Rauzy classes form a partition of the set of irreducible permutations. They were introduced as part of a renormalization algorithm for interval exchange transformations. We prove an explicit formula for the cardinality of each Rauzy class. Our proof uses a geometric interpretation of permutations and Rauzy classes in terms of translation surfaces and moduli spaces.

Using the Perron-Frobenius operator we establish a new functional central limit theorem for non-invertible measure preserving maps that are not necessarily ergodic. We apply the result to asymptotically periodic transformations and give a specific example using the tent map.

We study certain symmetries that arise when automorphisms S and T defined on a Lebesgue probability space (X, ℱ, μ) satisfy the equation $ST=T^{-1}S$. In an earlier paper [6] it was shown that this puts certain constraints on the spectrum of T. Here we show that it also forces constraints on the spectrum of $S^2$. In particular, $S^2$ has to have a multiplicity function which only takes even values on the orthogonal complement of the subspace $f\in L^2(X,ℱ,\mu ):f\left(T^{2}x\right)=f\left(x\right)$. For S and T ergodic satisfying this equation further constraints arise,...

We present a spectral theory for a class of operators satisfying a weak “Doeblin–Fortet” condition and apply it to a class of transition operators. This gives the convergence of the series ${\sum }_{k\ge 0}{k}^r{P}^{k}f$, $r\in \mathbb{N}$, under some regularity assumptions and implies the central limit theorem with a rate in $n^{-\frac{1}{2}}$ for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.

We present a spectral theory for a class of operators satisfying a weak “Doeblin–Fortet" condition and apply it to a class of transition operators. This gives the convergence of the series ∑k≥0krPkƒ, $r\in \mathbb{N}$, under some regularity assumptions and implies the central limit theorem with a rate in $n^{-\frac{1}{2}}$ for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.

We prove the norm convergence of multiple ergodic averages along cubes for several commuting transformations, and derive corresponding combinatorial results. The method we use relies primarily on the "magic extension" established recently by B. Host.