# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### Optimal strategies of time averaged optimization of cyclic motion

We consider dynamic inequalities with locally bounded derivatives on the circle and optimize the time averaged profit along admissible motions which are defined for all nonegative time. We describe the nature of optimal strategies, generic swichings between them when the problem depends additionally from a one dimensional parameter, and also the respective singularities of the averaged profit like a function of the parameter.

### Bound States in Curved Quantum Layers

We consider a nonrelativistic quantum particle constrained to a curved hard-wall layer of constant width built over a noncompact surface embedded in ${R}^{3}$. Under the assumption that the surface curvatures vanish at infinity, we find sufficient conditions which guarantee the existence of geometrically induced bound states.

### Lower bounds of the topological entropy

One major significance of the topological entropy is its strong relation to other dynamical invariants such as Lyapunov exponents, topological pressure, fractal dimension, and Hausdorff dimension, which provides our primary motivation. Almost all previous investigations of the topological entropy have been concerned with upper bounds. Exact formulas have been derived under strong smoothness assumptions only. In this talk we will give lower bounds of the topological entropy of smooth dynamical systems on Riemannian manifolds which are sharp in some cases. They are formulated in terms of the phase space dimension and of the exponential growth rates of a singular value function of the tangent map. These rates correspond to the deformation of k-volumes and can for instance be estimated in terms of Lyapunov exponents. Examples address Henon maps, linear maps, the geodesic flow on a (not necessarily compact) Riemannian manifold without conjugate points, and skew product systems.

### Elliptic tori, periodic solutions and the three body-problem

We prove, under suitable nonresonance conditions, that Hamiltonian systems with invariant elliptic tori possess periodic orbits of longer and longer period which accumulate on such tori. We apply this result to the spatial planetary three-body problem, obtaining periodic orbits accumulating on linearly stable invariant tori for such system. Also, we show that the spatial planetary three-body problem exhibits periodic orbits of any prescribed frequency, provided that the masses of the planets are small enough with respect to the inverse of the period.

### Arakelov theory of non compact Shimura varieties

In this talk I will give an introduction to Arakelov Geometry, focusing on the particular problems that appear when extending the theory to toroidal compactifications of noncompact Shimura varieties.

### Spaces of Positive Sections

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