deterministic incompressible fluid dynamics the energy induces on the group of volume preserving diffeomorphisms of t 2 a riemannian structure which has a positive renormalized ricci tensor

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

feigenbaum coullet tresser universality and milnor s hairiness conjecture we prove the feigenbaum coullet tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type this gives the first computer free proof of the original feigenbaum observation of the universal parameter scaling laws we use the hyperbolicity theorem to prove milnor s conjectures on self similarity and hairiness of the mandelbrot set near the corresponding parameter values we also conclude that the set of real infinitely renormalizable quadratics of type bounded by some n 1 has hausdorff dimension strictly between 0 and 1 in the course of getting these results we supply the space of quadratic like germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimension one foliation of the connectedness locus

regularity of a fourth order nonlinear pde with critical exponent in this paper we demonstrate the regularity of minimizers for a variational problem a special case of which arises in spectral theory and conformal geometry the associated euler lagrange equation is fourth order semilinear the leading term is the bilaplacian and lower order terms appear at critical powers

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

hyperbolicity sinks and measure in one dimensional dynamics letf be ac2 map of the circle or the interval and let f denote the complement of the basins of attraction of the attracting periodic orbits we prove that f is a hyperbolic expanding set if and obviously only if every periodic point is hyperbolic and f doesn t contain the critical point this is the real one dimensional version of fatou s hyperbolicity criteria for holomorphic endomorphisms of the riemann sphere we also explore other applications of the techniques used for the result above proving for instance that for everyc2 immersionf of the circle i e a map of the circle onto itself without critical points either its julia set has measure zero or it is the whole circle and thenf is ergodic i e positively invariant borel sets have zero or full measure

rigidity and non local connectivity of julia sets of some quadratic polynomials for an infinitely renormalizable quadratic map f c z mapsto z 2 c with the sequence of renormalization periods km and rotation numbers tm pm qm we prove that if limsup k m 1 log p m 0 then the mandelbrot set is locally connected at c we prove also that if limsup t m 1 1 q m 1 and qm then the julia set of fc is not locally connected and the mandelbrot set is locally connected at c provided that all the renormalizations are non primitive satellite this quantifies a construction of a douady and j hubbard and weakens a condition proposed by j milnor

stochastic analysis on the path space of a riemannian manifold i markovian stochastic calculus abstract on the brownian flow of a compact riemannian manifold an intrinsic stochastic calculus is defined this calculus is markovian the stochastic calculus of variation on the bundle of orthonormal frames appears only as an intermediate tool the bismut connection plays a paramount role in the final construction

non existence of infinitesimally invariant measures on loop groups let g be a compact lie group l g the associated loop group the canonical symplectic form on l g set h the hamiltonian function for which the associated hamiltonian vector field is the infinitesimal rotation then h generates a canonical semi definite riemannian structure on l g which induces a riemannian structure on the free loop group l g g l0 g this metric corresponds to the sobolev norm h1 using orthonormal frame methodology the positivity and finiteness of the ricci curvature of l0 g is proved by studying the dissipation towards high modes of a unitary group valued sde it is proved that the loop group does not have any infinitesimally invariant measure

relating notions of convergence in geometric analysis abstract we relate l p convergence of metric tensors or volume convergence to a given smooth metric to intrinsic flat and gromov hausdorff convergence for sequences of riemannian manifolds we present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general even when the sequence is conformal g j f j 2 g 0 to a fixed manifold we then prove a theorem demonstrating that when sequences of metric tensors on a fixed manifold m are bounded 1 1 j g 0 g j k g 0 and either the volumes converge vol j m vol 0 m or the metric tensors converge in the l p sense then the riemannian manifolds m g j converge in the measured gromov hausdorff and volume preserving intrinsic flat sense to m g 0

the isoperimetric inequality and q curvature abstract a well known question in differential geometry is to control the constant in isoperimetric inequality by intrinsic curvature conditions in dimension 2 the constant can be controlled by the integral of the positive part of the gaussian curvature in this paper we showed that on simply connected conformally flat manifolds of higher dimensions the role of the gaussian curvature can be replaced by the branson s q curvature we achieve this by exploring the relationship between a p weights and integrals of the q curvature

puzzle geometry and rigidity the fibonacci cycle is hyperbolic we describe a new and robust method to prove rigidity results in complex dynamics the new ingredient is the geometry of the critical puzzle pieces under control of geometry and complex bounds two generalized polynomial like maps which admit a topological conjugacy quasiconformal outside the filled in julia set are indeed quasiconformally conjugate the proof uses a new abstract removability type result for quasiconformal maps following ideas of heinonen and koskela and of kallunki and koskela optimized for applications in complex dynamics we prove as the first application of this new method that for even criticalities distinct from two the period two cycle of the fibonacci renormalization operator is hyperbolic with dimensional unstable manifold

renormalized differential geometry on path space structural equation curvature abstract the theory of integration in infinite dimensions is in some sense the backbone of probability theory on this backbone the stochastic calculus of variations has given rise to the flesh of differential calculus its first step is the construction at each point of the probability space of a cameron martin like tangent space in which the desired differential calculus can be developed this construction proceeds along the lines of first order differential geometry in this paper we address the following questions what could be the meaning of curvature of the probability space how and why how can curvatures be defined and computed why could a second order differential geometry be relevant to stochastic analysis we try to answer these questions for the probability space associated to the brownian motion of a compact riemannian manifold why a basicenergy identity for anticipative stochastic integralswill be obtained as a byproduct of our computation of curvature how there are essentially four bottlenecks in the development of differential geometry on wiener riemann manifolds i the difficulty of finding an atlas of local charts such that the changes of charts preserve the class of the wiener like measures together with their associated cameron martin like tangent spaces ii the difficulty of finding cylindrical approximations preserving the natural geometrical objects iii the difficulty of renormalizing the divergent series to which the summation operations of finite dimensional differential geometry give rise in the non intrinsic context of local charts iv the nonavailability of the computational procedures analogous to the local coordinates systems of the classical differential geometry in the context of path space the ito filtration provides a much richer structure than that available in the framework of an abstract wiener riemann manifold our work is a systematic attempt to replace the machinery of local charts with amethodology of moving frames in our context stochastic parallel transport provides a canonical moving frame on the path space the concept of a cylindrical approximation has to be reshaped in our new situation into some geometric limit theorems establishing that the riemannian geometric objects of the cylindrical approximations induce by a limiting procedure geometric objects on the path space those limit theorems are reminiscent of the classical theorems which say that a stratonovich sde is the limit of an appropriate sequence of ode the canonical coordinate system provided by the moving frame will make it possible to proceed to the needed renormalizations byintrinsicstochastic integrals in this context the anticipative stochastic integral theory of nualart and pardoux will play a decisive role finally the moving frame will provide aneffective algorithm of computation for this differential geometry in infinite dimensions in our study we encounter a new type of renormalization thehypoelliptic renormalization which corresponds to the fact that the bracket of smooth vector fields taking their values in the cameron martin space can get out of this cameron martin space this hypoelliptic problem induces the nonrenormalizability of some geometrical objects it leads also to a concept oftangent processesto probability spaces extending that based on the cameron martin theorem for tangent processes a formula of integration by parts still holds furthermore the tangent processes form a lie algebra under the bracket on the other hand tangent processes cannot be stochastically integrated this operation is well defined only for cameron martin type vector fields

a structure theorem in one dimensional dynamics on considere la classe 7b a des applications c f 0 1 0 1 telles que f 0 f 1 0 et f a un unique point critique c 0 1 si le point critique de f 7b a est non plat alors f n a pas d intervalle errant

integration by parts and quasi invariance for heat kernel measures on loop groups abstract integration by parts formulas are established both for wiener measure on the path space of a loop group and for the heat kernel measures on the loop group the wiener measure is defined to be the law of a certain loop group valued brownian motion and the heat kernel measures are time t t 0 distributions of this brownian motion a corollary of either of these integrations by parts formulas is the closability of the pre dirichlet form considered by b k driver and t lohrenz 1996 j functional anal 140 381 448 we also show that the heat kernel measures are quasi invariant under right under right and left translations by finite energy loops

the boundary of chaos for interval mappings a goal in the study of dynamics on the interval is to understand the transition to positive topological entropy there is a conjecture from the 1980s that the only route to positive topological entropy is through a cascade of period doubling bifurcations we prove this conjecture in natural families of smooth interval maps and use it to study the structure of the boundary of mappings with positive entropy in particular we show that in families of mappings with a fixed number of critical points the boundary is locally connected and for analytic mappings that it is a cellular set

quasi invariance of brownian measures on the group of circle homeomorphisms and infinite dimensional riemannian geometry abstract a cameron martin type theorem is proved for the canonical brownian motion on the group of homeomorphisms of the circle

bounds for maps of an interval with one critical point of inflection type ii recently quite a few papers appeared proving complex bounds local connectivity of julia sets absence of invariant linefields and quasisymmetric rigidity for real polynomial maps let us first discuss the unimodal case in 1990 martens ma proved that there are real bounds for a certain sequence of first return maps to suitable central intervals in 1990 sullivan showed that if the unimodal maps are real analytic infinitely renormalizable and have bounded combinatorics then their first return maps extend to the complex plane as polynomial like maps see su and also ms in 1994 the present authors proved that all real analytic unimodal maps allow such polynomial like complex extensions for the non central first return maps the proof even gave explicit numerical lower bounds for the moduli of certain annuli ls1 for the quadratic case different proofs were later provided by gs1 ly for the smooth case see ko in 1996 sands see sa improved our numerical bounds for maps of the form g x so that the schwarzian derivative of g is negative this polynomial like structure is one of the main ingredients in many recent results on smooth unimodal maps for example it is heavily used in the proof

integrability of scalar curvature and normal metric on conformally flat manifolds abstract on a manifold r n e 2 u d x 2 we say u is normal if the q curvature equation that u satisfies n 2 u q g e n u can be written as the integral form u x 1 c n r n log y x y q g y e n u y d y c in this paper we show that the integrability assumption on the negative part of the scalar curvature implies the metric is normal as an application we prove a bi lipschitz equivalence theorem for conformally flat metrics

heat kernel analysis on infinite dimensional heisenberg groups abstract we introduce a class of non commutative heisenberg like infinite dimensional lie groups based on an abstract wiener space the ricci curvature tensor for these groups is computed and shown to be bounded brownian motion and the corresponding heat kernel measures t t 0 are also studied we show that these heat kernel measures admit 1 gaussian like upper bounds 2 cameron martin type quasi invariance results 3 good l p bounds on the corresponding radon nikodym derivatives 4 integration by parts formulas and 5 logarithmic sobolev inequalities the last three results heavily rely on the boundedness of the ricci tensor

on the metric properties of multimodal interval maps and c 2 density of axiom a in this paper we shall prove that axiom a maps are dense in the space of c 2 interval maps endowed with the c 2 topology as a step of the proof we shall prove real and complex a priori bounds for first return maps to certain small neighborhoods of the critical points of real analytic multimodal interval maps with non degenerate critical points we shall also discuss rigidity for interval maps without large bounds

a proof of thurston s topological characterization of rational functions the criterion proved in this paper was stated by thurston in november 1982 thurston lectured on its proof on several occasions notably at the nsf summer conference in duluth 1983 where one of the authors jhh was present using the notes of various attendants at these lectures we have reconstructed a proof that we have made as precise as we could since this required a certain amount of work on our part we thought it might be of some use to present this proof to the reader we thank dennis sullivan for useful conversations and fritz yon haesseler and especially ben wittner for help with the writing and valuable suggestions after the first version was written clifford earle pointed out that better estimates than what we had were to be found in b

brownian measures on jordan virasoro curves associated to the weil petersson metric abstract in this paper existence of the brownian measure on jordan curves with respect to the weil petersson metric is established the step from brownian motion on the diffeomorphism group of the circle to brownian motion on jordan curves in c requires probabilistic arguments well beyond the classical theory of conformal welding due to the lacking quasi symmetry of canonical brownian motion on diff s 1 a new key step in our construction is the systematic use of a kahlerian diffusion on the space of jordan curves for which the welding functional gives rise to conformal martingales together with a douady earle type conformal extension of vector fields on the circle to the disk

smoothness of heat kernel measures on infinite dimensional heisenberg like groups abstract we study measures associated to brownian motions on infinite dimensional heisenberg like groups in particular we prove that the associated path space measure and heat kernel measure satisfy a strong definition of smoothness

density minoration of a strongly non degenerated random variable abstract we obtain a lower bound for the density of a d dimensional random variable on the wiener space under exponential moment condition of the divergence of covering vector fields