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Hamiltonian loops from the ergodic point of view

Leonid Polterovich (1999)

Journal of the European Mathematical Society

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y . A loop h : S 1 → G is called strictly ergodic if for some irrational number the associated skew product map T : S 1 × Y → S 1 × Y defined by T ( t , y ) = ( t + α ; h ( t ) y ) 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...

Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval II

Franz Hofbauer (1993)

Studia Mathematica

We construct examples of expanding piecewise monotonic maps on the interval which have a closed topologically transitive invariant subset A with Darboux property, Hausdorff dimension d ∈ (0,1) and zero d-dimensional Hausdorff measure. This shows that the results about Hausdorff and conformal measures proved in the first part of this paper are in some sense best possible.

Herman’s last geometric theorem

Bassam Fayad, Raphaël Krikorian (2009)

Annales scientifiques de l'École Normale Supérieure

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...

Homotopy and dynamics for homeomorphisms of solenoids and Knaster continua

Jarosław Kwapisz (2001)

Fundamenta Mathematicae

We describe the homotopy classes of self-homeomorphisms of solenoids and Knaster continua. In particular, we demonstrate that homeomorphisms within one homotopy class have the same (explicitly given) topological entropy and that they are actually semi-conjugate to an algebraic homeomorphism in the case when the entropy is positive.

Hopf's ratio ergodic theorem by inducing

Roland Zweimüller (2004)

Colloquium Mathematicae

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.

Identification of periodic and cyclic fractional stable motions

Vladas Pipiras, Murad S. Taqqu (2008)

Annales de l'I.H.P. Probabilités et statistiques

We consider an important subclass of self-similar, non-gaussian stable processes with stationary increments known as self-similar stable mixed moving averages. As previously shown by the authors, following the seminal approach of Jan Rosiński, these processes can be related to nonsingular flows through their minimal representations. Different types of flows give rise to different classes of self-similar mixed moving averages, and to corresponding general decompositions of these processes. Self-similar...

If the [T,Id] automorphism is Bernoulli then the [T,Id] endomorphism is standard

Christopher Hoffman, Daniel Rudolph (2003)

Studia Mathematica

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...

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