Conjugacy of a Discrete Semidynamical System in a Neighbourhood of the Nontrivial Invariant Manifold

The conjugacy of a discrete semidynamical system and its partially decoupled discrete semidynamical system in a Banach space is proved in a neighbourhood of the nontrivial invariant manifold.

PULLBACK ATTRACTORS OF NONAUTONOMOUS SEMIDYNAMICAL SYSTEMS

A nonautonomous semidynamical system is a skew-product semi-flow consisting of a cocycle mapping on a state space which is driven by a semidynamical system on a base space. It is shown that the driving system can be extended backwards in time on a compact invariant set, such as a global attractor, as a set-valued semidynamical system. Pullback attraction and invariance with respect to the cocycle mapping are then reformulated in terms of the set-valued backwards extension of the driving system.

C0-continuity of the Fröbenius-Perron semigroup

We consider the Fröbenius-Perron semigroup of linear operators associated to a semidynamical system defined in a topological spaceXendowed with a finite or aσ-finite regular measure. We prove that if there exists afaithful invariant measurefor the semidynamical system, then the Fröbenius-Perron semigroup of linear operators isC0-continuous in the spaceLμ 1(X). We also give a geometrical condition which ensuresC0-continuity of the Fröbenius-Perron semigroup of linear operators in the spaceLμ p(X)for1≤p<∞, as well as in the spaceLloc 1.

Dynamical systems defined on point processes

This paper introduces a new stochastic process in which the iterates of a dynamical system evolving in discrete time coincide with the events of a Poisson process. The autocovariance function of the stochastic process is studied and a necessary and sufficient condition for it to vanish is deduced. It is shown that the mean function of this process comprises a continuous-time semidynamical system if the underlying dynamical map is linear. The flow of probability density functions generated by the stochastic process is analysed in detail, and the relationship between the flow and the solutions of the linear Boltzmann equation is investigated. It is shown that the flow is a semigroup if and only if the point process defining the stochastic process is Poisson, thereby providing a new characterization of the Poisson process.

Dynamical systems defined on point processes

This paper introduces a new stochastic process in which the iterates of a dynamical system evolving in discrete time coincide with the events of a Poisson process. The autocovariance function of the stochastic process is studied and a necessary and sufficient condition for it to vanish is deduced. It is shown that the mean function of this process comprises a continuous-time semidynamical system if the underlying dynamical map is linear. The flow of probability density functions generated by the stochastic process is analysed in detail, and the relationship between the flow and the solutions of the linear Boltzmann equation is investigated. It is shown that the flow is a semigroup if and only if the point process defining the stochastic process is Poisson, thereby providing a new characterization of the Poisson process.