center manifolds and contractions on a scale of banach spaces we give a new proof for the existence of a ck center manifold at a nonhyperbolic equilibrium point of a finite dimensional vector field of class ck the problem is reduced to a fixed point problem on a scale of banach spaces these banach spaces consist of mappings with a certain maximal exponential growth at infinity we give conditions under which there is a unique fixed point depending differentiably on the parameters the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of banach spaces

maximal dissipativity of kolmogorov operators with cahn hilliard type drift term we prove the existence of invariant measures for kolmogorov operators lf associated with semilinear stochastic partial differential equations with cahn hilliard type drift term based on gradient estimates on the pseudo resolvent associated with lf and a priori estimates for the moments of we prove maximal dissipativity of lf in the space l1

on the completion of the mapping class group of genus two abstract in this paper we will study the lie algebra of the prounipotent radical of the relative completion of the mapping class group of genus two in particular we will partially determine a minimal presentation of the lie algebra by determining the generators and bounding the degree of the relations of it

ck invariant manifolds for maps of banach spaces abstract in this work we give a unified proof for the existence under appropriate gap conditions of the standard invariant manifolds for a ck map of a banach space near a fixed point where k 1 is an integer

existence uniqueness and invariant measures for stochastic semilinear equations on hilbert spaces a class of stochastic evolution equations with additive noise and weakly continuous drift is considered first regularity properties of the corresponding ornstein uhlenbeck transition semigrouprt are obtained we show thatrt is a compactc0 semigroup in all sobolev spaceswn p which are built on its invariant measure then we show the existence uniqueness compactness and smoothing properties of the transition semigroup for semilinear equations inlp spaces and spacesw1 p as a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to it is shown also that the density of this measure with respect to is inlp for allp 1

perturbed homoclinic solutions in reversible 1 1 resonance vector fields we consider a smooth reversible vector field in r 4 such that the origin is a fixed point the differential at the origin has two double pure imaginary eigenvalues iq for the critical value 0 of the parameter we show by a normal form analysis that the vector field can be approximated by an integrable field in r 4 for which we know all solutions specially interesting ones are the homoclinics to 0 and homoclinics to periodic solutions depending on the sign of a leading nonlinear coefficient we prove in particular the persistence of these homoclinics for the full vector field

center manifolds without a phase space we establish center manifold theorems that allow one to study the bifurcation of small solutions from a trivial state in systems of functional equations posed on the real line the class of equations includes most importantly nonlinear equations with nonlocal coupling through convolution operators as they arise in the description of spatially extended dynamics in neuroscience these systems possess a natural spatial translation symmetry but local existence or uniqueness theorems for a spatial evolution associated with this spatial shift or even a well motivated choice of phase space for the induced dynamics do not seem to be available due to the infinite range forward and backward coupling through nonlocal convolution operators we perform a reduction relying entirely on functional analytic methods despite the nonlocal nature of the problem we do recover a local differential equation describing the dynamics on the set of small bounded solutions exploiting that the translation invariance of the original problem induces a flow action on the center manifold we apply our reduction procedure to problems in mathematical neuroscience illustrating in particular the new type of algebra necessary for the computation of taylor jets of reduced vector fields

conjugate dynamics on center manifolds for stochastic partial differential equations abstract in this paper we prove that for a random differential equation with the driving noise constructed from a q wiener process and the wiener shift there exists a local center unstable stable center unstable center stable manifold and a local stable foliation an unstable foliation on the center unstable manifold and a stable foliation on the center stable manifold the smoothness of which depend on the vector fields of the equation also we show that any two arbitrarily local center manifolds constructed as above are conjugate

strong feller property and irreducibility for diffusions on hilbert spaces it is shown that the transition semigroup p t t 0 corresponding to a nonlinear stochastic evolution equation is strong feller and irreducible provided the nonlinearities are lipschitz continuous and the diffusion term is nondegenerate this result ensures the uniqueness of the invariant measure for p t t 0

propagating fronts and the center manifold theorem we prove the existence of propagating front solutions for the swift hohenberg equation sh using the center manifold theorem we reduce the problem to a two dimensional system of ordinary differential equations they describe stationary solutions and front solutions of the partial differential equation sh we identify the well known amplitude equation as the lowest order approximation to the equation of motion on the center manifold

on the sections of universal hyperelliptic curves abstract in this paper we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves c h g n k h g n k when g 3 and the image of the l adic cyclotomic character g k z is infinite furthermore we will study the nonabelian phenomena associated to the universal hyperelliptic curves for example we will show that the section conjecture holds for c h g k h g k and the unipotent analogue of the conjecture holds for the pointed cases this work is an extension of hain s original work on the rational points of universal curves to the hyperelliptic case

existence and uniqueness of invariant measures for a class of transition semigroups on hilbert spaces we prove a smoothing property and the irreducibility of transition semigroups corresponding to a class of semilinear stochastic equations on a separable hilbert space h existence and uniqueness of invariant measures are discussed as well

invariant manifolds for volterra integral equations of convolution type abstract in this paper we develop some elements of a qualitative theory for nonlinear volterra integral equations of convolution type our starting point is a local semiflow associated with the equation and acting on a space of compactly supported forcing functions within that framework we discuss the variation of constants formula the saddle point property the center manifold and hopf bifurcation some equations from population biology get special attention

invariant manifolds in singular perturbation problems for ordinary differential equations based on fenichel s geometric idea invariant manifold theory is applied to singular perturbation problems this approach clarifies the nature of outer and inner solutions a specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds a method to approximate the connections is also presented

bounded solutions to singularly perturbed systems of o d e abstract we consider existence and stability properties of bounded solutions to singular systems of o d e the method we suggest considers the concept of singular maps between scales of banach spaces and their properties this theory is then applied to obtain existence and transversality of bounded solutions to singular systems of o d e

smooth convergence of random center manifolds for spdes in varying phase spaces abstract in this paper we study the limiting behavior of center manifolds for a class of singularly perturbed stochastic partial differential equations in terms of the phase spaces we first prove the existence and smoothness of random center manifolds for these equations then we establish the smooth convergence of these random center manifolds as the phase spaces approach to their singular limit

center manifolds for smooth invariant manifolds this is the published version also available here http dx doi org 10 1090 s0002 9947 00 02443 0 first published in trans amer math soc in 2000 published by the american mathematical society

sub stable and weak stable manifolds associated with finitely non resonant spectral subspaces in this work we study c k delta 2 leq k leq infty 0 leq delta leq 1 maps of a banach space near a fixed point we show the existence and uniqueness of a class of c k delta local invariant sub manifolds of the stable manifold which correspond to a spectral subspace satisfying a finite non resonance condition of order l leq k and an overriding condition of order l leq k condition 3 of theorem 1 we study the dependence of these invariant manifolds on a parameter that lies in a banach space we also show that a c k delta local weak stable manifold that satisfies these two conditions is unique in the class of c k delta maps the uniqueness is due to the fact that our method does not require a cut off function an infinite dimensional banach space does not always admit smooth cut off functions

existence of pulses in excitable media with nonlocal coupling abstract we prove the existence of fast traveling pulse solutions in excitable media with non local coupling existence results had been known until now in the case of local diffusive coupling and in the case of a discrete medium with finite range non local coupling our approach replaces methods from geometric singular perturbation theory that had been crucial in previous existence proofs by a pde oriented approach relying on exponential weights fredholm theory and commutator estimates

improved moment estimates for invariant measures of semilinear diffusions in hilbert spaces and applications we study regularity properties for invariant measures of semilinear diffusions in a separable hilbert space based on a pathwise estimate for the underlying stochastic convolution we prove a priori estimates on such invariant measures as an application we combine such estimates with a new technique to prove the l1 uniqueness of the induced kolmogorov operator defined on a space of cylindrical functions finally examples of stochastic burgers equations and thin film growth models are given to illustrate our abstract result

rational points of universal curves in positive characteristics rational points of universal curves in positive characteristics by tatsunari watanabe department of mathematics duke university

properties of center manifolds the center manifold has a number of puzzling properties associated with the basic questions of existence uniqueness differentiability and analyticity which may cloud its profitable application in e g bifurcation theory this paper aims to deal with some of these subtle properties regarding existence and uniqueness it is shown that the cut off function appear ing in the usual existence proof is responsible for the selection of a single center manifold thereby hiding the inherent nonuniqueness conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection this intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium in the case of nonuniqueness the difference between any two center manifolds will be less than el exp e2x l with el and e2 constants which explains why the formal taylor expansions of different center manifolds are the same while the expansions do not converge the differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost center manifolds are usually not analytic however an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant using this manifold a new and very simple proof of the lyapunov subcenter theorem is given