G. Pianigiani

First return map and invariant measures

We give sufficient conditions for the existence of absolutely continuous invariant measures, finite or σ-finite, for maps on the interval. We givea priori bound for the number of different ergodic measures. The results are obtained via the first return map.

A Bogolyubov-Type Theorem with a Nonconvex Constraint in Banach Spaces

We prove an analogue of the classical Bogolyubov theorem, with a nonconvex constraint. In the case we consider, the constraint is the solution set of a Cauchy problem for a differential inclusion with a nonconvex right-hand side satisfying a Lipschitz condition. Our approach is based on a relaxation argument, as in the Filippov--Wazewski theorem.

Conditionally invariant measures and exponential decay

Let a measure ,u be given in A; what can be said about the distribution of n,? And does the measure p(Tp”E)/p(T-“A) converge to some limit measure pm(E), and if it does, is pu, independent of the initial measure ,u’? And what are the properties of p,? Pianigiani and Yorke [5] studied extensively such problems for mappings T which are “expansive” in a sense stated below. In particular they show the existence and the uniqueness of a smooth ,uu, and they prove that the distribution of n, is nearly exponential. In the first part of this paper we study the above questions for maps which are not expansive. In the second part of the paper we study systems depending upon a parameter. Such systems arise naturally from physical and biological problems. For example, the Lorenz model for a fluid flow is a system of differential equations depending on a parameter R, the Rayleigh number. Associated with this model is a family of maps T,: [0, l] -+ IR, see [6]. Interesting properties of the original system can be understood in terms of these maps. Let T, : A + IR”, A c IR”, E E [0, eO], E, > 0 be a family of maps and, for each E E [0, so], let n,(x) be the kickout time function. Define D,: R+ + [0, l] by:

Baire category and the weak bang-bang property for continuous differential inclusions

For continuous differential inclusions the classical bang-bang property is known to fail, yet a weak form of it is established here, in the case where the right hand side is a multifunction whose values are closed convex and bounded sets with nonempty interior contained in a reflexive and separable Banach space. Our approach is based on the Baire category method.

Approximate selections in

The existence of continuous approximate selections is proved for a class of upper semicontinuous multifunctions taking closed

{\alpha}

\alpha

-convex metric spaces and topological degree

-convex values in a metric space equipped with an appropriate notion of

The Baire method for the prescribed singular values problem

\alpha

On the Arrow–Hahn utility representation method

-convexity. The approach is based on the definition of pseudo-barycenter of an ordered

Emarks on Differential Inclusions Without Existence or Continuous Dependence

n