## 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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### References:

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