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An average intersection estimate for families of diffeomorphisms

Axel Kodat, Michael Shub

Abstract

We show that for any sufficiently rich compact family $\mathcal{H}$ of $C^1$ diffeomorphisms of a closed Riemannanian manifold $M$, the average geometric intersection number over $h \in \mathcal{H}$ between $h(V)$ and $W$, for $V, W$ any complementary dimensional submanifolds of $M$, is approximately (i.e. up to a uniform multiplicative error depending only on $\mathcal{H}$) the product of their volumes. We also give a construction showing that such families always exist.

An average intersection estimate for families of diffeomorphisms

Abstract

We show that for any sufficiently rich compact family of diffeomorphisms of a closed Riemannanian manifold , the average geometric intersection number over between and , for any complementary dimensional submanifolds of , is approximately (i.e. up to a uniform multiplicative error depending only on ) the product of their volumes. We also give a construction showing that such families always exist.
Paper Structure (19 sections, 28 theorems, 150 equations)

This paper contains 19 sections, 28 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Assumptions
  4. A digression on $\mathcal{H}$
  5. Acknowledgements
  6. Preliminaries
  7. Normal Jacobians
  8. The solution manifold $\widetilde{\mathcal{V}}$
  9. The intersection manifolds $\mathcal{V}$
  10. The bundles $\mathcal{G}_k \to \widetilde{\mathcal{V}}$
  11. Proof of Main Theorem
  12. The co-area formula
  13. Fiber integral bounds
  14. Generalization
  15. Construction of $(\mathcal{H}, \psi)$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Suppose $(\mathcal{H}, \psi)$ is compact $C^1$ family of diffeomorphisms of $M$ such that Then there is a constant $C = C(\mathcal{H}, \psi) \geq 1$ such that for all complementary dimensional submanifolds $V, W \subset M$.

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Poincaré's formula for homogeneous spaces
  • Example 1.4: ${\mathbb T}^2 \curvearrowright {\mathbb T}^2$
  • Proposition 1.5
  • proof
  • Corollary 1.6
  • proof
  • Claim 2.1
  • proof
  • ...and 54 more