We show that the entropy of a finitely generated pseudogroup (resp., of a foliation on a compact Riemannian manifold) can be calculated by suitable counting separated pseudo-orbits (resp., pseudoleaves).

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.

The

In this talk the qualitative analysis of statics and dynamics of textures in liquid crystals is performed with help of meanders(V.Arnold) and train tracks (W.Thurston).It is argued ,that similar analysis can be applied to 2+1 gravity.The train tracks alone are sufficient for the description of time evolution of liquid crystal textures and gravity and the master equation which describes such evolution could be used in principle instead of more familiar Wheeler-DeWitt equation for gravity .The solution of the master equation is possible but requires the large scale numerical work. To by-pass this difficulty the approximation of train tracks by the meandritic labyrinths is used which allows to study possible phases of such systems using Peierls-like arguments.

For every negatively curved 3-manifold, I constructed a canonical compact space laminated by riemann surfaces havng the following properties (i) for every g, the set of all compact leaves of genus greater than g is dense (ii) a generic leaf is dense, (iii) the space is stable in the following sense : if we deform the metric on the mainfold there is an homeomorphism between the two laminated spaces which preserve the leaves. Of course, these properties are reminiscence of that of the geodesic flow, who lives, so to say, on the boundary of this space. The general construction is part of a technique to construct laminations associated to what I call Monge-Ampere problems and which uses holomorphic curves rechniques. The fact that properties (i) (ii) (iii) are true, heavily depends on the negative curvature assumption.

The main result is a characterisation of irreducible Riemannian symmetric spaces and Euclidean buildings of rank $\geq2$ among singular spaces of nonpositive curvature by their asymptotic geometry: Namely, these are precisely the geodesically complete Hadamard spaces whose Tits boundary is a connected irreducible spherical Tits building.

We discuss generalizations to higher degrees of the following fundamental fact due to W. Thurston: any gap of a quadratic lamination is either pre-periodic, or pre-critical.

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.

Let A be a C1 Anosov diffeomorphism of a compact manifold M, and f a C1 circle diffeomorphism such that the product map A x f is a partially hyperbolic diffeomorphism. Then there is a C1 neighborhood of Axf that contains an open and dense set of partially hyperbolic diffeomorphisms with the accesibility property. If, in addition, A and f are C^2, A has a smooth invariant measure, and f has a smooth invariant measure, then there is a C2 neighborhood of A x f that contains an open and dense set of stably ergodic diffeomorphisms. This partially answers a question posed by Pugh and Shub.

Foliated complexes and hierarchical accessibility of finitely presented groups.

In this joint work with Thomas Delzant we define a class $\cal C$ of subgroups of a finitely presented group $G$ which we call elementary. In, particular this can be elementary subgroups (virtually abelian or finite) of a discrete group of isometries of the hyperbolic space or elementary (virtually cyclic or finite) subgroups of a word (Gromov) hyperbolic group etc. To the group $G$ we associate an invariant $c(G)$ called complexity which is strongly decreasing with respect to splittings of $G$ as amalgamated free product or $HNN$-extension over elementary subgroups. This implies that the group $G$ admits a finite hierarchy of graph of groups decompositions over its elementary subgroups. This is in a sense similar to well-known hierarchies of $3$-dimensional Haken manifolds along a system of incompressible surfaces.

We show that ``most" non-Haken 3-manifolds contain only finitely many isotopy classes of Heegaard surfaces of any bounded genus.

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.

In this talk I consider polynomial differential equations in $\C^2$ of the form $dw \over dz}={P_n/ Q_n$, where $P_n$, $Q_n$ are polynomials with complex coefficients of degree at most $n$. These equations can be identified with the space ${\bf A}_n$ of coefficients of the polynomials $P_n$ and $Q_n$. Let $\varphi$ be a solution of equation~(\ref{eq0.1}). A cycle on $\varphi$ is a non trivial element of the fundamental group $\pi_1(\varphi)$ of $\varphi$. Then, generic equations of the class ${\bf A}_n, \ n \geq 3$ have a countable set of homologically independent limit cycles. The exceptional set (the subset of $ {\bf A}_n$ excluded by the genericity assumptions) is given by a finite number of analytical and real algebraic conditions of real codimension at least two.

Foliations, finitely generated groups and pseudogroups of transformations of manifolds (or, topological spaces) carry some dynamics. In particular, geometric (or, topological) entropy of such systems can be defined [GLW]. For instance, if $G$ is a group of homeomorphisms of a compact metric space $(X, d)$ and $G_1$ is a finite symmetric generating set, then points $x$ and $y$ of $X$ are said to be $(n, \epsilon )$-separated when $d(gx, gy)\ge\epsilon$ for some $g = g_1\cdot\dots\cdot g_n$, where $g_i\in G_1$. If $N(n, \epsilon )$ denotes the maximal cardinality of a subset of $X$ consisting of pairwise $(n, \epsilon )$-separated points, then the {\it entropy} $h(G, G_1)$ w.r.t. $G_1$ can be defined as $$h(G, G_1) = \lim_{\epsilon\to 0}\limsup_{n\to\infty}\frac{1}{n}\log N(n, \epsilon ).$$ The similar definition can be written for pseudogroups and, when applied to holonomy pseudogroups, for foliations.

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

Consider generic orientable measured foliation on a closed orientable surface of genus $g$. We study how do the leaves of the foliation wind around the surface.

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.