Homoclinic tangencies and singular hyperbolicity are involved in the Palis conjecture for vector fields. Typical three dimensional vector fields are well understood by recent works. We study the dynamics of higher dimensional vector fields that are away from homoclinic tangencies. More precisely, we prove that for \emph{any} dimensional vector field that is away from homoclinic tangencies, all singularities contained in its robustly transitive singular set are all hyperbolic and have the same index. Moreover, the robustly transitive set is {$C^1$-generically }partially hyperbolic if the vector field cannot be accumulated by ones with a homoclinic tangency.
Let $\mathcal{E}$ be the set of endomorphisms of the $n$-torus. We exhibit an example of a map such that is robustly transitive if $\mathcal{E}$ is endowed with the $C^2$ topology but is not robustly transitive if $\mathcal{E}$ is endowed with the $C^1$ topology.
We say that a compact invariant set $Λ$ of a $C^1$-vector field $X$ on a compact boundaryless Riemannian manifold $M$ is robustly shadowable if it is locally maximal with respect to a neighborhood $U$ of $Λ$, and there exists a $C^1$-neigborhood $\mathcal{U}$ of $X$ such that for any $Y \in \mathcal{U}$, the continuation $Λ_Y$ of $Λ$ for $Y$ and $U$ is shadowable for $Y_t$. In this paper, we prove that any chain transitive set of a $C^1$-vector field on $M$ is hyperbolic if and only if it is robustly shadowable.
We improve previous results by exhibiting a construction that contains all known examples. A suficient condition for the existence of robustly transitive maps displaying singularities on a certain large class of compact manifolds is given.
We construct symplectomorphisms in dimension $d\geq 4$ having a semi-local robustly transitive partially hyperbolic set containing $C^2$-robust homoclinic tangencies of any codimension $c$ with $0<c\leq d/2$.
We show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic, but nevertheless have constant entropy and isomorphic unique measures of maximal entropy.
For diffeomorphisms or for non-singular flows, there are many results relating properties persistent under C1 perturbations and global structures for the dynamics ( such as hyperbolicity, partial hyperbolicity, dominated splitting). However, a difículty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point. In [BdL] with Christan Bonatti we propose a a general procedure for adapting the usual hyperbolic structures to the singularities. In this paper, using this tool, we recover the results in [BDP] for flows, showing that robustly cha…
In this paper, we define $C^1$-robust transitivity for actions of $\RR^2$ on closed connected orientable manifolds. We prove that if the ambient manifold is three dimensional and the dense orbit of a robustly transitive action is not planar, then it is "degenerate" and the action is defined by an Anosov flow.
We present an example of a $\mathcal{C}^1$-robustly transitive skew-product with non-trivial, non-hyperbolic action on homology. The example is conservative, ergodic, non-uniformly hyperbolic and its fiber directions cannot be decomposed into two dominated expanded/contracted bundles.
We construct a family of partially hyperbolic skew-product diffeomorphisms on $\mathbb{T}^3$ that are robustly transitive and admitting two physical measures with intermingled basins. In particularly, all these diffeomorphisms are not topologically mixing. Moreover, for every such example, it exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.
Consider the set $E$ of endomorphisms of the n-torus endowed with the $C^1$ topology. A point in $E$ that is persistently singular and robustly transitive is exhibited.
We prove that every robustly transitive and every stably ergodic symplectic diffeomorphism on a compact manifold admits a dominated splitting. In fact, these diffeomorphisms are partially hyperbolic.
It is well known that smooth (or continuous) vector fields cannot be topologically transitive on the sphere $§^2$. Piecewise-smooth vector fields, on the other hand, may present non-trivial recurrence even on $§^2$. Accordingly, in this paper the existence of topologically transitive piecewise-smooth vector fields on $§^2$ is proved, see Theorem \ref{teorema-principal}. We also prove that transitivity occurs alongside the presence of some particular portions of the phase portrait known as {\it sliding region} and {\it escaping region}. More precisely, Theorem \ref{main:transitivity} states tha…
In this article we intend to contribute in the understanding of the ergodic properties of the set RT of robustly transitive local diffeomorphisms on a compact manifold M without boundary. We prove that there exists a C^1 residual subset R_0 of RT such that any f in R_0 has a residual subset of M with dense pre-orbits. Moreover, C^1 generically in the space of local diffeomorphisms with no splitting and all points with dense pre-orbit, there are uncountably many ergodic expanding invariant measures with full support and exhibiting exponential decay of correlations. In particular, these results …
We analyze a class of deformations of Anosov diffeomorphisms: these $C^0$-small, but $C^1$-macroscopic deformations break the topological conjugacy class but leave the high entropy dynamics unchanged. More precisely, there is a partial conjugacy between the deformation and the original Anosov system that identifies all invariant probability measures with entropy close to the maximum. We also establish expansiveness around those measures. This class of deformations contains many of the known nonhyperbolic robustly transitive diffeomorphisms. In particular, we show that it includes a class of no…
We address the problem of necessary conditions and topological obstructions for the existence of robustly transitive maps on surfaces. Concretely, we show that partial hyperbolicity is a necessary condition in order to have $C^1$ robustly transitive endomorphisms with critical points on surfaces, and the only surfaces that admits robustly transitive maps are either the torus or the klein bottle. Moreover, we show that every robustly transitive endomorphism is homotopic to a linear map having at least one eigenvalue with modulus larger than one.
We prove that any diffeomorphism of a compact manifold can be approximated in topology C1 by another diffeomorphism exhibiting a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by one which is essentially hyperbolic (it has a finite number of transitive hyperbolic attractors with open and dense basin of attraction).
We study various types of shadowing properties and their implication for C1 generic vector fields. We show that, generically, any of the following three hypotheses implies that an isolated set is topologically transitive and hyperbolic: (i) the set is chain transitive and satisfies the (classical) shadowing property, (ii) the set satisfies the limit shadowing property, or (iii) the set satisfies the (asymptotic) shadowing property with the additional hypothesis that stable and unstable manifolds of any pair of critical orbits intersect each other. In our proof we essentially rely on the proper…
Robust transitivity without sectional-hyperbolicity
Instituto de Matemática, Universidade Federal do Rio de Janeiro（里约热内卢联邦大学数学研究所）；Hangzhou International Innovation Institute of Beihang University（北京航空航天大学杭州创新研究院）；Faculty of Applied Mathematics, AGH University of Science and Technology（AGH科技大学应用数学学院）
We study $C^1$-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices and prove that they exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets with residual basin of attraction. This represents a partial positive answer to conjectures in \cite{am}, the Palis conjecture \cite{pa} and extend the Araujo's thesis to higher dimensions \cite{a}.
In this paper we show the relation between robust transitivity and robust ergodicity for conservative diffeomorphisms. In dimension 2 robustly transitive systems are robustly ergodic. For the three dimensional case, we define it almost robust ergodicity and prove that generically robustly transitive systems are almost robustly ergodic, if the Lyapunov exponents are nonzero. We also show in higher dimensions, that under some conditions robust transitivity implies robust ergodicity.
Singular hyperbolicity is a weakened form of hyperbolicity that has been introduced for vector fields in order to allow non-isolated singularities inside the non-wandering set. A typical example of a singular hyperbolic set is the Lorenz attractor. However, in contrast to uniform hyperbolicity, singular hyperbolicity does not immediately imply robust topological properties, such as the transitivity. In this paper, we prove that open and densely inside the space of $C^1$ vector fields of a compact manifold, any singular hyperbolic attractors is robustly transitive.
We study the specification property for partially hyperbolic dynamical systems. In particular, we show that if a partially hyperbolic diffeomorphism has two saddles with different indices, and stable manifold of one of them coincides with the strongly stable leaf, then it does not satisfy the specification property. As an application, we prove that there exists a $C^1$-open and dense subset $\mathcal P$ in the set of robustly non-hyperbolic transitive diffeomorphisms on a three dimensional closed manifold such that diffeomorphisms in $\mathcal P$ do not satisfy the specification property.
In the paper, we show that for a generic $C^1$ vector field $X$ on a closed three dimensional manifold $M$, any isolated transitive set of $X$ is singular hyperbolic. It is a partial answer of the conjecture in \cite{MP}.
We prove that in a compact manifold of dimension $n\geq 2$, a $C^{1+α}$ volume-preserving diffeomorphisms that are robustly transitive in the $C^1$-topology have a dominated splitting. Also we prove that for 3-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field can not have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in 3-dimensional manifolds are Anosov. For this, we prove some ``pasting'' lemma, which allows to make perturbations in conservative systems.
Perturbations of time-one maps of transitive Anosov flows are studied. We show that most perturbations have no intermingled basins of hyperbolic physical measures.
We address the problem of giving necessary and sufficient conditions in order to have robustly transitive endomorphisms admitting persistent critical sets. We exhibit different type of open examples of robustly transitive maps in any isotopic class of endomorphisms acting on the two dimensional torus admitting persistent critical points. We also provide some necessary condition for robust transitivity in this setting.
We study the ergodic theory of non-conservative C^1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C^1-generic diffeomorphisms are nonuniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set L of any C^1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the w…
We prove that C^1-robustly transitive diffeomorphisms on surfaces with boundary do not exist, and we exhibit a class of diffeomorphisms of surfaces with boundary which are C^k-robustly transitive, with k greater or equal than 2. This class of diffeomorphisms are examples where a version of Palis' conjecture on surfaces with boundary, about homoclinic tangencies and uniform hyperbolicity, does not hold in the C^2-topology. This follows showing that blow-up of pseudo-Anosov diffeomorphisms on surfaces without boundary, become C^2-robustly topologically mixing diffeomorphisms on a surfaces with b…
The coexistence of singularities and regular orbits in chain transitive sets has been a major obstacle in understanding the hyperbolic/partial hyperbolic nature of robust dynamics. Notably, the vector fields with all periodic orbits robustly hyperbolic (star flows), are hyperbolic in absence of singularities. Morales, Pacifico and Pujals proposed a partial hyperbolicity called "singular hyperbolicity" that characterizes an open and dense subset of three dimensional star flows. In higher dimensions, Bonatti and da Luz characterize an open and dense set of star vector fields by multi-singular hy…
We provide examples of transitive partially hyperbolic dynamics (specific but paradigmatic examples of homoclinic classes) which blend different types of hyperbolicity in the one-dimensional center direction. These homoclinic classes have two disjoint parts: an "exposed" piece which is poorly homoclinically related with the rest and a "core" with rich homoclinic relations. There is an associated natural division of the space of ergodic measures which are either supported on the exposed piece or on the core. We describe the topology of these two parts and show that they glue along nonhyperbolic…
A vector field X is called a star flow if every periodic orbit, of any vector field C1-close to X, is hyperbolic. It is known that the chain recurrence classes of a generic star flow X on a 3 or 4 manifold are either hyperbolic or singular hyperbolic (see [MPP] for 3-manifolds and [GLW] on 4-manifolds). As it is defined, the notion of singular hyperbolicity forces the singularities in the same class to have the same index. However, in higher dimension (i.e $\geq 5$) \cite{BdL} shows that singularities of different indices may be robustly in the same chain recurrence class of a star flow. There…
We show that robustly transitive endomorphisms of a closed manifolds must have a non-trivial dominated splitting or be a local diffeomorphism. This allows to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation.
We prove a perturbation (pasting) lemma for conservative (and symplectic) systems. This allows us to prove that $C^{\infty}$ volume preserving vector fields are $C^1$-dense in $C^{1}$ volume preserving vector fields (After the conclusion of this work, Ali Tahzibi pointed out to us that this result was proved in 1979 by Carlos Zuppa, although his proof is different from ours.). Moreover, we obtain that $C^1$ robustly transitive conservative flows in three-dimensional manifolds are Anosov and we conclude that there are no geometrical Lorenz-like sets for conservative flows. Also, by-product of t…
We prove that a divergence-free and C1-robustly transitive vector field has no singularities. Moreover, if the vector field is C4 then the linear Poincare flow associated to it admits a dominated splitting over M.
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.
We construct partially hyperbolic diffeomorphisms having semi-local robustly transitive sets with $C^1$-robust cycles of any co-index. These constructions also provide a new method to create $C^2$-robust homoclinic, equidimensional and heterodimensional tangencies of large codimension in dimension $d\geq 4$. The method to generate robust homoclinic and equidimensional tangencies also works in the symplectic setting. Thus, these are mechanisms for the robust non-hyperbolicity of symplectomorphisms in higher dimension.
We construct examples of robustly transitive and stably ergodic partially hyperbolic diffeomorphisms $f$ on compact $3$-manifolds with fundamental groups of exponential growth such that $f^n$ is not homotopic to identity for all $n>0$. These provide counterexamples to a classification conjecture of Pujals.
We show that the robustly transitive diffeomorphisms constructed by Bonatti and Viana have unique equilibrium states for natural classes of potentials. In particular, we characterize the SRB measure as the unique equilibrium state for a suitable geometric potential. The techniques developed are applicable to a wide class of DA diffeomorphisms, and persist under $C^1$ perturbations of the map. These results are an application of general machinery developed by the first and last named authors.
