The Teichmüller geodesic flow is the flow obtained by quasiconformal deformation of Riemann surface structures. The goal of this lecture is to show the strong connection between the geometry of the Hodge bundle (a vector bundle over the moduli space of Riemann surfaces) and the dynamics of the Teichmüller geodesic flow. In particular, we shall provide geometric criterions (based on the variational formulas derived by G. Forni) to detect some special orbits (“totally degenerate”) of the Teichmüller...

It is shown that Markov operators with equicontinuous dual operators which overlap supports have at most one invariant measure. In this way we extend the well known result proved for Markov operators with the strong Feller property by R. Z. Khas'minski.

En théorie des groupes, le théorème de Kurosh est un résultat de structure concernant les sous-groupes d’un produit libre de groupes. Le théorème principal de cet article est un résultat analogue dans le cadre des relations d’équivalence boréliennes à classes dénombrables, que nous démontrons en développant une théorie de Bass-Serre dans ce cadre particulier.

For invertible transformations we introduce various notions of topological entropy. For compact invariant sets these notions are all the same and equal the usual topological entropy. We show that for non-invariant sets these notions are different. They can be used to detect the direction in time in which the system evolves to highest complexity.

We shed some light on the inter-connections between different characterizations leading to the classical Meixner family. This allows us to give free analogs of both Sheffer's and Al-Salam and Chihara's characterizations in the classical case by the use of the free derivative operator. The paper closes with a discussion of the q-deformed case, |q| < 1.

A technique is presented for multiplexing two ergodic measure preserving transformations together to derive a third limiting transformation. This technique is used to settle a question regarding rigidity sequences of weak mixing transformations. Namely, given any rigidity sequence for an ergodic measure preserving transformation, there exists a weak mixing transformation which is rigid along the same sequence. This establishes a wide range of rigidity sequences for weakly mixing dynamical systems....

Let $U₁,...,U_d$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $\mathbb{Z}{₊}^d$ and the associated collection of rectangular parallelepipeds in ${ℝ}^d$ with sides parallel to the axes and dimensions of the form $n₁\times \cdots \times n_d$ with $(n₁,...,n_d)\in \Gamma .$ The associated multiparameter geometric and ergodic maximal operators $M_{}$ and $M_{\Gamma }$ are defined respectively on $L¹\left({ℝ}^d\right)$ and L¹(Ω) by $M_{}g\left(x\right)=su{p}_{x\in R\in }1/\left|R\right|{\int }_R\left|g\left(y\right)\right|dy$ and $M_{\Gamma }f\left(\omega \right)=su{p}_{(n₁,...,n_d)\in \Gamma }1/n₁\cdots n_d{\sum }_{j₁=0}^{n₁-1}\cdots {\sum }_{j_d=0}^{n_d-1}\left|f\left(U{₁}^{j₁}\cdots U_d^{j_d}\omega \right)\right|$. Given a Young function Φ, it is shown that $M_{}$ satisfies the weak type estimate $|x\in {ℝ}^d:M_{}g\left(x\right)>\alpha |\le C_{}{\int }_{{ℝ}^d}\Phi (c_{}\left|g\right|/\alpha )$ for...

Let I be a compact real interval and let f:I → I be continuous. We describe an interval analogy of the irrational circle rotation that occurs as a subsystem of the dynamical system (I,f)-we call it an irrational twist system. Using a coding we show that any irrational twist system is strictly ergodic. We also prove that irrational twist systems exist as subsystems of a large class of systems (I,f) having a cycle of odd period greater than one.