Thick attractors of step skew products

A diffeomorphism is said to have a thick attractor provided that its Milnor attractor has positive but not full Lebesgue measure. We prove that there exists an open set in the space of boundary preserving step skew products with a fiber [0,1], such that any map in this set has a thick attractor.     

On the construction of minimal skew products

It is shown that under fairly general conditions on a compact metric spaceY there are minimal homeomorphisms onZ×Y of the form(z,y)→(σz, h _z (y)) where (Z, σ) is a arbitrary metric minimal flow andz→h _z is a continuous map fromZ to the space of homeomorphisms ofY. Similar results are obtained for strict ergodicity, topolotical weak mixing and some relativized concepts.     

Structure of attractors for skew product semiflows

In this work we study the continuity and structural stability of the uniform attractor associated with non-autonomous perturbations of differential equations. By a careful study of the different definitions of attractor in the non-autonomous framework, we introduce the notion of lifted-invariance on the uniform attractor, which becomes compatible with the dynamics in the global attractor of the associated skew product semiflow, and allows us to describe the internal dynamics and the characterization of the uniform attractors. The associated pullback attractors and their structural stability under perturbations will play a crucial role.     

On the ergodicity of a class of skew products

Letϕ: [0, 1]→R have continuous derivativeon the closed interval [0, 1], ∫ ₀ ¹ ϕ(x)dx=0, and letα be irrational. Ifϕ(1) ≠ϕ(0), then (x, y) ↦ (x + α, y + ϕ (x)) is ergodic onR/Z ×R.     

Topological and ergodic aspects of partially hyperbolic diffeomorphisms and nonhyperbolic step skew products

We review some ergodic and topological aspects of robustly transitive partially hyperbolic diffeomorphisms with one-dimensional center direction. We also discuss step skew-product maps whose fiber maps are defined on the circle which model such dynamics. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents as well as intermingled horseshoes having different types of hyperbolicity. We discuss some recent advances concerning the topology of the space of invariant measures and properties of the spectrum of Lyapunov exponents.     

On the ranks of idempotent matrices over skew semifields

As is well known, every positive idempotent matrix is of rank 1. It is proved that idempotent matrices without zeros have this property over many skew semifields, and all these skew semifields are described.     

Spectral types of skewed Bernoulli shift

For the transformation for , it is proved that a real-valued function of modulus is not a multiplicative coboundary if the discontinuities of are -adic points and . It is also proved that the weakly mixing skew product transformations arising from Bernoulli shifts have Lebesgue spectrum.

     

On skew polynomial rings over locally nilpotent rings

We will show that skew polynomial rings in several variables over locally nilpotent rings cannot contain nonzero idempotent elements. We will also prove that such rings are Brown–McCoy radical.     

Counting the relatively finite factors of a Bernoulli shift

LetT acting on (Ω, ℱ, μ) be afinite entropy Bernoulli shift. AT invariant factor is “relatively finite” if a.e. fiber of has a finite, hence constantk, number of points. We say two factors , “sit the same” if there is a measurable measure preserving map ϕ with and . We show here that up to sitting the same there are only finitely many relatively finite factors withk point fibers in a Bernoulli shift, and that they are classified by a certain algebraic structure in the symmetric group onk-points. This work was supported in part by the Miller Institute for Basic Research, U. C. Berkeley.     

On soluble skew linear groups over finite-dimensional division algebras

Let D be a division ring of finite degree d and let n be a positive integer. If G is any soluble subgroup of , we prove that G has derived length at most

9+log₂d+(11/3)log₂n     

Blaschke products and the rank of backward shift invariant subspaces over the bidisk

Let H²(D²) be the Hardy space over the bidisk. For sequences of Blaschke products {φn(z):−∞<n<∞} and {ψn(w):−∞<n<∞} satisfying some additional conditions, we may define a Rudin type invariant subspace M. We shall determine the rank of H²(D²)?M for the pair of operators and .     

Sums of products of hypergeometric Bernoulli numbers

We give a formula for sums of products of hypergeometric Bernoulli numbers. This formula is proved by using special values of multiple analogues of hypergeometric zeta functions.     

Skew products of Bernoulli shifts with rotations

This paper investigates certain skew products of Bernoulli shifts with rotations or permutations, and shows that these transformations, if weak mixing, are also Bernoulli. This work was partially supported by the U. S. Air Force Office of Scientific Research under contract F44620-C-0063, P001, and by National Science Foundation under grant GP 21509. This work was partially supportedby the National Science Foundation under grants GP 21509 and GJ-776.     

Free products of finitary skew linear groups

We give two very different proofs of the following result. Let {G _γ:λɛΛ : λ ∈ Λ} be a family of finitary skew linear groups of the same characteristic p ≧ 0. Then the free product of the G _γ is isomorphic to some finitary skew linear group of characteristic p. This extends recent work of R. J. H. Minty on the skew linear case and of O. Puglisi on the finitary linear case.