The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In the paper we show that a version of this conjecture is true for tables bounded by small perturbations of ellipses of small eccentricity.

We find a normal form which describes the high energy dynamics of a class of piecewise smooth Fermi-Ulam ping pong models. Depending on the value of a single real parameter, the dynamics can be either hyperbolic or elliptic. In the first case, we prove that the set of orbits undergoing Fermi acceleration has zero measure but full Hausdorff dimension. We also show that for almost every orbit, the energy eventually falls below a fixed threshold. In the second case, we prove that, generically, we have stable periodic orbits for arbitrarily high energies and that the set of Fermi accelerating orbits may have infinite measure.

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We show that any sufficiently (finitely) smooth Z 2symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid, i.e. all deformations among domains in the same class which preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak (see [22]).2 Remarkably, Sunada (see [23]) exhibits isospectral sets (i.e. sets of isospectral manifolds) of arbitrarily large cardinality.3 Results of this kind are usually referred to as infinitesimal spectral rigidity.

We consider the static wall approximation to the dynamics of a particle bouncing on a periodically oscillating infinitely heavy plate while subject to a potential force. We assume the case of a potential given by a power of the particle's height and sinusoidal motions of the plate. We find that for powers smaller than 1 the set of escaping orbits has full Hausdorff dimension for all motions and obtain existence of elliptic island of period 2 for arbitrarily high energies for a full-measure set of motions. Moreover we obtain conditions on the potential to ensure that the total (Lebesgue) measure of elliptic islands of period 2 is either finite or infinite. * jacopods@math.umd.edu Ij 1 j2 Figure 1: Choosing branches of F k for k = 2.Proposition III.1. If ∀ n ∈ N m n < Cm n , then the Hausdorff dimension of J is 1.