## Tesi etd-05082007-131542 |

Thesis type

Tesi di laurea specialistica

Author

Beraldo, Dario

URN

etd-05082007-131542

Title

Geodesic flows on moduli spaces, continued fractions and Lyapunov exponents

Struttura

SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di studi

MATEMATICA

Commissione

**Relatore**Marmi, Stefano

Parole chiave

- Lyapunov spectrum
- translation surface
- geodesic flow
- teichmuller space
- renormalization
- interval exchange map
- continued fraction
- linear cocycles
- mutiplicative ergodic theorem

Data inizio appello

25/05/2007;

Consultabilità

completa

Riassunto analitico

Translation surfaces (i.e. closed Riemann surfaces with a holomorphic 1-form) arise in many problems of topology, dynamics, algebraic geometry and so on. <br>They are naturally organized into families (called strata of the moduli space), according to genus and nature of singularities: each stratum is a complex orbifold of finite dimension. Consequently, there exists a well-defined geodesic flow, called Teichmuller flow, whose (ergodic) properties are worth to be studied. This can be achieved by representing translation surfaces as suspensions over interval exchange maps, whose combinatorical properties play a crucial role, and then defining the appropriate renormalization operators (Rauzy-Veech and Zorich renormalization maps).<br>In genus one, everything is well-known: the Teichmuller flow is just the geodesic flow on the modular surface, which has much to do with continued fractions.<br>In this thesis we present some analogies between the simple case of genus one and the general case. In particular we present the Zorich cocycle as a generalization of the Gauss map, and we prove that its Lyapunov spectrum is simple (a theorem due to A. Avila, M. Viana, 2005-2006).<br>

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