Given a finite-dimensional real vector space $V$, a probability measure $μ$ on $\operatorname{PGL}(V)$ and a $μ$-invariant subspace $W$, under a block-Lyapunov contraction assumption, we prove existence and uniqueness of lifts to $P(V)\setminus P(W)$ of stationary probability measures on the quotient $P(V/W)$. In the other direction, i.e. under block-Lyapunov expansion, we prove that stationary measures on $P(V/W)$ have lifts if any only if the group generated by the support of $μ$ stabilizes a subspace $W'$ not contained in $W$ and exhibiting a faster growth than on $W \cap W'$. These refine the description of stationary probability measures on projective spaces as given by Furstenberg, Kifer and Hennion, and under the same assumptions, extend corresponding results by Aoun, Benoist, Bruère, Guivarc'h, and others.
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…
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.
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 consider the action of $SL(2,\mathbb{R})$ on a vector bundle $\mathbf{H}$ preserving an ergodic probability measure $ν$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $\hatν$ is any lift of $ν$ to a probability measure on the projectivized bunde $\mathbb{P}(\mathbf{H})$ that is invariant under the upper triangular subgroup, then $\hat ν$ is supported in the projectivization $\mathbb{P}(\mathbf{E}_1)$ of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycl…
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…
We study the geometry of the action of SL(2,R) on the projective line in order to present a new and simpler proof of the Herman-Avila-Bochi formula. This formula gives the average Lyapunov exponent of a class of 1-families of SL(2,R)-cocycles.
Consider the 1-dimensional Hurwitz space parameterizing covers of P^1 branched at four points. We study its intersection with divisor classes on the moduli space of curves. As an application, we calculate the slope of the Teichmuller curve parameterizing square-tiled cyclic covers and recover the sum of its Lyapunov exponents obtained by Forni, Matheus and Zorich. Motivated by the work of Eskin, Kontsevich and Zorich, we exhibit a relation among the slope of Hurwitz spaces, the sum of Lyapunov exponents and the Siegel-Veech constant for the moduli space of quadratic differentials.
Given an i.i.d. sequence $\{A_n(ω)\}_{n\ge 1}$ of invertible matrices and a random matrix $B(ω)$, we consider the random matrix sequences inductively defined by $S_n(ω) = A_n(ω)S_{n-1}(ω)$ and $T_n(ω) = B(σ^{n-1}ω)S_{n-1}(ω)+A_n(ω)T_{n-1}(ω)$, and study several limit theorems involving $T_n(ω)$ as well as the asymptotic behaviour of the action of $T_n(ω)$ on the projective space and on the unit circle.
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.
In this paper we investigate the relation between measure expansiveness and hyperbolicity. We prove that non atomic invariant ergodic measures with all of its Lyapunov exponents positive is positively measure-expansive. We also prove that local diffeomorphisms robustly positively measure-expansive is expanding. Finally, we prove that if a $C^1$ volume preserving diffeomorphism that. can not be accumulated by positively measure expansive diffeomorphis have a dominated sppliting.
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 consider one-step cocycles of $2 \times 2$ matrices, and we are interested in their Lyapunov-optimizing measures, i.e., invariant probability measures that maximize or minimize a Lyapunov exponent. If the cocycle is dominated, that is, the two Lyapunov exponents are uniformly separated along all orbits, then Lyapunov-optimizing measures always exist, and are characterized by their support. Under an additional hypothesis of nonoverlapping between the cones that characterize domination, we prove that the Lyapunov-optimizing measures have zero entropy. This conclusion certainly fails without t…
It is given notions of singular hyperbolicity and sectional Lyapunov exponents of orders beyond the classical ones, namely, other dimensions besides the dimension 2 and the full dimension of the central subbundle of the singular hyperbolic set. It is obtained a characterization of dominated splittings, partial and singular hyperbolicity in this broad sense, by using Lyapunov exponents and the notion of infinitesimal Lyapunov functions. Furthermore, it is given alternative requirements to obtain singular hyperbolicity. As an application we obtain some results related to singular hyperbolic sets…
Let $f$ be a $C^r$$(r>1)$ diffeomorphism of a compact Riemannian manifold $M$, preserving an ergodic hyperbolic measure $μ$ with positive entropy, and let $\mathcal{A}$ be a Hölder continuous cocycle of injective bounded linear operators acting on a Banach space $X$. We prove that there is a sequence of horseshoes for $f$ and dominated splittings for $\mathcal{A}$ on the horseshoes, such that not only the measure theoretic entropy of $f$ but also the Lyapunov exponents of $\mathcal{A}$ with respect to $μ$ can be approximated by the topological entropy of $f$ and the Lyapunov exponents of $\mat…
Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Hölder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Hölder continuous functions relative to the uniform topology. This result is obtained as a consequence of a uniform large deviations type estimate in the space of cocycles. As a byproduct of our approach, we also establish other statistical properties for the iterates of such cocycles, namely a centra…
We consider a meromorphic family of endomorphisms of degree at least 2 of a complex projective space that is parameterized by the unit disk.We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated non-Archimedean dynamical system when the system degenerates. The convergence holdsin the hybrid space constructed by Berkovich and further studied by Boucksom and Jonsson. We also infer from our analysis an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of endomorphisms.
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.
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 $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 consider Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We refine a lower bound obtained by Eskin, Kontsevich, Moeller and Zorich showing that the sum of the first k exponents is greater or equal than the sum of the degree of any rank k holomorphic subbundle of the flat bundle and the asymptotic degree of its equivariant developing map. We also show that this inequality is an equality if the base curve is compact. We moreover relate the asymptotic degree to the dynamical degree defined by Daniel and Deroin. We then use the p…
We study the fiber Lyapunov exponents of step skew-product maps over a complete shift of $N$, $N\ge2$, symbols and with $C^1$ diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and zero exponents. Examples of such systems arise from the projective action of $2\times 2$ matrix cocycles and our results apply to an open and dense subset of elliptic $\mathrm{SL}(2,\bR)$ cocycles. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov ex…
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 denote by $M_k$ the set of $k$ by $k$ matrices with complex entries. We consider quantum channels $ϕ_L$ of the form: given a measurable function $L:M_k\to M_k$ and a measure $μ$ on $M_k$ we define the linear operator $ϕ_L:M_k \to M_k$, by the law $ρ\,\to\,ϕ_L(ρ) = \int_{M_k} L(v) ρL(v)^\dagger \, \dm(v).$ On a previous work the authors show that for a fixed measure $μ$ it is generic on the function $L$ the $Φ$-Erg property (also irreducibility). Here we will show that the purification property is also generic on $L$ for a fixed $μ$. Given $L$ and $μ$ there are two related stochastic process…
For a non-conformal repeller $Λ$ of a $C^{1+α}$ map $f$ preserving an ergodic measure $μ$ of positive entropy, this paper shows that the Lyapunov dimension of $μ$ can be approximated gradually by the Carathéodory singular dimension of a sequence of horseshoes. For a $C^{1+α}$ diffeomorphism $f$ preserving a hyperbolic ergodic measure $μ$ of positive entropy, if $(f, μ)$ has only two Lyapunov exponents $λ_u(μ)>0>λ_s(μ)$, then the Hausdorff or lower box or upper box dimension of $μ$ can be approximated by the corresponding dimension of the horseshoes $\{Λ_n\}$. The same statement holds true if $…
Ledrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. In this paper, we show that a self-affine measure on the plane satisfies the Ledrappier-Young formula if the corresponding iterated function system (IFS) satisfies the strong separation condition and the linear parts satisfy the dominated splitting condition. We give sufficient conditions, inspired by Ledrappier and by Falconer and Kempton, that the dimensions of such a self-affine measure is equal to the Lyapunov dimension. We show some a…
We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalize results that were known in the inve…
