Prevalence in the space of finitely smooth maps

In the present paper, we give a definition of prevalent ("metrically prevalent" ) sets in nonlinear function spaces. A subset of a Euclidean space is said to be metrically prevalent if its complement has measure zero. There is no natural way to generalize the definition of a set of measure zero in a finite-dimensional space to the infinite-dimensional case [6]. Therefore, it is necessary to give a special definition of a metrically prevalent set (set of full measure) in an infinite-dimensional space. There are various ways to do so. We suggest one of the possible ways to define the class of metrically prevalent sets in the space of smooth maps of one smooth manifold into another. It is shown in this paper that the class of metrically prevalent sets has natural properties; in particular, the intersection of finitely many metrically prevalent sets is metrically prevalent. The main result of the paper is a prevalent version of Thorn's transversality theorem. It is common practice in singularity theory and the theory of dynamical systems to say that a property holds for "almost every" map (or flow) if it holds for a residual set, i.e., a set that contains a countable intersection of open dense sets in the corresponding function space. However, even in finite-dimensional spaces such a set can have arbitrarily small (say, zero) Lebesgue measure. We prove that Thorn's transversality theorem holds for an essentially "thicker" set than a residual set. It seems reasonable to revise from the prevalent point of view the classical results of singularity theory and theory of dynamical systems, including the multijet transversality theorem, Mather's stability theorem, Kupka-Smale's theorem for dynamical systems, etc. We shall do this elsewhere. The notion of prevalence in linear Banach spaces was introduced and investigated in [8]. One of the possible ways to define a class of prevalent sets in the space of smooth maps of manifolds, which essentially differs from that presented in this paper, is given in [7]. Definitions of typicalness based on the Lebesgue measure in a finite-dimensional space were suggested by Kolmogorov [10] and Arnold [11]. These definitions were cited and discussed in [9]. Here we only point out that the finite-dimensional analog of Arnold's definition allows prevalent sets to have arbitrarily small measure, whereas the prevalent sets in the sense of the finite-dimensional analog of the definition given in the present paper are necessarily of full measure. Our definition is a modification of that due to Arnold. I wish to thank Yu. S. Illyashenko for constant attention to this work and useful discussions and R. I. Bogdanov for help in the preparation of this paper.