{"quality_controlled":"1","title":"Prevalence in the space of finitely smooth maps","_id":"8528","author":[{"orcid":"0000-0002-6051-2628","full_name":"Kaloshin, Vadim","first_name":"Vadim","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","last_name":"Kaloshin"}],"doi":"10.1007/bf02466014","abstract":[{"lang":"eng","text":"In the present paper, we give a definition of prevalent (\"metrically prevalent\" ) sets in nonlinear function\r\nspaces. A subset of a Euclidean space is said to be metrically prevalent if its complement has measure zero.\r\nThere is no natural way to generalize the definition of a set of measure zero in a finite-dimensional space\r\nto the infinite-dimensional case [6]. Therefore, it is necessary to give a special definition of a metrically\r\nprevalent set (set of full measure) in an infinite-dimensional space. There are various ways to do so. We\r\nsuggest one of the possible ways to define the class of metrically prevalent sets in the space of smooth maps\r\nof one smooth manifold into another. It is shown in this paper that the class of metrically prevalent sets\r\nhas natural properties; in particular, the intersection of finitely many metrically prevalent sets is metrically\r\nprevalent. The main result of the paper is a prevalent version of Thorn's transversality theorem.\r\nIt is common practice in singularity theory and the theory of dynamical systems to say that a property\r\nholds for \"almost every\" map (or flow) if it holds for a residual set, i.e., a set that contains a countable\r\nintersection of open dense sets in the corresponding function space. However, even in finite-dimensional\r\nspaces 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\r\nthe prevalent point of view the classical results of singularity theory and theory of dynamical systems,\r\nincluding 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\r\nintroduced and investigated in [8]. One of the possible ways to define a class of prevalent sets in the space\r\nof smooth maps of manifolds, which essentially differs from that presented in this paper, is given in [7].\r\nDefinitions of typicalness based on the Lebesgue measure in a finite-dimensional space were suggested\r\nby Kolmogorov [10] and Arnold [11]. These definitions were cited and discussed in [9]. Here we only point\r\nout that the finite-dimensional analog of Arnold's definition allows prevalent sets to have arbitrarily small\r\nmeasure, whereas the prevalent sets in the sense of the finite-dimensional analog of the definition given in\r\nthe present paper are necessarily of full measure. Our definition is a modification of that due to Arnold.\r\nI wish to thank Yu. S. Illyashenko for constant attention to this work and useful discussions and\r\nR. I. Bogdanov for help in the preparation of this paper. "}],"oa_version":"None","issue":"2","publication_status":"published","publication":"Functional Analysis and Its Applications","extern":"1","type":"journal_article","date_updated":"2021-01-12T08:19:54Z","volume":31,"language":[{"iso":"eng"}],"publication_identifier":{"issn":["0016-2663","1573-8485"]},"intvolume":"        31","day":"30","year":"1997","publisher":"Springer Nature","article_type":"original","status":"public","page":"95-99","date_published":"1997-03-30T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","month":"03","date_created":"2020-09-18T10:50:54Z","keyword":["Applied Mathematics","Analysis"],"citation":{"ieee":"V. Kaloshin, “Prevalence in the space of finitely smooth maps,” <i>Functional Analysis and Its Applications</i>, vol. 31, no. 2. Springer Nature, pp. 95–99, 1997.","ama":"Kaloshin V. Prevalence in the space of finitely smooth maps. <i>Functional Analysis and Its Applications</i>. 1997;31(2):95-99. doi:<a href=\"https://doi.org/10.1007/bf02466014\">10.1007/bf02466014</a>","mla":"Kaloshin, Vadim. “Prevalence in the Space of Finitely Smooth Maps.” <i>Functional Analysis and Its Applications</i>, vol. 31, no. 2, Springer Nature, 1997, pp. 95–99, doi:<a href=\"https://doi.org/10.1007/bf02466014\">10.1007/bf02466014</a>.","short":"V. Kaloshin, Functional Analysis and Its Applications 31 (1997) 95–99.","apa":"Kaloshin, V. (1997). Prevalence in the space of finitely smooth maps. <i>Functional Analysis and Its Applications</i>. Springer Nature. <a href=\"https://doi.org/10.1007/bf02466014\">https://doi.org/10.1007/bf02466014</a>","ista":"Kaloshin V. 1997. Prevalence in the space of finitely smooth maps. Functional Analysis and Its Applications. 31(2), 95–99.","chicago":"Kaloshin, Vadim. “Prevalence in the Space of Finitely Smooth Maps.” <i>Functional Analysis and Its Applications</i>. Springer Nature, 1997. <a href=\"https://doi.org/10.1007/bf02466014\">https://doi.org/10.1007/bf02466014</a>."}}