This article is devoted to the study of the translation flow on self-similar tilings associated with a substitution of Pisot type. We construct a geometric representation and give necessary and sufficient conditions for the flow to have pure discrete spectrum. As an application we demonstrate that, for certain beta-shifts, the natural extension is naturally isomorphic to a toral automorphism.

Let $G$ be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $Y$. A loop $h:S^1\to G$ is called strictly ergodic if for some irrational number the associated skew product map $T:S^1\times Y\to S^1\times Y$ defined by $T(t,y)=(t+\alpha ;h(t\left)y\right)$ is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of $G$ can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected...

Let A be a topologically transitive invariant subset of an expanding piecewise monotonic map on [0,1] with the Darboux property. We investigate existence and uniqueness of conformal measures on A and relate Hausdorff and conformal measures on A to each other.

We present a proof of Herman’s Last Geometric Theorem asserting that if $F$ is a smooth diffeomorphism of the annulus having the intersection property, then any given $F$-invariant smooth curve on which the rotation number of $F$ is Diophantine is accumulated by a positive measure set of smooth invariant curves on which $F$ is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable...

We present a very quick and easy proof of the classical Stepanov-Hopf ratio ergodic theorem, deriving it from Birkhoff's ergodic theorem by a simple inducing argument.

For any 1-1 measure preserving map T of a probability space we can form the [T,Id] and $[T,T^{-1}]$ automorphisms as well as the corresponding endomorphisms and decreasing sequence of σ-algebras. In this paper we show that if T has zero entropy and the [T,Id] automorphism is isomorphic to a Bernoulli shift then the decreasing sequence of σ-algebras generated by the [T,Id] endomorphism is standard. We also show that if T has zero entropy and the [T²,Id] automorphism is isomorphic to a Bernoulli shift then the...

We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Üstünel and Zakai (Probab. Theory Related Fields103 (1995) 409–429) on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples...

We consider the set of $C^1$ expanding maps of the circle which have a unique absolutely continuous invariant probability measure whose density is unbounded, and show that this set is dense in the space of $C^1$ expanding maps with the $C^1$ topology. This is in contrast with results for $C^2$ or $C^{1+\epsilon }$ maps, where the invariant densities can be shown to be continuous.

Let $\beta >1$ be a non-integer. We consider expansions of the form ${\sum }_{i=1}^{\infty }{d}_i/{\beta }^i$, where the digits ${\left(d_i\right)}_{i\ge 1}$ are generated by means of a Borel map $K_{\beta }$ defined on ${\{0,1\}}^{\mathbb{N}}\times [0,\lfloor \beta \rfloor \left(\beta -1\right)]$. We show existence and uniqueness of a $K_{\beta }$-invariant probability measure, absolutely continuous with respect to $m_p\otimes \lambda$, where $m_p$ is the Bernoulli measure on ${\{0,1\}}^{\mathbb{N}}$ with parameter $p$ ($0<p<1$) and $\lambda$ is the normalized Lebesgue measure on $[0,\lfloor \beta \rfloor (\beta -1\left)\right]$. Furthermore, this measure is of the form $m_p\otimes {\mu }_{\beta ,p}$, where ${\mu }_{\beta ,p}$ is equivalent to $\lambda$. We prove that the measure of maximal entropy and $m_p\otimes \lambda$ are mutually singular. In...