We prove that there exists an open and dense subset of the incompressible 3-flows of class C^2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincaré flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of Bochi-Mañé and of Newhouse for flows with singularities. That is we obtain for a residual subset of the C^1 incompressible flows on 3-manifolds that: (i) either all Lyapunov exponents are zero or the flow is Anosov, and (ii) either the flow is Anosov or else the elliptic periodic points are dense in the manifold.
We obtain sufficient conditions for an invariant splitting over a compact invariant subset of a $C^1$ flow $X_t$ to be dominated. In particular, we reduce the requirements to obtain sectional hyperbolicity and hyperbolicity.
We relate dominated splitting for a linear multiplicative cocyle with dominated splitting for the exterior powers of this cocycle. For a C1 vector field X on a 3-manifold, we can obtain singular-hyperbolicity using only the tangent map DX of X and a family of indefinite and non-degenerate quadratic forms without using the associated flow X_t and its derivative DX_t. As a consequence, we show the existence of adapted metrics for singular-hyperbolic sets for three-dimensional C1 vector fields.
We propose a weak form of domination, called partially dominated splitting and the main result is that there is a partially dominated splitting over a nonsingular compact invariant set for a flow if, and only if, the associated linear Poincare flow has a dominated splitting
Hyperbolicity and dominated splitting are two of the most important concepts in the global analysis of differentiable dynamics. In this paper we give several equivalent characterizations of the dominated splitting and in particular we show a criterion for dynamical systems being dominated splitting in terms of hyperbolicity.18 pages
For $C^1$ diffeomorphisms, we prove that the Pesin's entropy formula holds for some invariant measure supported on any topological attractor that admits a dominated splitting without mixed behavior. We also prove Shub's entropy conjecture for diffeomorphisms having such kind of splittings.
We construct a diffeomorphism $f$ on 2-torus with a dominated splitting $E \oplus F$ such that there exists an open neighborhood $\mathcal{U} \ni f$ satisfying that for any $g \in \mathcal{U}$, neither $E_g$ nor $F_g$ is integrable.
For differentiable dynamical systems with dominated splittings, we give upper estimates on the measure-theoretic tail entropy in terms of Lyapunov exponents. As our primary application, we verify the upper semi-continuity of metric entropy in various settings with domination.
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.
We introduce a new notion called generalized dominated splitting which is weaker than classical dominated splitting. We use this notion to generalize a result of Zhang\cite{Zh}: every diffeomorphism with nontrivial global generalized dominated splitting can not be minimal.
In this paper, we provide a technique result on the existence of Gibbs cu-states for diffeomorphisms with dominated splittings. More precisely, for given $C^2$ diffeomorphim $f$ with dominated splitting $T_ΛM = E\oplus F$ on an attractor $Λ$, by considering some suitable random perturbation of $f$, we show that for any zero noise limit of ergodic stationary measures, if it has positive integrable Lyapunov exponents along invariant sub-bundle $E$, then its ergodic components contain Gibbs cu-states associated to $E$. With this technique, we show the existence of SRB measures and physical measur…
Let $f$ be a $C^1$-diffeomorphism and $μ$ be a hyperbolic ergodic $f$-invariant Borel probability measure with positive measure-theoretic entropy. Assume that the Oseledec splitting $$T_xM=E_1(x) \oplus\cdots\oplus E_s(x) \oplus E_{s+1}(x) \oplus\cdots\oplus E_l(x) $$ is dominated on the Oseledec basin $Γ$. We give extensions of Katok's Horseshoes construction. Moreover there is a dominated splitting corresponding to Oseledec subspace on horseshoes.
Given a linear cocycle over an ergodic homeomorphism on a compact metric space, we show that the existence of a uniform gap between the $p$-th and $(p+1)$-th Lyapunov exponent on a $C^0$-neighbourhood implies the existence of a dominated splitting of index $p$.
Recent works related to Palis conjecture of J. Yang, S. Crovisier, M. Sambarino and D. Yang showed that any aperiodic class of a $C^1$-generic diffeomorphism far away from homoclinic bifurcations (or homoclinic tangencies) is partially hyperbolic. We show in this paper that, generically, a non-trivial dominated splitting implies partial hyperbolicity for an aperiodic class if it is Lyapunov stable. More precisely, for $C^1$-generic diffeomorphisms, if a Lyapunov stable aperiodic class has a non-trivial dominated splitting $E\oplus F$, then one of the two bundles is hyperbolic (either $E$ is co…
In this paper we obtain two criteria of stable ergodicity outside the partially hyperbolic scenario. In both criteria, we use a weak form of hyperbolicity called chain-hyperbolicity. It is obtained one criterion for diffeomorphisms with dominated splitting and one criterion for weakly partially hyperbolic diffeomorphisms. As an application of one of these criteria, we obtain the $C^1$-density of stable ergodicity inside a certain class of weakly partially hyperbolic diffeomorphisms.
We study the Ruelle dynamical determinant of a real analytic diffeomorphism on a compact surface, assuming that the tangent space over the nonwandering set admits a dominated splitting. Combining previous work of Pujals and Sambarino with methods introduced by Rugh, we show that the determinant is either entire or holomorphic in a (possibly multiply) slit plane.
Consider the space of analytic, quasi-periodic cocycles on the higher dimensional torus. We provide examples of cocycles with nontrivial Lyapunov spectrum, whose homotopy classes do not contain any cocycles satisfying the dominated splitting property. This shows that the main result in the recent work "Complex one-frequency cocycles" by A. Ávila, S. Jitomirskaya and C. Sadel does not hold in the higher dimensional torus setting.
Let Diff^1(M) be the set of all C^1-diffeomorphisms f : M \rightarrow M, where M is a compact boundaryless d-dimensional manifold, d \geq 2. We prove that there is a residual subset R of Diff^1(M) such that if f \in R and if H(p) is the homoclinic class associated with a hyperbolic periodic point p, then either H(p) admits a dominated splitting of the form E\oplusF1 \oplus...\oplusFk \oplusG, where Fi is not hyperbolic and one-dimensional, or f|H(p) has no symbolic extensions.
Versions of the Oseledets multiplicative ergodic theorem for cocycles acting on infinite-dimensional Banach spaces have been investigated since the pioneering work of Ruelle in 1982 and are a topic of continuing research interest. For a cocycle to induce a continuous splitting in which the growth in one subbundle exponentially dominates the growth in another requires additional assumptions; a necessary and sufficient condition for the existence of such a dominated splitting was recently given by J. Bochi and N. Gourmelon for invertible finite-dimensional cocycles in discrete time. We extend th…
Let $f:M\rightarrow M$ be a $C^1$ diffeomorphism with a dominated splitting on a compact Riemanian manifold $M$ without boundary. We state and prove several sufficient conditions for the topological entropy of $f$ to be positive. The conditions deal with the dynamical behaviour of the (non-necessarily invariant) Lebesgue measure. In particular, if the Lebesgue measure is $δ$-recurrent then the entropy of $f$ is positive. We give counterexamples showing that these sufficient conditions are not necessary. Finally, in the case of partially hyperbolic diffeomorphisms, we give a positive lower boun…
For any C1 diffeomorphism with dominated splitting we consider a nonempty set of invariant measures which describes the asymptotic statistics of Lebesgue-almost all orbits. They are the limits of convergent subsequences of averages of the Dirac delta measures supported on those orbits. We prove that the metric entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating subbundle. As a consequence, if those exponents are non negative, and if the exponents on the dominated subbundle are non positive, those measures satisfy the Pesin Entropy Form…
We study generic volume-preserving diffeomorphisms on compact manifolds. We show that the following property holds generically in the $C^1$ topology: Either there is at least one zero Lyapunov exponent at almost every point, or the set of points with only non-zero exponents forms an ergodic component. Moreover, if this nonuniformly hyperbolic component has positive measure then it is essentially dense in the manifold (that is, it has a positive measure intersection with any nonempty open set) and there is a global dominated splitting. For the proof we establish some new properties of independe…
We prove that any smooth volume-preserving action of a lattice $Γ$ in $\textrm{SL}(n,\mathbb{R})$, $n\ge 3$, on a closed $n$-manifold which possesses one element that admits a dominated splitting should be standard. In other words, the manifold is the $n$-flat torus and the action is smoothly conjugate to an affine action. Note that an Anosov diffeomorphism, or more generally, a partial hyperbolic diffeomorphism admits a dominated splitting. We have a topological global rigidity when $α$ is $C^{1}$. Similar theorems hold for an action of a lattice in $\textrm{Sp}(2n,\mathbb{R})$ with $n\ge 2$ …
A theorem of J. Bochi and N. Gourmelon states that an invertible linear cocycle admits a dominated splitting if and only if the singular values of its iterates become separated at a uniform exponential rate. It is not difficult to show that for cocycles of non-invertible linear maps over an invertible dynamical system -- which we refer to as semi-invertible cocycles -- this criterion fails to imply the existence of a dominated splitting. In this article we show that a simple modification of Bochi and Gourmelon's singular value criterion is equivalent to the existence of a dominated splitting i…
We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Guéritaud-Guichard-Kassel-Wienhard and Kapovich-Leeb-Porti by different methods. We also give characterizations in terms of multicones and cone-types inspired in the work of Avila-Bochi-Yoccoz and Bochi-Gourmelon. Finally we provide a new proof of the higher rank Morse Lemma of Kapovich-Leeb-Porti.
We prove three formulas for computing topological pressure of $C^1$-generic conservative diffeomorphism and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there is no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there is no equilibrium states. For $C^1$ generic conservative diffeomorphism on compact surfaces with no dominated splitting and $ϕ_m(x):=-\frac{1}{m}\log \Vert D_x f^m\Vert, m \in \mathbb{N}$, we show that there exist equilibrium states with zer…
We prove that the statistical properties of random perturbations of a nonuniformly hyperbolic diffeomorphism are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain $C^2$-open class of nonuniformly hyperbolic diffeomorphisms introduced in [Alves, J; Bonatti, C. and Viana, V., SRB measures for partially hyperbolic systems with mostly expanding central direction, Invent. Math., 140 (2000), 351-398] are stochastically stable. Our setting encompasses that of partially hy…
In this paper we consider a non-atomic invariant hyperbolic measure $μ$ of a $C^1$ diffeomorphsim on a compact manifold, in whose Oseledec splitting the stable bundle dominates the unstable bundle on $μ$ a.e. points. We show an \textit{exponentially} shadowing and an \textit{exponentially} closing lemma, and as applications we show two classical results. One is that there exists a hyperbolic periodic point such that the closure of its unstable manifold has \textit{positive} measure and it has a homoclinic point from which one can deduce a horseshoe. Moreover, such hyperbolic periodic points ar…
A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local $C^1$-perturbations. Our main theorem shows that, when one works in the $C^1$ topology, the entropy of such horseshoes can be made arbitrarily close to an upper bound deriving from Ruelle's inequality, i.e., the sum of the positive Lyapunov exponents (or the same for the inverse diffeomorphism, whichever is smaller). Adapting classical techniques, we use perturbations that are local and can be chosen to preserve volume or symplectic form or a homoclini…
We prove that a generic linear cocycle over a minimal base dynamics of finite dimension has the property that the Oseledets splitting with respect to any invariant probability coincides almost everywhere with the finest dominated splitting. Therefore the restriction of the generic cocycle to a subbundle of the finest dominated splitting is uniformly subexponentially quasiconformal. This extends a previous result for SL(2,R)-cocycles due to Avila and the author.
Let $M$ be a compact manifold and $f:\,M\to M$ be a $C^1$ diffeomorphism on $M$. If $μ$ is an $f$-invariant probability measure which is absolutely continuous relative to Lebesgue measure and for $μ$ $a.\,\,e.\,\,x\in M,$ there is a dominated splitting $T_{orb(x)}M=E\oplus F$ on its orbit $orb(x)$, then we give an estimation through Lyapunov characteristic exponents from below in Pesin's entropy formula, i.e., the metric entropy $h_μ(f)$ satisfies $$h_μ(f)\geq\int χ(x)dμ,$$ where $χ(x)=\sum_{i=1}^{dim\,F(x)}λ_i(x)$ and $λ_1(x)\geqλ_2(x)\geq...\geqλ_{dim\,M}(x)$ are the Lyapunov exponents at $x…
We study generic diffeomorphisms with a homoclinc class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one dimensional then the diffeomorphism is partially hyperbolic and from this we deduce that the diffeomorphism is transitive.
We consider families of diffeomorphisms with dominated splittings and preserving a Borel probability measure, and we study the regularity of the Lyapunov exponents associated to the invariant bundles with respect to the parameter. We obtain that the regularity is at least the sum of the regularities of the two invariant bundles (for regularities in $[0,1]$), and under suitable conditions we obtain formulas for the derivatives. Similar results are obtained for families of flows, and for the case when the invariant measure depends on the map. We also obtain several applications. Near the time on…
In this work, we are interested in the study of the upper Lyapunov exponent $λ^+(θ)$ associated to the periodic family of cocycles defined by $$A_θ(x):=A(x)R_θ,\qquad x\in X,$$ where $A\::\: X\to \mathbb{GL}^+(2,\mathbb{R})$ is a linear cocycle orientation--preser\-ving and $R_θ$ is a rotation of angle $θ\in\mathbb{R}$. We show that if the cocycle $A$ has dominated splitting, then there exists a non empty open set $\mathcal{U}$ of parameters $θ$ such that the cocycle $A_θ$ has dominated splitting and the function $\mathcal{U}\niθ\mapstoλ^+(θ)$ is real analytic and strictly concave. As a conseq…
We provide a new approach to stable ergodicity of systems with dominated splittings, based on a geometrical analysis of global stable and unstable manifolds of hyperbolic points. Our method suggests that the lack of uniform size of Pesin's local stable and unstable manifolds - a notorious problem in the theory of non-uniform hyperbolicity - is often less severe than it appeas to be.
Inspired by the work of Pujals and Sambarino on dominated splitting, we present billiards with a modified reflection law which constitute simple examples of dynamical systems with limit sets with dominated splitting and where the dynamics is a rational or irrational rotation.
Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient Euclidian space. We use elementary tangential calculus to derive the governing equations in terms of exterior differential operators in Cartesian coordinates. The resulting equations can be seen as the Navier-Stokes equations posed on an evolving manifold. We consider a splitting of the surface Navier-Stokes system into coupled equations for the tangential and …
We consider the structure of substantially dissipative complex Hénon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the one-dimensional assumption that there are no critical points on the Julia set. Indeed, we prove the corresponding description of the Fatou set, namely that it consists of only finitely many components, each either attracting or parabolic periodic. In particular there are no rotation domains, and no wandering components. Moreover, we show that $J = J^\star$ and the dynamics on $J$ is hyperbolic away from parabolic cycle…
We show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates. Then we consider sets $Σ$ in $GL(d,\mathbb{R})$ with the property that any cocycle with values in $Σ$ has a dominated splitting. We characterize these sets in terms of existence of invariant multicones, thus extending a 2-dimensional result by Avila, Bochi, and Yoccoz. We give an example showing how these multicones can fail to have convexity properties.
In this paper, we first develop a notion of dominated splitting for $\mathbb M(2,\mathbb C)$-sequences and show it is a stable property under $\|\cdot \|_\infty$-perturbation. Then we show an energy parameter belongs to the spectrum of a Jacobi operator, possibly singular, if and only if the associated Jacobi cocycle does not admit dominated splitting. This generalizes the results obtained by the second author [Z] in the scenario of Schrödinger operators. Finally, we consider dynamically defined Jacobi operators whose base dynamics is only assumed to be topologically transitive. We show an ene…
In the present paper we study the C1-robustness of the three properties: average shadowing, asymptotic average shadowing and limit shadowing within two classes of conservative flows: the incompressible and the Hamiltonian ones. We obtain that the first two properties guarantee dominated splitting (or partial hyperbolicity) on the whole manifold, and the third one implies that the flow is Anosov.
We show that, for every compact n-dimensional manifold, n\geq 1, there is a residual subset of Diff^1(M) of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities: Either it is contained in the closure of an infinite set of sinks or sources (Newhouse phenomenon), or it presents some weak form of hyperbolicity called dominated splitting (this is a generalization of a bidimensional result of Mane [Ma3}). In particular, we show that any C^1 -robustly transitive diffeomorphism admits a dominated splitting.
We prove the following dichotomy for vector fields in a C1-residual subset of volume-preserving flows: for Lebesgue almost every point all Lyapunov exponents equal to zero or its orbit has a dominated splitting. As a consequence if we have a vector field in this residual that cannot be C1-approximated by a vector field having elliptic periodic orbits, then, there exists a full measure set such that every orbit of this set admits a dominated splitting for the linear Poincare flow. Moreover, we prove that a volume-preserving and C1-stably ergodic flow can be C1-approximated by another volume-pre…
