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Semigroup actions on tori and stationary measures on projective spaces

Yves Guivarc'h, Roman Urban (2005)

Studia Mathematica

Let Γ be a subsemigroup of G = GL(d,ℝ), d > 1. We assume that the action of Γ on ℝ d is strongly irreducible and that Γ contains a proximal and quasi-expanding element. We describe contraction properties of the dynamics of Γ on ℝ d at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space ℙ d - 1 . In the case where Γ is a subsemigroup of GL(d,ℝ) ∩ M(d,ℤ) and Γ has the above properties, we deduce that the Γ-orbits...

Semisimple extensions of irrational rotations

Mariusz Lemańczyk, Mieczysław K. Mentzen, Hitoshi Nakada (2003)

Studia Mathematica

We show that semisimple actions of l.c.s.c. Abelian groups and cocycles with values in such groups can be used to build new examples of semisimple automorphisms (ℤ-actions) which are relatively weakly mixing extensions of irrational rotations.

Sequence entropy and rigid σ-algebras

Alvaro Coronel, Alejandro Maass, Song Shao (2009)

Studia Mathematica

We study relationships between sequence entropy and the Kronecker and rigid algebras. Let (Y,,ν,T) be a factor of a measure-theoretical dynamical system (X,,μ,T) and S be a sequence of positive integers with positive upper density. We prove there exists a subsequence A ⊆ S such that h μ A ( T , ξ | ) = H μ ( ξ | ( X | Y ) ) for all finite partitions ξ, where (X|Y) is the Kronecker algebra over . A similar result holds for rigid algebras over . As an application, we characterize compact, rigid and mixing extensions via relative sequence...

Sequence entropy pairs and complexity pairs for a measure

Wen Huang, Alejandro Maass, Xiangdong Ye (2004)

Annales de l’institut Fourier

In this paper we explore topological factors in between the Kronecker factor and the maximal equicontinuous factor of a system. For this purpose we introduce the concept of sequence entropy n -tuple for a measure and we show that the set of sequence entropy tuples for a measure is contained in the set of topological sequence entropy tuples [H- Y]. The reciprocal is not true. In addition, following topological ideas in [BHM], we introduce a weak notion and a strong notion of complexity pair for a...

Sequences of algebraic integers and density modulo  1

Roman Urban (2007)

Journal de Théorie des Nombres de Bordeaux

We prove density modulo 1 of the sets of the form { μ m λ n ξ + r m : n , m ∈ ℕ } , where λ , μ ∈ ℝ is a pair of rationally independent algebraic integers of degree d ≥ 2 , satisfying some additional assumptions, ξ ≠ 0 , and r m is any sequence of real numbers.

Sets of k -recurrence but not ( k + 1 ) -recurrence

Nikos Frantzikinakis, Emmanuel Lesigne, Máté Wierdl (2006)

Annales de l’institut Fourier

For every k ∈ ℕ , we produce a set of integers which is k -recurrent but not ( k + 1 ) -recurrent. This extends a result of Furstenberg who produced a 1 -recurrent set which is not 2 -recurrent. We discuss a similar result for convergence of multiple ergodic averages. We also point out a combinatorial consequence related to Szemerédi’s theorem.

Sets of nondifferentiability for conjugacies between expanding interval maps

Thomas Jordan, Marc Kesseböhmer, Mark Pollicott, Bernd O. Stratmann (2009)

Fundamenta Mathematicae

We study differentiability of topological conjugacies between expanding piecewise C 1 + ϵ interval maps. If these conjugacies are not C¹, then their derivative vanishes Lebesgue almost everywhere. We show that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Moreover, by employing the thermodynamic formalism, we show that this Hausdorff dimension can be determined explicitly in terms of the Lyapunov spectrum....

Sets of β -expansions and the Hausdorff measure of slices through fractals

Tom Kempton (2016)

Journal of the European Mathematical Society

We study natural measures on sets of β -expansions and on slices through self similar sets. In the setting of β -expansions, these allow us to better understand the measure of maximal entropy for the random β -transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading...

Sets with doubleton sections, good sets and ergodic theory

A. Kłopotowski, M. G. Nadkarni, H. Sarbadhikari, S. M. Srivastava (2002)

Fundamenta Mathematicae

A Borel subset of the unit square whose vertical and horizontal sections are two-point sets admits a natural group action. We exploit this to discuss some questions about Borel subsets of the unit square on which every function is a sum of functions of the coordinates. Connection with probability measures with prescribed marginals and some function algebra questions is discussed.

Simple systems are disjoint from Gaussian systems

Andrés del Junco, Mariusz Lemańczyk (1999)

Studia Mathematica

We prove the theorem promised in the title. Gaussians can be distinguished from simple maps by their property of divisibility. Roughly speaking, a system is divisible if it has a rich supply of direct product splittings. Gaussians are divisible and weakly mixing simple maps have no splittings at all so they cannot be isomorphic. The proof that they are disjoint consists of an elaboration of this idea, which involves, among other things, the notion of virtual divisibility, which is, more or less,...

Smooth Extensions of Bernoulli Shifts

Zbigniew S. Kowalski (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

For homographic extensions of the one-sided Bernoulli shift we construct σ-finite invariant and ergodic product measures. We apply the above to the description of invariant product probability measures for smooth extensions of one-sided Bernoulli shifts.

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