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Fractal representation of the attractive lamination of an automorphism of the free group

Pierre Arnoux, Valérie Berthé, Arnaud Hilion, Anne Siegel (2006)

Annales de l’institut Fourier

In this paper, we extend to automorphisms of free groups some results and constructions that classically hold for morphisms of the free monoid, i.e., the so-called substitutions. A geometric representation of the attractive lamination of a class of automorphisms of the free group (irreducible with irreducible powers (iwip) automorphisms) is given in the case where the dilation coefficient of the automorphism is a unit Pisot number. The shift map associated with the attractive symbolic lamination...

Fragmentable mappings and CHART groups

Warren B. Moors (2016)

Fundamenta Mathematicae

The purpose of this note is two-fold: firstly, to give a new and interesting result concerning separate and joint continuity, and secondly, to give a stream-lined (and self-contained) proof of the fact that "tame" CHART groups are topological groups.

Fraïssé structures and a conjecture of Furstenberg

Dana Bartošová, Andy Zucker (2019)

Commentationes Mathematicae Universitatis Carolinae

We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between S ( G ) , the Samuel compactification, and E ( M ( G ) ) , the enveloping semigroup of the universal minimal flow. We resolve Furstenberg’s problem for several automorphism groups and give a detailed study in the case of G = S ∞ , leading us to define and investigate several new types...

Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems

S. Bezuglyi, K. Medynets (2008)

Colloquium Mathematicae

We consider the full group [φ] and topological full group [[φ]] of a Cantor minimal system (X,φ). We prove that the commutator subgroups D([φ]) and D([[φ]]) are simple and show that the groups D([φ]) and D([[φ]]) completely determine the class of orbit equivalence and flip conjugacy of φ, respectively. These results improve the classification found in [GPS]. As a corollary of the technique used, we establish the fact that φ can be written as a product of three involutions from [φ].

General multifractal analysis of local entropies

Floris Takens, Evgeny Verbitski (2000)

Fundamenta Mathematicae

We address the problem of the multifractal analysis of local entropies for arbitrary invariant measures. We obtain an upper estimate on the multifractal spectrum of local entropies, which is similar to the estimate for local dimensions. We show that in the case of Gibbs measures the above estimate becomes an exact equality. In this case the multifractal spectrum of local entropies is a smooth concave function. We discuss possible singularities in the multifractal spectrum and their relation to phase...

Generalized Conley-Zehnder index

Jean Gutt (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space ( W , Ω ¯ ) , having chosen a given reference...

Generalized recurrence, compactifications, and the Lyapunov topology

Ethan Akin, Joseph Auslander (2010)

Studia Mathematica

We study generalized recurrence for closed relations on locally compact spaces. This includes continuous maps and real flows. The main tools are Lyapunov functions and their compactifications. Under certain conditions it is shown that the Lyapunov functions determine the topology of the space.

Generic diffeomorphisms on compact surfaces

Flavio Abdenur, Christian Bonatti, Sylvain Crovisier, Lorenzo J. Díaz (2005)

Fundamenta Mathematicae

We discuss the remaining obstacles to prove Smale's conjecture about the C¹-density of hyperbolicity among surface diffeomorphisms. Using a C¹-generic approach, we classify the possible pathologies that may obstruct the C¹-density of hyperbolicity. We show that there are essentially two types of obstruction: (i) persistence of infinitely many hyperbolic homoclinic classes and (ii) existence of a single homoclinic class which robustly exhibits homoclinic tangencies. In the course of our discussion,...

Generic measures for geodesic flows on nonpositively curved manifolds

Yves Coudène, Barbara Schapira (2014)

Journal de l’École polytechnique — Mathématiques

We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold and supported by trajectories not bounding a flat strip. This is done by showing that Dirac measures on periodic orbits are dense in that set.In the case of a compact surface, we get the following sharp result:...

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 properties of learning systems

Tomasz Szarek (2000)

Annales Polonici Mathematici

It is shown that the set of learning systems having a singular stationary distribution is generic in the family of all systems satisfying the average contractivity condition.

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