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Galois orbits and equidistribution: Manin-Mumford and André-Oort.

Andrei Yafaev (2009)

Journal de Théorie des Nombres de Bordeaux

We overview a unified approach to the André-Oort and Manin-Mumford conjectures based on a combination of Galois-theoretic and ergodic techniques. This paper is based on recent work of Klingler, Ullmo and Yafaev on the André-Oort conjecture, and of Ratazzi and Ullmo on the Manin-Mumford conjecture.

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.

Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to β -shifts

Veronica Baker, Marcy Barge, Jaroslaw Kwapisz (2006)

Annales de l’institut Fourier

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.

Geometric rigidity of × m invariant measures

Michael Hochman (2012)

Journal of the European Mathematical Society

Let μ be a probability measure on [ 0 , 1 ] which is invariant and ergodic for T a ( x ) = a x 𝚖𝚘𝚍 1 , and 0 < 𝚍𝚒𝚖 μ < 1 . Let f be a local diffeomorphism on some open set. We show that if E and ( f μ ) E μ E , then f ' ( x ) ± a r : r at μ -a.e. point x f - 1 E . In particular, if g is a piecewise-analytic map preserving μ then there is an open g -invariant set U containing supp μ such that g U is piecewise-linear with slopes which are rational powers of a . In a similar vein, for μ as above, if b is another integer and a , b are not powers of a common integer, and if ν is a T b -invariant...

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