We show that, whenever Γ is a countable abelian group and ∆ ≤ Γ is a finitely-generated subgroup, a generic measure-preserving action of ∆ on a standard atomless probability space (X,μ) extends to a free measure-preserving action of Γ on (X,μ). This extends a result of Ageev, corresponding to the case when ∆ is infinite cyclic.

Using the Perron-Frobenius operator we establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic. We apply the result to asymptotically periodic transformations and give an extensive specific example using the tent map.

In this chapter, we extend the techniques of Chapter 2.4 on Fano’s method (the local Fano method) to a more global construction. In particular, we show that, rather than constructing a local packing, choosing a scaling δ > 0, and then optimizing over this δ, it is actually, in many cases, possible to prove lower bounds on minimax error directly using packing and covering numbers (metric entropy...

We prove that a divergence-free and C1-robustly transitive vector field has no singularities. Moreover, if the vector field is C4 then the linear Poincaré flow associated to it admits a dominated splitting over M . MSC 2000: primary 37D30, 37D25; secondary 37A99. keywords: Volume-preserving flows; Robust transitivity; Dominated splitting; Ergodicity.

We consider group-valued cocycles over a partially hyperbolic diffeomorphism which is accessible volume-preserving and center bunched. We study cocycles with values in the group of invertible continuous linear operators on a Banach space. We describe properties of holonomies for fiber bunched cocycles and establish their Hölder regularity. We also study cohomology of cocycles and its connection...

We study the supremum of random Dirichlet polynomials DN (t) = ∑ N n=1 εnd(n)n , where (εn) is a sequence of independent Rademacher random variables, and d is a sub-multiplicative function. The approach is gaussian and entirely based on comparison properties of Gaussian processes, with no use of the metric entropy method.

We show that if a volume-preserving diffeomorphism belongs to the C1-interior of the set of all volume preserving diffeomorphisms having the periodic shadowing property then it is Anosov. Mathematics Subject Classification: 37C10, 37C50, 37D20

In this paper, the abstract computational principles underlying topographic maps are discussed. We give a definition of a “perfectly neighbourhood preserving” map, which we call a topographic homeomorphism, and we prove that this has certain desirable properties. It is argued that when a topographic homeomorphism does not exist (the usual case), many equally valid choices are available for quan...

In this paper, we tackle the study of so-called supplementarity measures. They are introduced axiomatically and are then related to incompatibility measures by antonyms. To do this, we have to establish what we mean by antonymous measure. We then prove that, under certain conditions, supplementarity and incompatibility measures are antonymous. Besides, with the aim of constructing antonymous me...

If K is a pre-compact subset of a normed space X and > 0, the quantity logN(K, BX) describes the complexity of K at the level of resolution . A 1972 conjecture of A. Pietsch – originally stated in the operatortheoretic laguage – asserts that the two “metric entropy functionals,” → logN(K, BX) and → logN(BX∗ , K◦), are equivalent in the appropriate sense, uniformly over normed spaces X and over ...