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Faithful zero-dimensional principal extensions

Tomasz Downarowicz, Dawid Huczek (2012)

Studia Mathematica

We prove that every topological dynamical system (X,T) has a faithful zero-dimensional principal extension, i.e. a zero-dimensional extension (Y,S) such that for every S-invariant measure ν on Y the conditional entropy h(ν | X) is zero, and, in addition, every invariant measure on X has exactly one preimage on Y. This is a strengthening of the authors' result in Acta Appl. Math. [to appear] (where the extension was principal, but not necessarily faithful).

Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents

Anton Zorich (1996)

Annales de l'institut Fourier

We construct a map on the space of interval exchange transformations, which generalizes the classical map on the interval, related to continued fraction expansion. This map is based on Rauzy induction, but unlike its relative kown up to now, the map is ergodic with respect to some finite absolutely continuous measure on the space of interval exchange transformations. We present the prescription for calculation of this measure based on technique developed by W. Veech for Rauzy induction.We study...

Generalized interval exchanges and the 2–3 conjecture

Shmuel Friedland, Benjamin Weiss (2005)

Open Mathematics

We introduce the notion of a generalized interval exchange φ 𝒜 induced by a measurable k-partition 𝒜 = A 1 , . . . , A k of [0,1). φ 𝒜 can be viewed as the corresponding restriction of a nondecreasing function f 𝒜 on ℝ with f 𝒜 ( 0 ) = 0 , f 𝒜 ( k ) = 1 . A is called λ-dense if λ(A i∩(a, b))>0 for each i and any 0≤ a< b≤1. We show that the 2–3 Furstenberg conjecture is invalid if and only if there are 2 and 3 λ-dense partitions A and B of [0,1), such that f 𝒜 ∘ f ℬ = f ℬ ∘ f 𝒜 . We give necessary and sufficient conditions for this equality to hold. We show that...

Générateurs indépendants pour les systèmes d'isométries de dimension un

Damien Gaboriau (1997)

Annales de l'institut Fourier

Un système fini d’isométries partielles de R est dit à générateurs indépendants si les composés non triviaux fixent au plus un point. On décrit un procédé simple et naturel pour obtenir des générateurs indépendants, sans modifier les orbites, pour tout système sans composante minimale homogène : en prenant la restriction de chaque générateur à un certain sous-intervalle de son domaine. Un système avec une composante minimale homogène ne possède pas de générateurs indépendants.

Generic points in the cartesian powers of the Morse dynamical system

Emmanuel Lesigne, Anthony Quas, Máté Wierdl (2003)

Bulletin de la Société Mathématique de France

The symbolic dynamical system associated with the Morse sequence is strictly ergodic. We describe some topological and metrical properties of the Cartesian powers of this system, and some of its other self-joinings. Among other things, we show that non generic points appear in the fourth power of the system, but not in lower powers. We exhibit various examples and counterexamples related to the property of weak disjointness of measure preserving dynamical systems.

Generic smooth cocycles of degree zero over irrational rotations

A. Iwanik (1995)

Studia Mathematica

If a rotation α of has unbounded partial quotients then “most” of its skew-product diffeomorphic extensions to the 2-torus × defined by C 1 cocycles of topological degree zero enjoy nontrivial ergodic properties. In fact they admit a cyclic approximation with speed o(1/n) and have nondiscrete (simple) spectrum. Similar results are obtained for C r cocycles if α admits a sufficiently good approximation by rationals. For a.e. α and generic C 1 cocycles the speed can be improved to o(1/(nlogn)). For generic...

Genericity of nonsingular transformations with infinite ergodic index

J. Choksi, M. Nadkarni (2000)

Colloquium Mathematicae

It is shown that in the group of invertible measurable nonsingular transformations on a Lebesgue probability space, endowed with the coarse topology, the transformations with infinite ergodic index are generic; they actually form a dense G δ set. (A transformation has infinite ergodic index if all its finite Cartesian powers are ergodic.) This answers a question asked by C. Silva. A similar result was proved by U. Sachdeva in 1971, for the group of transformations preserving an infinite measure. Exploring...

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