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Non-transitive points and porosity

T. K. Subrahmonian Moothathu (2013)

Colloquium Mathematicae

We establish that for a fairly general class of topologically transitive dynamical systems, the set of non-transitive points is very small when the rate of transitivity is very high. The notion of smallness that we consider here is that of σ-porosity, and in particular we show that the set of non-transitive points is σ-porous for any subshift that is a factor of a transitive subshift of finite type, and for the tent map of [0,1]. The result extends to some finite-to-one factor systems. We also show...

On the characterization of Hardy-Besov spaces on the dyadic group and its applications

Jun Tateoka (1994)

Studia Mathematica

C. Watari [12] obtained a simple characterization of Lipschitz classes L i p ( p ) α ( W ) ( 1 ≥ p ≥ ∞ , α > 0 ) on the dyadic group using the L p -modulus of continuity and the best approximation by Walsh polynomials. Onneweer and Weiyi [4] characterized homogeneous Besov spaces B p , q α on locally compact Vilenkin groups, but there are still some gaps to be filled up. Our purpose is to give the characterization of Besov spaces B p , q α by oscillations, atoms and others on the dyadic groups. As applications, we show a strong capacity inequality of the...

Path-wise solutions of stochastic differential equations driven by Lévy processes.

David R. E. Williams (2001)

Revista Matemática Iberoamericana

In this paper we show that a path-wise solution to the following integral equationYt = ∫0t  f(Yt) dXt,     Y0 = a ∈ Rd,exists under the assumption that Xt is a Lévy process of finite p-variation for some p ≥ 1 and that f is an α-Lipschitz function for some α > p. We examine two types of solution, determined by the solution's behaviour at jump times of the process X, one we call geometric, the other forward. The geometric solution is obtained by adding fictitious time and solving an associated...

Point derivations for Lipschitz functions andClarke's generalized derivative

Vladimir Demyanov, Diethard Pallaschke (1997)

Applicationes Mathematicae

Clarke’s generalized derivative f 0 ( x , v ) is studied as a function on the Banach algebra Lip(X,d) of bounded Lipschitz functions f defined on an open subset X of a normed vector space E. For fixed x ∈ X and fixed v ∈ E the function f 0 ( x , v ) is continuous and sublinear in f ∈ L i p ( X , d ) . It is shown that all linear functionals in the support set of this continuous sublinear function satisfy Leibniz’s product rule and are thus point derivations. A characterization of the support set in terms of point derivations is given.

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