# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### Some remarks on evolutions

We present some result concerning evolutions of Noetherian local rings of characteristic zero using Andre-Quillen homology of commutative rings.

### Residue Theorems and Lie Algebroid Invariants

The recent work of Harvey and Lawson has resulted in a family of residue theorems relating the singular sets (viewed as rectifiable currents) of a vector bundle map to the Chern-Weil characteristic forms of the bundles. Many of the classical singularity theorems (e.g., the Argument Principle, the Poincare-Hopf Index Theorem, and the Riemann-Roch Theorem) are special cases of these more general formulae.

In this talk, I will review briefly the techniques used by Harvey and Lawson and give an account of current work to extend these to the study of Lie algebroids.

### Poincaré recurrence, geodesic flows, hyperbolic geometry

Today, we are arguably in possession of a satisfactory approach to a quantitative description of recurrence. As one knows well: Poincaré only established that orbits return infinitely often, not how often or when they return.

This will be a colloquium type exposition of recent results that shed new light on the subject, also discussing joint work with Benoît Saussol, Christian Wolf, and Gianluigi Del Magno. The talk will include the view of the (maybe classical) hyperbolic geometer and briefly some consequences for number theory. No prerequisites from dynamical systems or ergodic theory will be necessary.

### 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.

### Coupled oscillators on a circle

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