We show that singular stochastic delay differential equations (SDDEs) induce cocycle maps on a field of Banach spaces. A general Multiplicative Ergodic Theorem fields spaces is proved and applied to linear SDDEs. In Part II this article, we use our results prove stable manifold theorem for non-linear

In this paper, we investigate multiplicative properties of the classical Dold–Kan correspondence. The inverse of the normalization functor maps commutative differential graded algebras to E∞algebras. We prove that it in fact sends algebras over arbitrary differential graded E∞-operads to E∞-algebras in simplicial modules and is part of a Quillen adjunction. More generally, this inverse maps hom...

A generalised Paley map is a Cayley map for the additive group of a finite field F , with a subgroup S = −S of the multiplicative group as generating set, cyclically ordered by powers of a generator of S. We characterise these as the orientably regular maps with orientation-preserving automorphism group acting primitively and faithfully on the vertices; allowing a non-faithful primitive action ...

The author is grateful for support from a PhD grant from the Engineering and Physical Sciences Research council of the UK and post-doctoral fellowship number SFRH/BPD/40762/2007 from the Fundação para a Ciéncia e a Tec-nologia of Portugal. This paper contains some work included in the authors PhD thesis ([5]). Abstract We introduce notions of compactness and weak compactness for multi-linear ma...

We introduce notions of compactness and weak compactness for multilinear maps from a product of normed spaces to a normed space, and prove some general results about these notions. We then consider linear maps T : A → B between Banach algebras that are “close to multiplicative” in the following senses: the failure of multiplicativity, defined by ST (a, b) = T (a)T (b) − T (ab) (a, b ∈ A), is co...

The objective of this note is to prove the almost sure invariance principle (ASIP) for a large class of random dynamical systems. The random dynamics is driven by an invertiblemeasure preserving transformation σ of (Ω,F ,P) called the base transformation. Trajectories in the phase space X are formed by concatenations f ω := fσn−1ω ◦ · · · ◦ fσω ◦ fω of maps from a family of maps fω : X → X, ω ∈...

Bifurcations in a system of coupled maps are investigated. Using symbolic dynamics it is proven that for coupled shift maps the well known space–time– mixing attractor becomes unstable at a critical coupling strength in favour of a synchronized state. For coupled non–hyperbolic maps analytical and numerical evidence is given that arbitrary small coupling changes the dynamical behaviour. The ano...