• Teichmueller dynamics
• Khinchin theorem
• interval exchange transformations
• Ergodic theory

This thesis is concerned with two themes which are strictly linked with each other, and therefore will be developed in parallel.

The first one is dynamics in Teichmueller space. The Teichmueller space of a (topological, closed and orientable) surface S is defined as the set of the complex structures one can endow S with, up to isotopies. Such a space can be given a structure of geodesic metric space.

The description of this structure requires the notion of flat structures on the underlying surface, i.e. flat Riemannian metrics with conical singularities, such that a foliation in straight lines in each direction is defined. The space of all flat structures is a sort of tangent bundle to the Teichmueller space, and the geodesic flow, knows as Teichmueller flow, has a simple description in these terms. It becomes interesting from a dynamical viewpoint when projected onto the moduli space, namely the set of the complex structures up to diffeomorphisms. Invariant subspaces under the flow are called strata; we are concerned in particular with dynamics in the strata made up by translation structures, a subspecies of the flat structures.

The second theme treated in this work are interval exchange maps (i.e.m.s)i.e. injective maps of an interval which are locally a translation except at finitely many singularities. They are completely determined by providing some combinatorial data as well as the length data of the sub-intervals. If one considers an adequate leftmost portion of the considered interval, the first return map of the i.e.m. on this portion is a new i.e.m.. This yields a dynamics on the parameter space for i.e.m.s, called Rauzy dynamics.

The themes above are linked on two levels. First of all, if one fixes a translation surface, the first return map induced by the flow in vertical direction on a horizontal segment is an i.e.m.; and a `generic' i.e.m. can always be obtained this way. But a link at a higher level is possible, too: the Teichmueller flow admits a transverse section such that the return map can be interpreted as Rauzy dynamics.

Chapter 0 of the thesis is an introduction: it includes the preliminary material from the theory of dynamical systems which will be used in this work, as well as a description of the simplest case of the theory, represented by flat tori and rotations of the circumference.

In Chapter 1 Teichmueller dynamics is formally, but rapidly, introduced; whereas Chapter 2 is concerned with the formalism related to interval exchange maps and Rauzy dynamics. Moreover, it is explained how it is possible to switch from this setting to the one of translation structures, and conversely.

The first half of Chapter 3 treats, still in an extremely concise manner, classical questions related to the themes above. In particular it deals with ergodicity of i.e.m.s and of the Teichmueller and Rauzy dynamics and briefly introduces the Kontsevich-Zorich cocycle. The chapter ends with a technical discussion needed for the results tackled in the following chapters: its protagonists will be the reduced triples for an i.e.m. T, namely triples (b,a;n) where b is a singular point for $T^{-1}$, a is a singular point for T, and n is a positive integer, such that no singularities for $T^{-1},...,T^-n$ appears between $T^n(b)$ and a.

Chapter 4 thus deals with a first generalisation of a theorem of A.Ya. Khinchin, found by Luca Marchese (2010). The Khinchin theorem in its classical formulation states a condition for a Diophantine inequality to have finitely, or infinitely many, solutions. Its generalisation to i.e.m.s states:
Let f(n) be a positive, decreasing sequence. We are concerned with the quantity of solutions (b,a;n) to the condition $|T^n(a)-b|<f(n)$ for a fixed i.e.m. T, where b is a singular point of $T^-1$, and a is a singular point for T.
If the sequence f(n) has a finite sum, then solutions are finitely many for almost any T; if nf(n) is still a decreasing sequence, with infinite sum, then solutions are infinitely many for almost any T.

This result will be partially proved. It is interesting not only as a property of singularities of an i.e.m., but also because it yields a weaker version of a theorem of Jon Chaika, which states a similar property for generic points.

Chapter 5 is again about translation surfaces. The theorem above is restated as a property of lengths of connections, namely segments connecting two singular points on a translation surface. Hence it is possible to gain another result of Chaika, which gives a property of 'strong recurrence' of foliations. And, eventually, this restatement of the generalised Khinchin theorem yields a logarithm law for the Teichmueller flow:
Let X be a translation surface, and let Sys(X) be the minimum length of a connection of X. Denote $g^t$ the Teichmueller flow. Then, for almost any X, it holds that $\limsup [-\log (Sys(g^t(X))]/(\log t)=1/2$.