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We address the following conjecture about the existence of common zeros for commuting vector fields in dimension three: if $X,Y$ are two $C^1$ commuting vector fields on a $3$-manifold $M$, and $U$ is a relatively compact open such that $X$ does not vanish on the boundary of $U$ and has a non vanishing Poincar\'e-Hopf index in $U$, then $X$ and $Y$ have a common zero inside $U$. We prove this conjecture when $X$ and $Y$ are of class $C^3$ and every periodic orbit of $Y$ along which $X$ and $Y$ are collinear is partially hyperbolic. We also prove the conjecture, still in the $C^3$ setting, assuming that the flow $Y$ leaves invariant a transverse plane field. These results shed new light on the $C^3$ case of the conjecture., Comment: 50 pages, 16 figures. FInal version to appear in Proceedings of the London Mathematical Society

This paper is the paper announced in [Be2, References [2]]. We show that every compact abelian group of homeomorphisms of $\mathbb{R}^3$ is either zero-dimensional or equivalent to a subgroup of the orthogonal group O(3). We prove a similar result if we replace $\mathbb{R}^3$ by $\mathbb{S}^3$, and we study regular homeomorphisms that are conjugate to their inverses.

Let F be a codimension-one, C^2-foliation on a manifold M without boundary. In this work we show that if the Godbillon--Vey class GV(F) \in H^3(M) is non-zero, then F has a hyperbolic resilient leaf. Our approach is based on methods of C^1-dynamical systems, and does not use the classification theory of C^2-foliations. We first prove that for a codimension--one C^1-foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points E(F) has positive Lebesgue measure. We then prove that if E(F) has positive measure for a C^1-foliation F, then F must have a hyperbolic resilient leaf, and hence its geometric entropy must be positive. The proof of this uses a pseudogroup version of the Pliss Lemma. The theorem then follows, as a C^2-foliation with non-zero Godbillon-Vey class has non-trivial Godbillon measure. These results apply for both the case when M is compact, and when M is an open manifold., Comment: This manuscript is a revision of the section 3 material from the previous version, and includes edits to the pictures in the text

We study the notion of sensitivity on $G$-spaces and through examples observe that $G$-sensitivity neither implies nor is implied by sensitivity. Further, we obtain necessary and sufficient conditions for a map to be $G$-sensitive. Next, we define the notion of Devaney's chaos on $G$-space and show that $G$-sensitivity is a redundant condition in the definition.

We study a wide class of metrics in a Lebesgue space with a standard measure, the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the "-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the "-entropy of the averages of some (and hence any) admissible metric over fragments of its trajectory is uniformly bounded., Comment: 37p. Ref.19

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold without boundary is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property holds in the setting of holomorphically foliated spaces.

This paper is devoted to a systematic study of the geometry of nondegenerate ${\mathbb R}^n$-actions on $n$-manifolds. The motivations for this study come from both dynamics, where these actions form a special class of integrable dynamical systems and the understanding of their nature is important for the study of other Hamiltonian and non-Hamiltonian integrable systems, and geometry, where these actions are related to a lot of other geometric objects, including reflection groups, singular affine structures, toric and quasi-toric manifolds, monodromy phenomena, topological invariants, etc. We construct a geometric theory of these actions, and obtain a series of results, including: local and semi-local normal forms, automorphism and twisting groups, the reflection principle, the toric degree, the monodromy, complete fans associated to hyperbolic domains, quotient spaces, elbolic actions and toric manifolds, existence and classification theorems.

We extend the one dimensional Denjoy-Sacksteder theorems to some diffeomorphism groups of smooth compact closed n-dimensional manifolds. More precisely, we show the existence of a non trivial stabilizer for actions of “quasi-conformal groups” admitting an “exceptional” minimal set which is of “strongly decreasing type”; our results include the classical Denjoy-Sacksteder theorems.

For [math] and [math] , consider the matrix [math] and the rotation matrix [math] . Let [math] denote some product of [math] instances of [math] and [math] of [math] , with the condition [math] ( [math] ). We analyze the measure of the set of [math] for which [math] ( [math] ). This can be regarded as a model problem for the Bochi–Fayad conjecture.

We show that if $r\geq 3$ and $\alpha$ is a faithful $Z^r$-Cartan action on a torus $T^d$ by automorphisms, then any closed subset of $(T^d)^2$ which is invariant and topologically transitive under the diagonal $\bZ^r$-action by $\alpha$ is homogeneous, in the sense that it is either the full torus $(T^d)^2$, or a finite set of rational points, or a finite disjoint union of parallel translates of some d-dimensional invariant subtorus. A counterexample is constructed for the rank 2 case., Comment: 40 pages

We prove that a transversely equicontinuous minimal lamination on a locally compact metric space $Z$ has a transversely invariant Radon measure. Moreover if the space $Z$ is compact, then the tranversely invariant Radon measure is shown to be unique up to a scaling., 10 pages

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky's property ($\tau$) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to answering the question of Lubotzky and Zuk, whether for families of subgroups, property ($\tau$) is inherited to the lattice of subgroups generated by the family. On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicite estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expander covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance., Comment: Corrections made based on the referee's comments

We show that if a connected closed orientable manifold $M$ admits a codimension one locally free smooth action $\phi$ of a connected nilpotent Lie group such that any orbit of $\phi$ is non-compact, then $M$ is homeomorphic to a nilmanifold. And as an example of such an action, we study also a homogeneous action.

Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes., 22 pages

In this article we collect a series of observations that constrain actions of many groups on compact manifolds. In particular, we show that "generic" finitely generated groups have no smooth volume preserving actions on compact manifolds while also producing many finitely presented, torsion free groups with the same property., References added, minor changes

Using methods and results of decomposition theory we construct minimal actions of groups of homeomorphisms of some classical fractals (the Sierpi\'nski carpet and its generalizations, and the Menger curve) with positive entropy. Suspending these group actions we get minimal foliated spaces which have positive geometric entropy and are modelled on these fractals.

Let $R_1, R_2,\ldots,R_m$ be rotations generating ${\mathbb{SO}}_{d+1}$ , $d\geq 2$ , and let $f_1, f_2,\ldots,f_m$ be small smooth perturbations of them. We show that $\{f_\alpha\}$ can be linearized simultaneously if and only if the associated random walk has zero Lyapunov exponents. As a consequence, we obtain stable ergodicity of actions of random rotations in even dimensions

We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens’ theorem for the full G G -shift for a finitely-generated torsion-free nilpotent group G G . Using bounds for the Möbius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape \[ ∑ | τ | ≤ N 1 e h | τ | ∼ C N α ( log ⁡ N ) β \sum _{\vert \tau \vert \le N}\frac {1}{e^{h\vert \tau \vert }}\sim CN^{\alpha }(\log N)^{\beta } \] where | τ | \vert \tau \vert is the cardinality of the finite orbit τ \tau and h h denotes the topological entropy. For the usual orbit-counting function we find upper and lower bounds, together with numerical evidence to suggest that for actions of non-cyclic groups there is no single asymptotic in terms of elementary functions.

If $\cal {G}$ is a finitely generated group with generators $\{g_1,\dots,g_j\}$ , then an infinite-order element $f \in \cal {G}$ is a distortion element of $\cal {G}$ provided that ${\lim \, \inf_{n \to \infty} |f^n|/n = 0,}$ where $|f^n|$ is the word length of $f^n$ in the generators. Let $S$ be a closed orientable surface, and let $\rm{Diff}(S)_0$ denote the identity component of the group of $C^1$ -diffeomorphisms of $S$ . Our main result shows that if $S$ has genus at least two and that if $f$ is a distortion element in some finitely generated subgroup of $\rm{Diff}(S)_0$ , then $\rm {supp}(\mu) \subset \rm {Fix}(f)$ for every $f$ -invariant Borel probability measure $\mu$ . Related results are proved for $S = T^2$ or $S^2$ . For $\mu$ a Borel probability measure on $S$ , denote the group of $C^1$ -diffeomorphisms that preserve $\mu$ by $\rm{Diff}_{\mu}(S)$ . We give several applications of our main result, showing that certain groups, including a large class of higher-rank lattices, admit no homomorphisms to $\rm{Diff}_{\mu}(S)$ with infinite image

Let $Y=(f,g,h):\mathbb{R}^{3} \to \mathbb{R}^{3}$ be a $C^{2}$ map and let $\spec(Y)$ denote the set of eigenvalues of the derivative $DY_p$, when $p$ varies in $\mathbb{R}^3$. We begin proving that if, for some $\epsilon>0,$ $\spec(Y)\cap (-\epsilon,\epsilon)=\emptyset,$ then the foliation $\mathcal{F}(k),$ with $k\in \{f,g,h\},$ made up by the level surfaces $\{k={\rm constant}\},$ consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek's Jacobian Conjecture for polynomial maps of $\mathbb{R}^n.$, Comment: 13 pages and 3 figures

Let $G$ be a simply connected Lie group and consider a Lie $G$ foliation $\mathscr F$ on a closed manifold $M$ whose leaves are all dense in $M$ . Then the space of ends ${\mathscr E}(F)$ of a leaf $F$ of $\mathscr F$ is shown to be either a singleton, a two points set, or a Cantor set. Further if $G$ is solvable, or if $G$ has no cocompact discrete normal subgroup and $\mathscr F$ admits a transverse Riemannian foliation of the complementary dimension, then ${\mathscr E}(F)$ consists of one or two points. On the contrary there exists a Lie $\widetilde{SL}(2,\bm{R})$ foliation on a closed 5-manifold whose leaf is diffeomorphic to a 2-sphere minus a Cantor set.

We introduce the so-called BT-category of borelian-topological spaces: it will be a natural frame for a measurable classification of usual foliations and laminations. We focus on the two-dimensional case: borelian laminations by surfaces. We prove two main results: (1) Any borelian lamination by planes is the suspension of a $\Z^2$-action on a Borel space iff this lamination is hyperfinite. (2) Any borelian lamination by surfaces is amenable if parabolic, i.e. if it admits a complex structure parabolic on each leaf. The third result is an improvement of (2) in case of laminations endowed with a transverse quasi-invariant measure $\mu$. The statement is the following: (3) Any borelian lamination by planes, cylinders and tori is $\mu$-amenable if and only if it admits a metric which is flat on $\mu$-almost all leafs., Comment: 32 pages, 1 figure

In this paper we consider the ergodic properties of linear actions on the plane of (2×2)-matrices with entries in the complex numbers, quaternions, or Clifford numbers.

If $M$ is a compact or convex-cocompact negatively curved manifold, we associate to any Gibbs measure on $\tm$ a quasi-invariant transverse measure for the horospherical foliation, and prove that this measure is uniquely determined by its Radon-Nikodym cocycle. (This extends the Bowen-Marcus unique ergodicity result for this foliation.) We shall also prove equidistribution properties for the leaves of the foliation w.r.t. these Gibbs measures. We use these results in the study of invaiant meausres for horospherical foliations on regular covers of $M$., 28 pages, 6 figures

An action of a topological group $G$ on a topological space $X$ is orbit nonproper if, for some $x\in X$, the map $g\mapsto gx:G\to X$ is nonproper. We describe the collection of connected, simply connected Lie groups admitting a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold.

Let $G$ be a non-unimodular solvable Lie group which is a semidirect product of $R^m$ and $R^n$. We consider a codimension one locally free volume preserving action of $G$ on a closed manifold. It is shown that, under some conditions on the group $G$, such an action is homogeneous. It is also shown that such a group $G$ has a homogeneous action if and only if the structure constants of $G$ satisfy certain algebraic conditions.

We consider endomorphism actions of arbitrary discrete semigroups on a connected metrizable topological group G. We give necessary and sufficient conditions for expansiveness of such actions when G is a Lie group or a compact finite-dimensional group., Comment: 21 pages, no figures

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $\Gamma $ by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $\Gamma $-action by means of a `fundamental homoclinic point', and the description of entropy in terms of the renormalized logarithmic growth-rate of the set of $\Gamma_n$-fixed points, where $(\Gamma_n, n\ge1)$ is a decreasing sequence of finite index normal subgroups of $\Gamma $ with trivial intersection., Comment: Using a recent result of Y. Choi the discussion of expansiveness was strengthened and shortened. The paper will appear in: Ergodic Theory and Dynamical Systems