Since several years, pseudo-orbits play an important role in the theory of classical dynamical systems. In particular, it is shown ([Mi] and [BS]) that pseudo-orbits can be used to calculate the topological entropy of transformations. More recently, the similar result was obtained [Hur] for the inverse-image entropy introduced earlier by Remi Langevin and the second author [LW1].
On the other hand, geometric entropy $h(\Cal F)$ of a foliation $\Cal F$ of a compact Riemannian manifold has been introduced [GLW] and shown to be a handful tool to study topology and dynamics of foliated manifolds ( [Hu1 - 3], [ GLW], [ LW2], [Eg], [IT], etc.). The entropy $h(\Cal F)$ can be calculated either by counting the number of points separated along the leaves or by counting the number of separated orbits of holonomy pseudogroups generated by nice coverings by charts distinguished by $\Cal F$. Also, pseudoleaves of foliations have been defined by Takashi Inaba [In] who has shown (among the other results) that expansive C$^1$-foliations (in the sense of [IT]) of codimension-one which have the pseudoleaf tracing property are topologically stable.
In this article, we show that the geometric entropy $h(\Cal F)$ of any foliation $\Cal F$ of a compact Riemannian manifold $M$ coincides with that calculated by suitable counting separated pseudoleaves (Theorem 2 in Section 4). To this end, we study the entropy of finitely generated pseudogroups of local transformations of local transformations of compact metric spaces and we show that it can be calculated by counting (again, in a suitable way) separated pseudo-orbits (Theorem 1 in Section 2). In Section 5, we sketch an easy proof of Theorem 2 for foliated bundles. Finally, in Section 6 we provide an example of a group acting on $S^1$ for which the "usual" formula for the entropy in terms of pseudo-orbits does not work: It gives the number strictly bigger than the entropy calculated in terms of separated orbits.
Some significance of these results could be observed if one would try to
calculate (or, to estimate) entropies of pseudogroups or foliations
with the aid of computers.
Under the assumption that the corresponding
quotient space (Julia set) is locally connected,
we prove the same statement for higher degrees
with one multiple ctitical point. The method
is purely topological and transparent.
It seems to allow a generalization
for any number of the critical points.
Corollaries are given.
Useing Dehn surgery techniques, we show that when filling a 3-manifold containing no closed essential surface but a torus boundary component, for all but finitely many fillings the resulting manifold contains only finitely many isotopy classes of Heegaard surfaces that are not surfaces for $X$. In 1990 Klaus Johannson had shown finiteness for Haken manifolds (including manifolds with boundary) of isotopy classes of surfaces of bounded genus. Combining his solution (for $X$) and our work provides a proof for the above statement.
This is a part of a broader project, still in progress, where the bounded genus assumption will be omitted.
This proves, for most 3-manifolds, the following conjecture:
Conjecture (Waldhausen '78): any closed 3-manifold contains only finitely
many isotopy classes of minimal genus Heegaard surfaces.
Attie and Hurder [AH] applied entropy to produce an example of a complete Riemannian manifold of bounded geometry which cannot be quasi-isometric to a leaf of a foliation of a compact manifold (see also [Hu]). A similar but simpler example of this sort has been obtained by Zeghib [Ze]. Following this line, we constructed [Wa] an example of a {\it virtual leaf}, i.e. a complete surface $N$ of bounded geometry which is not quasi-isometric to a leaf as above but which, for any $\delta > 0$, is quasi-isometric to a $\delta$-pseudoleaf on a fixed compact foliated manifold $(M, \Cal F)$. (Recall that a $\delta$-{\it pseudoleaf} is an immersed submanifold which cuts the leaves at an angle less than $\delta$.) The key observation leading to this construction is that the dynamics of pseudo-orbits of a finitely generated group can be more chaotic than the dynamics of true orbits.
In this lecture, we intend to decsribe our example of a virtual leaf and to give a brief review of those results concerning the entropy of foliations which are necessary to understand the reasoning.
[AH] O. Attie and S. Hurder, Manifolds which cannot be leaves of foliation \jour Topology \vol 35 \yr 1996\pages 335 -- 353\endref
[GLW] E. Ghys, R. Langevin and P. Walczak, Entropie g\'eom\'etrique des feuilletages\jour Acta Math. \vol 160 \yr 1988\pages 105 -- 142\endref
[Hu] S. Hurder, Coarse geometry of foliations \inbook Geometric Study of Foliations, Proc. Tokyo 1993\publ World Sci.\publaddr Singapore\yr 1994\pages 35 -- 96\endref
[Wa] P. Walczak, A virtual leaf info preprint \yr 1997\endref
[Ze] A. Zeghib, An example of a 2-dimensional no leaf
\inbook Geometric Study of Foliations, Proc. Tokyo 1993\publ World Sci.
\publaddr Singapore\yr 1994\pages475 -- 477\endref
It follows from the results of H.Masur and W.Veech that generically all the leaves asymptotically follow one and the same asymptotic cycle, determined by the foliation. We give the precise bound for the deviation from the asymptotic cycle; it is expressed in terms of the Lyapunov exponents of the Teichmuller geodesic flow.
The similar results are valid for orientable measured foliations having prescribed types of saddles. In collaboration with M.Kontsevich we classified the corresponding connected components of the strata of the moduli spaces of holomorphic and quadratic differentials.