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Geometric entropy of foliations. (Entropie géométrique des feuilletages.) (French) Zbl 0666.57021

Let \(X\) be a metric space. Let \(H_ 1\) be a finite collection of homeomorphisms between open sets of \(X\). Assume that \(H_ 1\) contains the identity map of \(X\) and satisfies that if \(f\in H_ 1\), then \(f^{-1}\in H_ 1\). \(H_ 1\) generates a pseudo-group \(H\) of local homeomorphisms. In this setting the entropy \(h(H,H_ 1)\) of \(H\) with respect to \(H_ 1\) is defined using a generalized notion of \((n,\varepsilon)\)-separated set. Let \(M\) be a compact smooth manifold and \({\mathcal F}\) a foliation on \(M\). Using a distinguished open set for \({\mathcal F}\), a notion of good covering \({\mathcal U}\) of \(M\) with respect to \({\mathcal F}\) is defined, and the (finite) family \(H_ 1({\mathcal U})\) of local homeomorphisms with respect to \({\mathcal U}\) is defined. \(H({\mathcal U})\) is defined to be a pseudo-group of local homeomorphisms generated by \(H_ 1({\mathcal U})\). Let \(g\) be a Riemannian metric on \(M\). Then a notion of \(({\mathcal F},g,r,\varepsilon)\)-separated set is defined. The geometric entropy \(h({\mathcal F},g)\) of \({\mathcal F}\) with respect to \(g\) is defined using \(({\mathcal F},g,r,\varepsilon)\)-separated sets.
Theorem. Let \(\{\phi_ t\}\) be a non-singular flow on \(M\) and \({\mathcal F}\) the 1-dimensional foliation determined by \(\{\phi_ t\}\). Then \(h({\mathcal F},g)=2h(\phi_ 1)\).
Theorem. In general, \(h({\mathcal F},g)=\sup \{h({\mathcal F},{\mathcal U})/\text{diam}({\mathcal U})\}\), where \({\mathcal U}\) is a good covering of \(M\) with respect to \({\mathcal F}\) and \(\text{diam}({\mathcal U})\) is the maximum of the diameters of the elements of \({\mathcal U}\).
Theorem. If \(h({\mathcal F},g)=0\), then \({\mathcal F}\) possesses a transverse invariant measure in the sense of J. Plante.
Theorem. Let \({\mathcal F}\) be a foliation of codimension 1 on \(M\). Then \(h({\mathcal F},g)=0\) if and only if \({\mathcal F}\) possesses no “feuille ressort”.
Corollary. If all leaves of \({\mathcal F}\) of codimension one are below exponential growth, then the geometric entropy of \({\mathcal F}\) is zero.
Corollary. If the geometric entropy of a foliation of codimension one is zero, then its Godbillon-Vey invariant is also zero.
Reviewer: K. Shiraiwa

MSC:

57R30 Foliations in differential topology; geometric theory
37A99 Ergodic theory
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