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 $ε>0,$ $\spec(Y)\cap (-ε,ε)=\emptyset,$ 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 $\mathbb{R}^n.$
It was shown that in robustly transitive, partially hyperbolic diffeomorphisms on three dimensional closed manifolds, the strong stable or unstable foliation is minimal. In this article, we prove ``almost all'' leaves of both stable and unstable foliations are dense in the whole manifold.
Metric entropies along a hierarchy of unstable foliations are investigated for $C^1$ diffeomorphisms with dominated splitting. The analogues of Ruelle's inequality and Pesin's formula, which relate the metric entropy and Lyapunov exponents in each hierarchy, are given.
This essay summarizes the state of the art on some aspects of the dynamics of polynomial diffeomorphsms in complex dimension two, and it presents a number of open questions.
Let $f$ be a $C^2$ partially hyperbolic diffeomorphisms of ${\mathbb T}^3$ (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism $A$ with eigenvalues $$λ_{s}<1<λ_{c}<λ_{u}.$$ Under the assumption that the set $$\{x: \,\mid\log \det(Tf\mid_{E^{cu}(x)})\mid \leq \log λ_{u} \}$$ has zero volume inside any unstable leaf of $f$ where $E^{cu} = E^c\oplus E^u$ is the center unstable bundle, we prove that the stable foliation of $f$ is $C^1$ robustly minimal, i.e., the stable foliation of any diffeomorphism $C^1$ sufficiently close to $f$ is minimal. In particula…
In this paper we focused our study on Derived From Anosov diffeomorphisms (DA diffeomorphisms ) of the torus $\mathbb{T}^3,$ it is, an absolute partially hyperbolic diffeomorphism on $\mathbb{T}^3$ homotopic to an Anosov linear automorphism of the $\mathbb{T}^3.$ We can prove that if $f: \mathbb{T}^3 \rightarrow \mathbb{T}^3 $ is a volume preserving DA diffeomorphism homotopic to linear Anosov $A,$ such that the center Lyapunov exponent satisfies $λ^c_f(x) > λ^c_A > 0,$ with $x $ belongs to a positive volume set, then the center foliation of $f$ is non absolutely continuous. We construct a new…
The two main results of this paper concern the regularity of the invariant foliation of a C0-integrable symplectic twist diffeomorphisms of the 2-dimensional annulus, namely that $\bullet$ the generating function of such a foliation is C1 ; $\bullet$ the foliation is H{ö}lder with exponent 1/2. We also characterize foliations by graphs that are straightenable via a symplectic homeomorphism and prove that every symplectic homeomorphism that leaves invariant all the leaves of a straightenable foliation has Arnol'd-Liouville coordinates, in which the Dynamics restricted to the leaves is conjugate…
The (measure-theoretical) entropy of a diffeomorphism along an expanding invariant foliation is the rate of complexity generated by the diffeomorphism along the leaves of the foliation. We prove that this number varies upper semi-continuously with the diffeomorphism ($\C^1$ topology), the invariant measure (weak* topology) and the foliation itself in a suitable sense. This has several important consequences. For one thing, it implies that the set of Gibbs $u$-states of $\C^{1+}$ partially hyperbolic diffeomorphisms is an upper semi-continuous function of the map in the $\C^1$ topology. Another…
We consider a partially hyperbolic C1-diffeomorphism f on a smooth compact manifold M with a uniformly compact f-invariant center foliation. We show that if the unstable bundle is one-dimensional and oriented, then the holonomy of the center foliation vanishes everywhere, the quotient space of the center foliation is a torus and f induces a hyperbolic automorphism on it, in particular, f is centrally transitive. We actually obtain further interesting results without restrictions on the unstable, stable and center dimension: we prove a kind of spectral decomposition for the chain recurrent set …
We consider diffeomorphisms in the $C^\infty$-closure of the conjugancy class of translations of the 2-torus. By a theorem of Fathi and Herman, a generic diffeomorphism in that space is minimal and uniquely ergodic. We define a new mixing-like property, which takes into account the "directions" of mixing, and we prove that generic elements of the space in question satisfy this property. As a consequence, we show that there is a residual set of strictly ergodic diffeomorphisms without invariant foliations of any kind. We also obtain an analytic version of these results.
We show that a partially hyperbolic $C^1$ -diffeomorphism $f : M \to M$ with a uniformly compact $f$ -invariant center foliation $F^c$ is dynamically coherent. Further, the induced homeomorphism $F : M/F^c \to M/F^c$ on the quotient space of the center foliation has the shadowing property, i.e. for every $\varepsilon> 0$ there exists $δ> 0$ such that every $δ$-pseudo orbit of center leaves is $\varepsilon$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient d…
A diffeomorphism $f:\mathbb{R}^2\to\mathbb{R}^2$ in the plane is Anosov if it has a hyperbolic splitting at every point of the plane. The two known topological conjugacy classes of such diffeomorphisms are linear hyperbolic automorphisms and translations (the existence of Anosov structures for plane translations was originally shown by W. White). P. Mendes conjectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the plane. We prove that this claim holds when the Anosov diffeomorphism is the time-one map of a flow, via a theorem about foliations invariant …
In this paper we are considering partially hyperbolic diffeomorphims of the torus, with $dim(E^c) > 1.$ We prove, under some conditions, that if the all center Lyapunov exponents of the linearization $A,$ of a \mbox{DA-diffeomorphism} $f,$ are positive and the center foliation of $f$ is absolutely continuous, then the sum of the center Lyapunov exponents of $f$ is bounded by the sum of the center Lyapunov exponents of $A.$ After, we construct a $C^1-$open class of volume preserving \mbox{DA-diffeomorphisms}, far from Anosov diffeomorphisms, with non compact pathological two dimensional center …
In this work we completely classify $C^\infty$ conjugacy for conservative partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain $C^\infty$ conjugacy to its linear Anosov automorphism. On a recent work Avila, Viana and Wilkinson proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute…
We prove that if the stable foliation and the unstable foliation of an Anosov diffeomorphism on a connected compact manifold are $C^3$, then the diffeomorphism has fixed points. This is a partial positive answer to a Smale conjecture for fixed points of Anosov diffeomorphisms.
Center foliations of partially hyperbolic diffeomorphisms may exhibit pathological behavior from a measure-theoretical viewpoint: quite often, the disintegration of the ambient volume measure along the center leaves consists of atomic measures. We add to this theory by constructing stable examples for which the disintegration is singular without being atomic. In the context of diffeomorphisms with mostly contracting center direction, for which upper leafwise absolute continuity is known to hold, we provide examples where the center foliation is not lower leafwise absolutely continuous.
Let $M$ be a compact manifold equipped with a pair of complementary foliations, say horizontal and vertical. In Catuogno, Silva and Ruffino ($Stoch$. $Dyn$., 2013) it is shown that, up to a stopping time $τ$, a stochastic flow of local diffeomorphisms $φ_t$ in $M$ can be written as a Markovian process in the subgroup of diffeomorphisms which preserve the horizontal foliation composed with a process in the subgroup of diffeomorphisms which preserve the vertical foliation. Here, we discuss topological aspects of this decomposition. The main result guarantees the global decomposition of a flow if…
A partially hyperbolic diffeomorphism $f$ has quasi-shadowing property if for any pseudo orbit ${x_k}_{k\in \mathbb{Z}}$, there is a sequence of points ${y_k}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_k)$ by a motion $τ$ along the center direction. We show that any partially hyperbolic diffeomorphism has quasi-shadowing property, and if $f$ has $C^1$ center foliation then we can require $τ$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^0$-perturbation. When…
In this work we obtain some metric and ergodic properties of $C^{1+}$ partially hyperbolic diffeomorphisms with one-dimensional topological neutral center, mainly regarding the behavior of its center foliation. Based on a trichotomy for the center conditional measures of any invariant ergodic measure, we show that if these conditionals have full support, then the center foliation is leafwise absolutely continuous, the diffeomorphism is Bernoulli in the $C^{1+}$ case, and an invariance principle occurs in the sense that M may be covered by a finite number of open sets where the system of center…
Let f:M->M be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of f. This is proved by thorough study of the accessible boundaries of the center-stable and the center-unstable leaves. Also we show that a finite cover of f fibers over an Anosov toral automorphism if one of the following conditions is met: 1. the center foliation of f has codimension 2, or 2. the center leaves of f are simply connected leaves and the unstable foliation of f is …
We show that the metric entropy of a $C^1$ diffeomorphism with a dominated splitting and the dominating bundle uniformly expanding is bounded from above by the integrated volume growth of the dominating (expanding) bundle plus the maximal Lyapunov exponent from the dominated bundle multiplied by its dimension. We also discuss different types of volume growth that can be associated with an expanding foliation and relationships between them, specially when there exists a closed form non-degenerate on the foliation. Some consequences for partially hyperbolic diffeomorphisms are presented.
In a non-compact setting, the notion of hyperbolicity, and the associated structure of stable and unstable manifolds (for unbounded orbits), is highly dependent on the choice of metric used to define it. We consider the simplest version of this, the analogue for the plane of Anosov diffeomorphisms, studied earlier by W. White and P. Mendes. The two known topological conjugacy classes of such diffeomorphisms are linear hyperbolic automorphisms and translations. We show that if the structure of stable and unstable manifolds is required to be preserved by these conjugacies, the number of distinct…
For a partiallyhyperbolic diffeomorphism on a 3-manifold, we show that any invariant foliation tangent to the center-unstable (or center-stable) bundle has no compact leaves.
We study perturbations of a partially hyperbolic toral automorphism L which is diagonalizable over C and has a dense center foliation. For a small perturbation of L with a smooth center foliation we establish existence of a smooth leaf conjugacy to L. We also show that if a small perturbation of an ergodic irreducible L has smooth center foliation and is bi-Holder conjugate to L, then the conjugacy is smooth. As a corollary, we show that for any symplectic perturbation of such an L any bi-Holder conjugacy must be smooth. For a totally irreducible L with two-dimensional center, we establish a n…
Let $f$ be a partially hyperbolic diffeomorphism on a closed (i.e., compact and boundaryless) Riemannian manifold $M$ with a uniformly compact center foliation $\mathcal{W}^{c}$. The relationship among topological entropy $h(f)$, entropy of the restriction of $f$ on the center foliation $h(f, \mathcal{W}^{c})$ and the growth rate of periodic center leaves $p^{c}(f)$ is investigated. It is first shown that if a compact locally maximal invariant center set $Λ$ is center topologically mixing then $f|_Λ$ has the center specification property, i.e., any specification with a large spacing can be cen…
We prove that generically in $\text{Diff}^{1}_{m}(M)$, if an expanding $f$-invariant foliation $W$ of dimension $u$ is minimal and there is a periodic point of unstable index $u$, the foliation is stably minimal. By this we mean there is a $C^{1}$-neighborhood $\mathcal{U}$ of $f$ such that for all $C^{2}$-diffeomorphisms $g\in \mathcal{U}$, the $g$-invariant analytic continuation of $W$ is minimal. In particular, all such $g$ are topologically mixing. Moreover, all such $g$ have a hyperbolic ergodic component of the volume measure $m$ which is essentially dense. This component is, in fact, Be…
We show that for a large space of volume preserving partially hyperbolic diffeomorphisms of the 3-torus with non-compact central leaves the central foliation generically is non-absolutely continuous.
We consider Anosov diffeomorphisms on $\mathbb{T}^3$ such that the tangent bundle splits into three subbundles $E^s_f \oplus E^{wu}_f \oplus E^{su}_f.$ We show that if $f$ is $C^r, r \geq 2,$ volume preserving, then $f$ is $C^1$ conjugated with its linear part $A$ if and only if the center foliation $\mathcal{F}^{wu}_f$ is absolutely continuous and the equality $λ^{wu}_f(x) = λ^{wu}_A,$ between center Lyapunov exponents of $f$ and $A,$ holds for $m$ a.e. $x \in \mathbb{T}^3.$ We also conclude rigidity of derived from Anosov diffeomorphism, assuming an strong absolute continuity property (Unifo…
We show that for the C^1-open set of partially hyperbolic diffeomorphisms constructed in (M. Shub and A. Wilkinson, "Pathological foliations and removable zero exponents," Invent. math. 139 (2000) 3, 495-508), Lebesgue measure on the 3-torus decomposes as atomic measure along the leaves of the central foliation.
Given any smooth Anosov map we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that in the case of expanding maps it reduces exactly to the usual space of functions of bounded variation which has proven particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations which provides new information that could prove useful in treating hyperbolic systems with singularities.
We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique non-trivial homologies. We prove the following two results: if the center foliation is one dimensional, then the topological entropy is locally a constant; and if the center foliation is two dimensional, then the topological entropy is continuous on the set of all $C^\8$ diffeomorphisms. The proof uses a topological invariant we introduced; Yomdin's theorem on upper semi-continuity; Katok's theorem on lower semi-continuity for two dimensional systems and a ref…
We address the following question: let F:(R^2,0)->(R^2,0) be an analytic local diffeomorphism defined in the neighborhood of the non resonant elliptic fixed point 0 and let Φbe a formal conjugacy to a normal form N. Supposing F leaves invariant the foliation by circles centered at 0, what is the analytic nature of Φand N?
Weprovethattheasymptoticsofergodicintegralsalonganinvariant foliation of a toral Anosov diffeomorphism, or of a pseudo-Anosov diffeomorphism on a compact orientable surface of higher genus, are determined (up to a logarithmic error) by the action of the diffeomorphism on the cohomology of the surface. As a consequence of our argument and of the results of Giulietti and Liverani [GL] on horospherical averages, toral Anosov diffeomorphisms have no Ruelle resonances in the open interval $(1,e^{h_{top}} )$.
We consider a smooth Anosov diffeomorphism with a smooth dynamical foliation. We show upper bounds on the essential spectral radius of its transfer operator acting on anisotropic Sobolev spaces. (Such bounds are related to the essential decorrelation rate for the SRB measure.) We compare our results to the estimates of Kitaev on the domain of holomorphy of dynamical Fredholm determinants for differentiable dynamics.
After a short review on foliations, we prove that a codimension 1 holomorphic foliation on $\mathbb P^3_{\mathbb C}$ with simple singularities is given by a closed rational 1-form. The proof uses Hironaka-Matsumura prolongation theorem of formal objects.
We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ are either dynamically coherent or have an invariant two-dimensional torus which is either contracting or repelling. We develop for this end some general results on codimension one foliations which may be of independent interest.
A holomorphic pre-foliation $\mathscr{F}=\ell\boxtimes\mathcal{F}$ of co-degree $1$ and degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is the data of a line $\ell$ of $\mathbb{P}^{2}_{\mathbb{C}}$ and a holomorphic foliation $\mathcal{F}$ on $\mathbb{P }^{2}_{\mathbb{C}}$ of degree $d-1.$ We study pre-foliations of co-degree $1$ on $\mathbb{P}^{2}_{\mathbb{ C}}$ with a flat Legendre transform (dual web). After having established some general results on the flatness of the dual $d$-web of a homogeneous pre-foliation of co-degree $1$ and degree $d$, we describe some explicit examples and we show th…
In this paper we study holomorphic foliations on $\mathbb{P}^2$ with only one singular point. If the singularity has algebraic multiplicity one, we prove that the foliation has no invariant algebraic curve. We also present several examples of such foliations in degree three.
Let $M$ be a manifold equipped (locally) with a pair of complementary foliations. In Catuogno, da Silva and Ruffino (Stoch. Dyn. 2013), it is shown that, up to a stopping time $τ$, a stochastic flow of local diffeomorphisms $φ_t$ in $M$ can be decomposed in diffeomorphisms that preserves this foliations. In this article we present techniques which allows us to extend the time of this decomposition. For this extension, we use two techniques: In the first one, assuming that the vector fields of the system commute with each other, we apply Marcus equation to jump nondecomposable diffeomorphisms. …
We study groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the notion of holonomy group of a leaf of a foliation. For dimension one, there is a well-established dictionary relating analytic/formal classification of the group, with its algebraic properties (finiteness, commutativity, solvability, ...). Such system of equivalences also characterizes the existence of suitable {\it integrating factors}, i.e., invariant vector fields and one-forms associated…
This paper provides a canonical compactification of the plane ${\mathbb R}^2$ by adding a circle at infinity associated to a countable family of singular foliations or laminations (under some hypotheses), generalizing an idea by Mather \cite{Ma}. Moreover any homeomorphism of ${\mathbb R}^2 $ preserving the foliations extends on the circle at infinity. Then this paper provides conditions ensuring the minimality of the action on the circle at infinity induced by an action on ${\mathbb R}^2 $ preserving one foliation or two transverse foliations. In particular the action on the circle at infinit…
We study groups of formal diffeomorphisms in several complex variables. For abelian, metabelian or nilpotent groups we investigate the existence of suitable formal vector fields and closed differential forms which exhibit an invariance property under the group action. Our results are applicable in the construction of suitable integrating factors for holomorphic foliations with singularities. We believe they are a starting point in the study of the connection between Liouvillian integration and transverse structures of holomorphic foliations with singularities in the case of arbitrary codimensi…
Let $F: \mathbb{R}^n\to\mathbb{R}^n$ be a $C^{\infty}$ map such that $DF(x)$ is invertible for every $x\in\mathbb{R}^n$. Although being a local diffeomorphism, $F$ is not necessarily globally injective if $n\geq2$. Finding additional assumptions implying the global injectivity of $F$ for $n\geq 2$ is object of intense study in several areas of Mathematics. In this paper we revisit some assumptions and relations between them in the bidimensional case and discuss the natural higher dimensional situation.
We study 3-dimensional dynamically coherent partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the transverse geometry and topology of the center stable and center unstable foliations, and the dynamics within their leaves. We find a structural dichotomy for these foliations, which we use to show that every such diffeomorphism on a hyperbolic or Seifert fibered 3-manifold is leaf conjugate to the time one map of a (topological) Anosov flow. This proves a classification conjecture of Hertz-Hertz-Ures in hyperbolic 3-manifolds and in the homotopy class of the ide…
