Morse Spectra, Homology Measures, Spaces of Cycles and Parametric Packing Problems

Abstract

An ”ensemble” Ψ = Ψ(X) of (finitely or infinitely many) particles in a space X, e.g. in the Euclidean 3-space, is customary characterised by the set function U ↦ entU(Ψ) = ent(Ψ∣U), U ⊂ X, that assigns the entropies of the U-reductions Ψ∣U of Ψ, to all bounded open subsets U ⊂ X. In the physicists’ parlance, this entropy is ”the logarithm of the number of the states of E that are effectively observable from U”, This ”definition”, in the context of mathematical statistical mechanics, is translated to the language of the measure/probability theory.1 But what happens if ”effectively observable number of states” is replaced by ”the number of effective/persistent degrees of freedom of ensembles of moving particles”? We suggest in this paper several mathematical counterparts to the idea of ” persistent degrees of freedom ” and formulate specific questions, many of which are inspired by Larry Guth’s results and ideas on the Hermann Weyl kind of asymptotics of the Morse (co)homology spectra of the volume energy function on the spaces of cycles in balls.2 And often we present variable aspects of the same idea in different sections of this paper. Hardly anything that can be called ”new theorem” can be found in our paper but we reshuffle many known results and expose them from a particular angle. This article is meant as an introductory chapter to something yet to be written with much of what we present here extracted from my yet unfinished manuscript Number of Questions. 1 Overview of Concepts and Examples. We introduce below the idea of ”parametric packing” and of related concepts which are expanded in detail in the rest of the paper. A. Let X be a topological space, e.g. a manifold, and I is a countable index set that may be finite, especially if X is compact. A collection of I-tuples of non-empty open (sometimes closed) subsets Ui ⊂ X, i ∈ I, is called a packing or an I-packing of X if these subsets do not intersect. Denote by Ψ(X; I) the space of these packings with some natural topology, where, observe there are several candidates for such a topology if X is noncompact. 1See: Lanford’s Entropy and equilibrium states in classical statistical mechanics, Lecture Notes in Physics, Volume 20, pp. 1-113, 1973 and Ruelle’s Thermodynamic formalism : the mathematical structures of classical equilibrium statistical mechanics, 2nd Edition, Cambridge Mathematical Library 2004, where the emphasis is laid upon (discrete) lattice systems. Also a categorical rendition of Boltzmann-Shannon entropy is suggested in ”In a Search for a Structure, Part 1: On Entropy”, www.ihes.fr/∼gromov/PDF/structre-serch-entropy-july5-2012.pdf 2Minimax problems related to cup powers and Steenrod squares, Geometric and Functional Analysis, 18 (6), 1917-1987 (2009).