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Tame Flows
Liviu I. Nicolaescu, University of Notre Dame, IN

Memoirs of the American Mathematical Society
2010; 130 pp; softcover
Volume: 208
ISBN-10: 0-8218-4870-4
ISBN-13: 978-0-8218-4870-8
List Price: US$73
Individual Members: US$43.80
Institutional Members: US$58.40
Order Code: MEMO/208/980
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The tame flows are "nice" flows on "nice" spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow \(\Phi: \mathbb{R}\times X\rightarrow X\) on pfaffian set \(X\) is tame if the graph of \(\Phi\) is a pfaffian subset of \(\mathbb{R}\times X\times X\). Any compact tame set admits plenty tame flows. The author proves that the flow determined by the gradient of a generic real analytic function with respect to a generic real analytic metric is tame.

Table of Contents

  • Introduction
  • Tame spaces
  • Basic properties and examples of tame flows
  • Some global properties of tame flows
  • Tame Morse flows
  • Tame Morse-Smale flows
  • The gap between two vector subspaces
  • The Whitney and Verdier regularity conditions
  • Smale transversality and Whitney regularity
  • The Conley index
  • Flips/flops and gradient like tame flows
  • Simplicial flows and combinatorial Morse theory
  • Tame currents
  • Appendix A. An "elementary" proof of the generalized Stokes formula
  • Appendix B. On the topology of tame sets
  • Bibliography
  • Index
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