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