# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### 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.

### Dynamics of polynomial diffeomorphisms of ${ℂ}^{n}$

In this talk we discuss the dynamics of regular polynomial diffeomorphisms of ${ℂ}^{n}$. These maps provide a natural generalization of the complex Hénon map to higher dimensions. Given such a map $f$ we construct a filtration of ${ℂ}^{n}$ which allows us to describe the invariant sets of dynamical interest (i.e., the Julia sets) of $f$. In particular, if $f$ is a hyperbolic map we obtain a complete classification of its orbits. We also provide estimates for the Hausdorff and box dimensions of the Julia sets of $f$. This is joint work with R. Shafikov.

### The inverse problem of linear age-structured population dynamics

We consider the problem of determining the individual survival and reproduction functions (or birth and death rates) from data on total population size and cumulative number of births in a linear age-structured population model. We give conditions that guarantee that this inverse problem has a unique solution. The proof uses a variant of the Müntz-Szasz theorem.
Seminário em colaboração com o Seminário de Matemática Aplicada e Análise Numérica.

### Efeito do crescimento acelerado no crescimento de redes

Foi observado que na maioria da redes reais o número médio de ligações por nodo cresce com o tempo. A Internet, WWW, redes de colaborações e muitas outras são exemplos que revelam este tipo de comportamento. Designamos esse tipo de crescimento por crescimento acelerado. Mostrou-se que o efeito desse crescimento acelerado influencia a distribuição de número de ligações e pode determinar a estrutura da rede. Para além de uma introdução ao tema das redes discutirei as consequências gerais da aceleração assim como as suas consequências em exemplos simples. Em particular, mostrarei que o crescimento acelerado explica bem a estrutura da Rede de Palavras (a rede formada pelas palavras da linguagem Humana).

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