We study the entropy and Lyapunov exponents of invariant measures $μ$ for smooth surface diffeomorphisms $f$, as functions of $(f,μ)$. The main result is an inequality relating the discontinuities of these functions. One consequence is that for a $C^\infty$ surface diffeomorphisms, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semi-continuity of the Hausdorff dimension on the set of ergodic invariant measures with entropy bounded away from zero. We also obtain a new criterion for the existence of SRB measures with positive entropy.
We consider symplectic cocycles over two classes of partially hyperbolic diffeomorphisms: having compact center leaves and time one maps of Anosov flows. We prove that the Lyapunov exponents are non-zero in an open and dense set in the Hölder topology.
In this paper we construct some "pathological" volume preserving partially hyperbolic diffeomorphisms on $\toro{3}$ such that their behaviour in small scales in the central direction (Lyapunov exponent) is opposite to the behavior of their linearization. These examples are isotopic to Anosov. We also get partially hyperbolic diffeomorphisms isotopic to Anosov (consequently with non-compact central leaves) with zero central Lyapunov exponent at almost every point.
In this paper we mainly address the problem of disintegration of Lebesgue measure and measure of maximal entropy along the central foliation of (conservative) Derived from Anosov (DA) diffeomorphisms. We prove that for accessible DA diffeomorphisms of $\mathbb{T}^3$, atomic disintegration has the peculiarity of being mono-atomic (one atom per leaf). We further provide open and non-empty condition for the existence of atomic disintegration. Finally, we prove some new relations between Lyapunov exponents of DA diffeomorphisms and their linearization.
We prove a finite smooth version of the entropic continuity of Lyapunov exponents of Buzzi-Crovisier-Sarig for $C^\infty$ surface diffeomorphisms [9]. As a consequence we show that any $C^r$, $r > 1$, smooth surface diffeomorphism $f$ with $h_{top}(f) > \frac{1}{r} \limsup_n \frac{1}{n} \log^+ \|df^n\|$ admits a measure of maximal entropy. We also prove the $C^r$ continuity of the topological entropy at $f$.
We establish sufficient conditions for the upper semicontinuity and the continuity of the entropy of Sinai probability measures invariant by partially hyperbolic diffeomorphisms and discuss their application in several examples.
Let $M$ be a 2$d-$dimensional compact connected Riemannian manifold and $ω$ be a symplectic form on $M$. In this paper, we prove that a symplectic diffeomorphism, with all Lyapunov exponent zero for almost everywhere, can be $C^1$ approximated by one with a positive Lyapunov exponent for a positive-measured subset of $M$. That is, the set \[ \left\{ f\in \mathcal{S}ym^1_ω(M)\,| \begin{array}{ll} &\mbox{The largest Lyapunov exponent }λ_1(f,\,x)>0\\ &\mbox{ for a positive measure set } \end{array} \right\} \] is dense in $\mathcal{S}ym^1_ω(M)$. \end{abstract} \end{center}
We prove that, for semi-invertible continuous cocycles, continuity of Lyapunov exponents is equivalent to continuity, in measure, of Oseledets subspaces.
The globally positive diffeomorphisms of the 2n-dimensional annulus are important because they represent what happens close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite. For these globally positive diffeomorphisms, an Aubry-Mather theory was developed by Garibaldi \& Thieullen that provides the existence of some minimizing measures. Using the two Green bundles G- and G+ that can be defined along the support of these minimizing measures, we will prove that there is a deep link between: -the angle between G- and G+ along the suppor…
We prove that for every smooth compact manifold $M$ and any $r \ge 1$, whenever there is an open domain in $\mathrm{Diff}^r(M)$ exhibiting a persistent homoclinic tangency related to a basic set with a sectionally dissipative periodic saddle, topologically generic diffeomorphisms in this domain have Lyapunov unstable Milnor attractors. This implies, in particular, that the instability of Milnor attractors is locally topologically generic in $C^1$ if $\mathrm{dim}\,M \ge 3$ and in $C^2$ if $\mathrm{dim}\,M = 2$. Moreover, it follows from the results of C. Bonatti, L. J. Díaz and E. R. Pujals th…
Bochi-Katok-Rodriguez Hertz proposed in [BKH21] a program on the flexibility of Lyapunov exponents for conservative Anosov diffeomorphisms, and obtained partial results in this direction. For conservative Anosov diffeomorphisms with strong hyperbolic properties we establish extended flexibility results for their Lyapunov exponents. We give examples of Anosov diffeomorphisms with the strong unstable exponent larger than the strong unstable exponent of the linear part. We also give examples of derived from Anosov diffeomorphisms with the metric entropy entropy larger than the entropy of the line…
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 paper we establish uniform large deviations estimates of exponential type and Hölder continuity of the Lyapunov exponents for random non-invertible cocycles with constant rank.
Let $M$ be a compact manifold and $\text{Diff}^1_m(M)$ be the set of $C^1$ volume-preserving diffeomorphisms of $M$. We prove that there is a residual subset $\mathcal {R}\subset \text{Diff}^1_m(M)$ such that each $f\in \mathcal{R}$ is a continuity point of the map $g\to h_m(g)$ from $\text{Diff}^1_m(M)$ to $\mathbb{R}$, where $h_m(g)$ is the metric entropy of $g$ with respect to volume measure $m$.
Let $d\geq 2$ be an integer and let $ω_1,\cdots ,ω_d$ be moduli of continuity in a specified class which contains the moduli of Hölder continuity. Let $f_k$, $k\in\{1,\cdots,d\}$, be $C^{1+ω_k}$ orientation preserving diffeomorphisms of the circle and $f_1,\cdots, f_d$ commute with each other. We prove that if the rotation numbers of $f_k$'s are independent over the rationals and $ω_1(t)\cdotsω_d(t)=tω(t)$ with $\lim_{t\rightarrow 0^+}ω(t)=0$, then $f_1,\cdots,f_d$ are simultaneously (topologically) conjugate to rigid rotations.
The Lyapunov exponents of locally constant GL(2;C)-cocycles over Bernoulli shifts depend continuously on the cocycle and on the invariant probability. The Oseledets decomposition also depends continuously on the cocycle, in measure.
We provide conditions which imply the continuity of the Lyapunov exponents for non-uniformly fiber-bunched cocycles in $SL(2,\mathbb{R})$. The main theorem is an extension of the result of Backes, Brown and Butler and gives a partial answer to a conjecture of Marcelo Viana.
We show that the integrated Lyapunov exponents of $C^1$ volume preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is trivial (all Lyapunov exponents equal to zero) or else dominated (uniform hyperbolicity in the projective bundle) almost everywhere. We deduce a sharp dichotomy for generic volume preserving diffeomorphisms on any compact manifold: almost every orbit either is projectively hyperbolic or has all Lyapunov exponents equal to zero. Similarly, for a residual subset of all $C^1$ symplectic diffeomorphisms on…
We prove the Hölder continuity of the Lyapunov exponent for quasi-periodic Schrödinger cocycles with a $C^2$ cos-type potential and any fixed Liouvillean frequency, provided the coupling constant is sufficiently large. Moreover, the Hölder exponent is independent of the frequency and the coupling constant.
In this paper, we consider certain partially hyperbolic diffeomorphisms with center of arbitrary dimension and obtain continuity properties of the topological entropy under $C^1$ perturbations. The systems considered have subexponential growth in the center direction and uniform exponential growth along the unstable foliation. Our result applies to partially hyperbolic diffeomorphisms which are Lyapunov stable in the center direction. It applies to another important class of systems which do have subexponential growth in the center direction, for which we develop a technique to use exponential…
We prove that the Bochi-Mañé theorem is false, in general, for linear cocycles over non-invertible maps: there are C0-open subsets of linear cocycles that are not uniformly hyperbolic and yet have Lyapunov exponents bounded from zero.
In this paper, we study two properties of the Lyapunov exponents under small perturbations: one is when we can remove zero Lyapunov exponents and the other is when we can distinguish all the Lyapunov exponents. The first result shows that we can perturb all the zero integrated Lyapunov exponents $\int_M λ_j(x)dω(x)$ into nonzero ones, for any partially hyperbolic diffeomorphism. The second part contains an example which shows the local genericity of diffeomorphisms with non-simple spectrum and three results: one discusses the relation between simple-spectrum property and the existence of compl…
We prove a conjecture of Viana which states that Lyapunov exponents vary continuously when restricted to $GL(2,\mathbb{R})$-valued cocycles over a subshift of finite type which admit invariant holonomies that depend continuously on the cocycle.
We outline the flexibility program in smooth dynamics, focusing on flexibility of Lyapunov exponents for volume-preserving diffeomorphisms. We prove flexibility results for Anosov diffeomorphisms admitting dominated splittings into one-dimensional bundles.
In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism $L$ with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to $L$. We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism $f_0$ over an Anosov automorphism of the 2-torus, we show that for any vo…
In this paper we study the relationship between Lyapunov exponents and the induced map on cohomology for $C^{1}-$diffeomorphisms on compact manifolds. We show that if the induced map on cohomology has spectral radius strictly larger than 1, then the diffeomorphism has an invariant ergodic measure with at least one positive Lyapunov exponent. Furthermore, if the diffeomorphism also preserves a continuous volume form then it has an invariant ergodic measure with at least one positive and one negative Lyapunov exponent, in agreement with Shub's entropy conjecture. We also consider diffeomorphisms…
This paper is concerned with the study of linear cocycles over uniformly ergodic Markov shifts on a compact space of symbols. We establish the joint Hölder continuity of the maximal Lyapunov exponent as a function of the cocycle and the transition kernel in the vicinity of any irreducible cocycle with simple maximal Lyapunov exponent. Our approach, via Furstenberg's formula, shows the Hölder continuous dependence on the data of the stationary measure of the projective cocycle and in particular provides a more computable Hölder exponent.
Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Holder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Di…
We prove that a locally constant $SL_{2}(\mathbb{R})$-valued cocycle over the shift generated by an irreducible collection of matrices is a continuity point for Lyapunov exponents in the $α$-Hölder topology for every $α> 0$. This gives negative answers to conjectures of Viana and the author; we pose a new conjecture to replace these conjectures. We show that an analogous continuity result also holds for $GL_{2}(\mathbb{R})$-valued cocycles that admit canonical holonomies.
Let $\Diff^{ r}_m(M)$ be the set of $C^{ r}$ volume-preserving diffeomorphisms on a compact Riemannian manifold $M$ ($\dim M\geq 2$). In this paper, we prove that the diffeomorphisms without zero Lyapunov exponents on a set of positive volume are $C^1$ dense in $\Diff^{ r}_m(M), r\geq 1$. We also prove a weaker result for symplectic diffeomorphisms $\mathcal{S}ym^{r}_ω(M), r\geq1 $ saying that the symplectic diffeomorphisms with non-zero Lyapunov exponents on a set of positive volume are $C^1$ dense in $\mathcal{S}ym^{r}_ω(M), r\geq1 $.
We show that if the base frequency is Diophantine, then the Lyapunov exponent of a $C^{k}$ quasi-periodic $SL(2,\mathbb{R})$ cocycle is $1/2$-Hölder continuous in the almost reducible regime, if $k$ is large enough. As a consequence, we show that if the frequency is Diophantine, $k$ is large enough, and the potential is $C^k$ small, then the integrated density of states of the corresponding quasi-periodic Schrödinger operator is $1/2$-Hölder continuous.
We characterize the maximal entropy measures of partially hyperbolic C^2 diffeomorphisms whose center foliations form circle bundles, by means of suitable finite sets of saddle points, that we call skeletons. In the special case of 3-dimensional nilmanifolds other than the torus, this entails the following dichotomy: either the diffeomorphism is a rotation extension of an Anosov diffeomorphism -- in which case there is a unique maximal measure, with full support and zero center Lyapunov exponents -- or there exist exactly two ergodic maximal measures, both hyperbolic and whose center Lyapunov …
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 consider a similar type of scenario for the disappearance of uniform of hyperbolicity as in Bjerklöv and Saprykina (2008, Nonlinearity 21), where it was proved that the minimum distance between invariant stable and unstable bundles has a linear power law dependence on parameters. In this scenario we prove that the Lyapunov exponent is sharp $\frac12$-Hölder continuous. In particular, we show that the Lyapunov exponent of Schrödinger cocycles with a potential having a unique non-degenerate minimum, is sharp $\frac12$-Hölder continuous below the lowest energy of the spectrum, in the large cou…
We consider a $C^1$ neighborhood of the time-one map of a hyperbolic flow and prove that the topological entropy varies continuously for diffeomorphisms in this neighborhood. This shows that the topological entropy varies continuously for all known examples of partially hyperbolic diffeomorphisms with one-dimensional center bundle.
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 consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.
We show that a $C^1-$generic non partially hyperbolic symplectic diffeomorphism $f$ has topological entropy equal to the supremum of the sum of the positive Lyapunov exponents of its hyperbolic periodic points. Moreover, we also prove that $f$ has topological entropy approximated by the topological entropy of $f$ restrict to basic hyperbolic sets. In particular, the topological entropy map is lower semicontinuous in a $C^1-$generic set of symplectic diffeomorphisms far from partial hyperbolicity.
In this paper we continue our study of polynomial diffeomorphisms of C^2. Let us recall that there is an invariant measure $μ$, which is the pluri-complex version of the harmonic measure of the Julia set for polynomial maps of C. In this paper we give an integral formula for the Lyapunov exponents of a polynomial automorphism with respect to $μ$ analogous to the Brolin-Manning formula polynomial maps of C. Our formula relates the Lyapunov exponents to the value of a Green function at a type of critical point which we define in this paper. We show that these critical points have a very natural …
We study, both numerically and theoretically, the relationship between the random Lyapunov exponent of a family of area preserving diffeomorphisms of the 2-sphere and the mean of the Lyapunov exponents of the individual members.
We study, for $C^1$ generic diffeomorphisms, homoclinic classes which are Lyapunov stable both for backward and forward iterations. We prove they must admit a dominated splitting and show that under some hypothesis they must be the whole manifold. As a consequence of our results we also prove that in dimension 2 the class must be the whole manifold and in dimension 3, these classes must have nonempty interior. Many results on Lyapunov stable homoclinic classes for $C^1$-generic diffeomorphisms are also deduced.
We study order-preserving C^1-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures of nearly all family members are one-rectifiable, that is, absolutely continuous with respect to the restriction of the one-dimensional Hausdorff measure to a countable union of Lipschitz graphs.
