Paweł Walczak

Dynamics of foliations, groups and pseudogroups

1 Dynamical systems.- 1.1 Pseudogroups.- 1.2 First examples.- 1.3 Foliations, laminations and holonomy.- 1.4 Markov pseudogroups.- 1.5 Hyperbolic spaces and groups.- 2 Growth.- 2.1 Growth types.- 2.2 Growth in groups.- 2.3 Orbit growth for pseudogroups.- 2.4 Expansion growth.- 3 Entropy.- 3.1 Entropy of classical systems.- 3.1.1 Topological entropy of a transformation.- 3.1.2 Invariant measures.- 3.1.3 Measure-theoretic entropy.- 3.1.4 Examples.- 3.1.5 Variational principle.- 3.2 Entropy of pseudogroups.- 3.3 Geometric entropy of foliations.- 3.4 Relating various entropies.- 3.5 Examples and constructions.- 3.5.1 Pullback.- 3.5.2 Gluing.- 3.5.3 Turbulizat ion.- 3.6 Entropy and resiliency.- 4 Invariant measures.- 4.1 Basic definitions and facts.- 4.2 Transverse invariant measures and homology.- 4.3 Measures and orbit growth.- 4.4 Transverse invariant measures in codimension 1.- 4.5 Vanishing entropy and invariant measures.- 4.6 Entropy, geodesic flow and invariant measures.- 4.7 Harmonic measures.- 4.8 Patterson-Sullivan measures.- 5 Hausdorff dimension.- 5.1 Definitions and basic facts.- 5.2 Julia sets.- 5.3 Dimension in foliated manifolds.- 5.4 Dimension of a hyperbolic boundary.- 5.5 Dimension of a limit set.- 6 Varia.- 6.1 Complexity growth.- 6.2 Expansive systems.- 6.3 Pseudo-orbits and pseudoleaves.- 6.4 Generic leaves.

Topics in extrinsic geometry of codimension-one foliations

-1. Integral Formulae (Introduction, Preliminaries, Integral Formulae for Codimension-one foliations).-2.Variation Formulae (Introduction, Auxiliary results, Variations of extrinsic geometric quantities, Variations of general functional, Variations of particular functional, Applications and examples).-3. Extrinsic Geometric Flows (Introduction, The systems of PDEs related to EGF, Auxiliary results, Existence and uniqueness results, A solution to general case, Global existence of EGF, Variation formulae for EGF, Extrinsic geometric solitons, Applications and examples).- References.

Dynamics of the geodesic flow of a foliation

The geodesic flow of a foliated Riemannian manifold ( M , F ) is studied. The invariance of some smooth measure is established under some geometrical conditions on F . The Lyapunov exponents and the entropy of this flow are estimated. As an application, the non-existence of foliations with ‘short’ second fundamental tensors is obtained on compact negatively curved manifolds.

Extrinsic curvatures of distributions of arbitrary codimension

Abstract In this article, using the generalized Newton transformation, we define higher order mean curvatures of distributions of arbitrary codimension and we show that they agree with the ones from Brito and Naveira [F. Brito, A.M. Naveira, Total extrinsic curvature of certain distributions on closed spaces of constant curvature, Ann. Global Anal. Geom., 18 (2000) 371–383]. We also introduce higher order mean curvature vector fields and we compute their divergence for certain distributions and using this we obtain total extrinsic mean curvatures.

Integral formulae for codimension-one foliated Randers spaces

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and their applications for codimension-one foliated Randers spaces. The goal is a generalization of Reeb’s formula (that the total mean curvature of the leaves is zero) and its companion (that twice total second mean curvature of the leaves equals to the total Ricci curvature in the normal direction). We also extend results by Brito, Langevin and Rosenberg (that total mean curvatures of arbitrary order for a codimension-one foliated Riemannian manifold of constant curvature don’t depend on a foliation). All of that is done by a comparison of extrinsic and intrinsic curvatures of the two Riemannian structures which arise in a natural way from a given Randers structure.

Geometry and its applications

Part I: Geometry.- The Ricci flow on some generalized Wallach spaces (N.A. Abiev, A. Arvanitoyeorgos, Y.G. Nikonorov, P. Siasos).- Gaussian mean curvature flow for submanifolds in space forms (A. Borisenko, V. Rovenski).- Cantor laminations and exceptional minimal sets in codimension one foliations (G. Hector).- Integral formulas in foliations theory (K. Andrzejewski, P. Walczak, V. Rovenski).- On prescribing the mixed scalar curvature of a foliations (V. Rovenski, L. Zelenko).- The partial Ricci flow for foliations (V. Rovenski).- Osculation in general (P. Walczak).- On stability of totally geodesic unit vector fields on three-dimensional Lie groups (A. Yampolsky).- Part II: Applications.- Rotational liquid film interacted with ambient gaseous media (I. Gaissinski, Y. Levy, V. Rovenski, V. Sherbaum).- On cycles and other geometric phenomena in phase portraits of some nonlinear dynamical systems (V. Golubyatnikov, Yu. A. Gaidov).- Remez-type inequality for smooth functions (Y. Iomdin).

Integral Formulas in Foliation Theory

In this chapter we give an overview of integral formulas, and some of their consequences, appearing in the study of extrinsic geometry of foliations and distributions on Riemannian manifolds.

A VIRTUAL LEAF

First, we study the growth rate of separation of pseudo-orbits of a finitely generated group acting on a compact space. Then, suspending a suitable group action and a suitable pseudo-orbit we get, after a minor modification, a foliation and a Riemannian manifold which is not quasi-isometric to any leaf of any compact foliated manifold but which is quasi-isometric to pseudoleaves of the constructed foliation intersecting its leaves at arbitrarily small angle. Such an object is called a virtual leaf here.

Pseudo-orbits, pseudoleaves and geometric entropy of foliations

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

Multi Ping-Pong and an Entropy Estimate in Groups

Abstract We provide an entropy estimate from below for a finitely generated group of transformation of a compact metric space which contains a ping-pong game with several players located anywhere in the group.