FIRST JOINT AMS-UMI MEETING PISA, JUNE 12-16 2002 SPECIAL SESSION ON DYNAMICAL SYSTEMS SCIENTIFIC COMMITTEE: A. Giorgilli (University of Milano-Bicocca) antonio@matapp.unimib.it S. Marmi (Scuola Normale Superiore, Pisa) marmi@sns.it J. Mather (Princeton) jnm@math.princeton.edu INVITED 50 MINUTES TALKS: From the U.S.: G. Forni, V. Kaloshin, J. Franks, R. Moeckel From Italy: D. Bambusi, G. Benettin, A. Berretti, L. Chierchia, C. Liverani INVITED 25 MINUTES TALKS: Y. Cheung, R.O. Ruggiero ABSTRACTS: Dario BAMBUSI, University of Milan Title: BIRKHOFF NORMAL FORM FOR SOME NONLINEAR PDEs Abstract: We consider the problem of extending to PDEs Birkhoff normal form theorem on Hamiltonian systems close to elliptic equilibria. As a model problem we take the nonlinear wave equation $$ u_{tt}-u_{xx}+g(x,u)=0\ , $$ with Dirichlet boundary conditions on $[0,\pi]$; $g$ is an analytic skewsymmetric function which vanishes for $u=0$ and is periodic with period $2\pi$ in the $x$ variable. We prove, under a nonresonance condition which is fulfilled for most $g$, that for any integer $M$ there exists a canonical transformation that puts the Hamiltonian in Birkhoff normal form up to a reminder of order $M$. The canonical transformation is well defined in a neighbourhood of the origin of a Sobolev type phase space of sufficiently high order. As a consequence a very precise qualitative description of the dynamics is obtained. The technique of proof is applicable to quite general equations in one space dimension. Giancarlo BENETTIN, University of Padova Title: REGULAR AND CHAOTIC MOTIONS OF THE EULER PERTURBED RIGID BODY Abstract: The Euler rigid body (the rigid body with a fixed point in absence of external torques) is a "degenerate" integrable dynamical system: it has three degrees of freedom, but only two of the actions enter the Hamiltonian. For the perturbed system (small potential, or equivalently fast rotating body), Nekhoroshev theory works but, at variance with the nondegenerate case, large scale chaotic motions are possible in connection with resonant initial data. The anomaly disappears for motions sufficiently close to proper rotations, which in any case turn out to be stable for times exponentially long in the perturbation parameter. Alberto BERRETTI, University of Rome Tor Vergata Title: PERIODIC AND QUASIPERIODIC ORBITS FOR THE STANDARD MAP Abstract: We consider both periodic and quasi-periodic solutions for the standard map, and we study the corresponding conjugating functions, i.e. the functions conjugating the motions to trivial rotations. We compare the invariant curves with rotation numbers $\omega$ satisfying the Bryuno condition and the sequences of periodic orbits with rotation numbers given by their convergents $\omega_{N}=p_{N}/q_{N}$. We prove the following results for N tending to infinity: (1) for rotation numbers $\omega_{N}$ we study the radius of convergence of the conjugating functions and we find lower bounds on them, which tend to a limit which is a lower bound on the corresponding quantity for $\omega$; (2) the periodic orbits consist of points which are more and more close to the invariant curve with rotation number $\omega$; (3) such orbits lie on analytical curves which tend uniformly to the invariant curve. Luigi CHIERCHIA, University of Rome III Title: THE STABILITY PROBLEM FOR PROPERLY DEGENERATE SYSTEMS WITH 2 DEGREES OF FREEDOM Abstract: Nearly-integrable, properly degenerate Hamiltonian systems with 2 degrees of freedom are considered. It is proven that, under general condition, total stability holds. Relations with Celestial Mechanics are discussed. Giovanni FORNI, Northwestern University Title: QUANTITATIVE EQUIDISTRIBUTION OF HOROCYCLES ON NON-COMPACT HYPERBOLIC SURFACES Abstract: It is known by a result of P.Sarnak that periodic horocycles on a finite volume non-compact hyperbolic surface equidistribute under the action of the geodesic flow with a polynomial asymptotics whose coefficients are given by invariant distributions. In a joint work with L. Flaminio we have generalized this result to arbitrary horocycle arcs. In the case of compact hyperbolic surfaces the same method yields precise asymptotic formulas for the ergodic averages of the horocycle flow. John FRANKS, Northwestern University Title: NILPOTENT GROUPS ACTING ON S^1 AND R Abstract: It is an old result that any finitely generated torsion-free nilpotent group can act by homeomorphisms on the line or $S^1.$ By contrast Plante and Thurston proved that every nilpotent subgroup of $\Diff^2(S^1)$ is Abelian. One of our main results is that $\Diff^1(S^1)$ contains every finitely-generated, torsion-free nilpotent group. We also show that some, but not all, nilpotent groups are isomorphic to subgroups of $\Diff^\infty(R)$. This talk represents joint work with Benson Farb. Vadim KALOSHIN, M.I.T. Title: SUPEREXPONENTIAL GROWTH OF THE NUMBER OF PERIODIC ORBITS IS GENERIC, BUT OF MEASURE ZERO Abstract: Let M be a compact manifold, and f a Cr-smooth map from M to M. Define the number of isolated periodic points of period n by Pn(f) = #{isolated points x in M such that fn(x)=x}. Artin and Mazur (1965), using Nash's approximations, proved that for a dense set of Cr-smooth maps, the number of periodic points Pn(f) grows at most exponentially fast, i.e. for some C > 0 (*) Pn(f) <= exp(C n), for any positive integer n. For more than thirty years there was no other proof known. During the talk we shall give an elementary proof of this result due to the author. Various questions about relation of property (*), the dynamical zeta-function (Smale 1967), and topological entropy (Bowen 1978), and other for generic diffeomorphisms were raised. It turns out that Artin-Mazur property (*) is not generic, which was shown by the author using a theorem of Gonchenko, Shilnikov, and Turaev. Therefore, all the above questions have negative answer. Other aspects of the problem will be discussed. Carlangelo LIVERANI, University of Rome Tor Vergata Title: ON CONTACT FLOWS Abstract: I will discuss a new approach to studying the decay of correlations for flows. In the case of contact flows some new results are obtained. Rick MOECKEL, University of Minnesota Title: ISOLATING BLOCKS FOR THE PLANAR THREE-BODY PROBLEM Abstract: The planar three-body problem has an invariant three-sphere near the collinear central configuration. I will describe how to construct an explicit isolating block around it similar to the one employed by Conley and Easton in the restricted three-body problem. 25 MINUTES INVITED TALKS Yitwah CHEUNG, Northwestern University Title: Hausdorff dimension of the union of divergent trajectories Abstract: It is known by the work of Dani '86 and Kleinboch- Margulis '96 that the union of bounded trajectories of a partially hyperbolic homogeneous flow is a set of full Hausdorff dimension. This contrasts the situation of a (quasi)unipotent flow where the Hausdorff codimension is necessarily at least one, by virtue of Ratner theory. We consider analogous questions for divergent trajectories. For (quasi)unipotent flows it is known that divergent trajectories do not exist (Margulis '74, Dani '84.) For a partially hyperbolic flow, however, Dani '85 showed that, in the case of rank 1, all divergent trajectories are degenerate (and hence form a set of codimension at least one) while in the higher rank case, he showed that nondegenerate divergent trajectories always exist. (All statements above will be made precise in the talk.) We compute the Hausdorff dimension of the union of divergent trajectories in a special case. Let G be the product of the Lie group SL(2,R) with itself n times and Gamma the product of SL(2,Z) with itself n times, considered as a lattice in G. Let X be the space of right cosets of Gamma and consider the action of g_t=diag(exp(t),exp(-t)) on X by right multiplication in each factor. Let D be the subset of X consisting of those points whose orbit leaves every compact set. Our main result asserts that the Hausdorff codimension of D is 1/2, (if n>1). Rafael Oswaldo RUGGIERO, PUC University, Rio de Janeiro Title: On variational and topological properties of C^1 invariant Lagrangian tori (joint work with M. J. Dias Carneiro) Abstract: We study C^1, invariant, two-dimensional tori of constant energy without singularities of the Euler-Lagrange flow of a convex, superlinear Lagrangian defined in the tangent space of a closed surface. Assume that the energy level of such a torus is regular (the Euler-Lagrange flow is never vertical). Then we show that the torus is a graph of the canonical projection if and only if it is minimizing (i.e., the lift in the universal covering of each orbit of the Euler-Lagrange flow in the torus is a global, time-dependent minimizer of the lifted Lagrangian action). This result wa known for C^3 invariant, Lagrangian tori of Finsler metrics, which implied the statement for C^3 invariant Lagrangian tori of unimodal Lagrangians with high energy level. We also show that graphs have energy E>=c(L), where c(L) is Mane's critical value; and that the presence of Reeb components in such tori implies that the energy is c(L). Moreover, if we assume that the Lagrangian is unimodal, we show that Lagrangian invariant tori with no singularities have no Reeb components, regardless of the energy level.