## The Lyapunov exponents of generic volume-preserving and symplectic maps.(English)Zbl 1101.37039

The authors show that for a residual (dense $$G_\delta$$) subset of all volume-preserving or symplectic $$C^1$$ diffeomorphisms, either all the Lyapunov exponents are zero for almost every orbit, or the diffeomorphism exhibits a large amount of hyperbolicity. The way of achieving this result is to analyze the continuous dependence of Lyapunov exponents on the dynamical system.
For a precise formulation, we need to recall some definitions. Let $$M$$ be a compact Riemannian manifold of dimension $$d\geq 2$$, and $$\mu$$ be the measure induced by some volume form. $$\text{Diff}^1_\mu(M)$$ denotes the space of all $$\mu$$-preserving $$C^1$$ diffeomorphisms. Let $$f\in \text{Diff}^1_\mu(M)$$. By the theorem of Oseledets, for $$\mu$$-a.e. point $$x\in M$$, there exist $$k(x)\in \mathbb N$$ real numbers $$\widehat \lambda_1(f,x)> \dots \widehat \lambda_{k(x)}(f,x)$$, called Lyapunov exponents, and a splitting $$T_xM =E^1_x \oplus \dots \oplus E^{k(x)}_x$$ of the tangent space at $$x$$, called Oseledets splitting, all depending measurably on the point $$x$$ such that
$\lim_{n\to \pm \infty} \frac{1}{n} \log \|Df_x^n(v)\|= \widehat \lambda_j(f,x) \quad \text{for all } v\in E^j_x\setminus 0.$ The Oseledets splitting is said to be trivial if all Lyapunov exponents vanish.
Let $$\Gamma$$ be an $$f$$-invariant set, and suppose that for each $$x\in \Gamma$$ there are $$Df$$-invariant subspaces $$E^1_x$$ and $$E^2_x$$ of constant dimension such that $$T_x M=E^1_x \oplus E^2_x$$. Given $$m\in \mathbb N$$, we say that $$T_{\Gamma} M=E^1 \oplus E^2$$ is an $$m$$-dominated splitting if for every $$x\in \Gamma$$
$\|Df_x^m\mid_{E^2_x}\|\;\|(Df_x^m\mid_{E^1_x})^{-1}\| <1/2.$
The Oseledets splitting is said to be dominated if the above condition with appropriate $$m$$ holds for every two Lyapunov subspaces. Domination is a very strong property. In particular, it implies that the angles between Oseledets subspaces are bounded away from zero, and the Oseledets splitting extends to a continuous splitting on the closure.
Theorem 1. There exists a residual set $$R \subset\text{Diff}^1_\mu(M)$$ such that for every $$f\in R$$ and $$\mu$$-almost every $$x\in M$$, the Oseledets splitting is either trivial or dominated at $$x$$.
Let $$\lambda_1(f,x)\geq \dots \geq \lambda_d(f,x)$$ be the Lyapunov exponents. For $$0\leq j\leq d-1$$ define
$\text{LE}_j(f)=\int_M [\lambda_1(f,x)+ \dots + \lambda_j(f,x)] d\mu(x).$
It is known that the mapping $$f\in \text{Diff}^1_\mu \mapsto \text{LE}_j(f)$$ is upper semi-continuous. The following theorem indicates that lower semi-continuity is much more delicate,
Theorem 2. If the map $$f\in \text{Diff}^1_\mu \mapsto (\text{LE}_1(f),\dots, \text{LE}_{d-1}(f))\in \mathbb R^d$$ is continuous at $$f=f_0$$, then for $$\mu$$-almost every $$x\in M$$, the Oseledets splitting of $$f_0$$ is either trivial or dominated.
It is not known if the necessary condition above is also sufficient for continuity.
Theorem 1 is a direct consequence of Theorem 2 due to the following fact: the set of continuity points of a semi-continuous function on a Baire space is always a residual subset of the space.
The corresponding results for symplectic diffeomorphisms are the following. Let $$(M^{2q},\omega)$$ be a compact symplectic manifold without boundary, endowed by a Riemannian metric, and let $$\mu$$ be the volume measure corresponding to $$\omega$$. Let $$\text{Sympl}^1_\omega(M)$$ be the space of $$C^1$$ symplectic diffeomorphisms on $$M$$. Consider the splitting $$T_x M=E^+_x \oplus E^0_x \oplus E^-_x$$, and let $$E^+$$, $$E^-$$, $$E^0$$ be the sums of Oseledets spaces, associated to positive, negative and zero Lyapunov exponents. Recall that the Lyapunov exponents of $$f\in \text{Sympl}^1_\omega(M)$$ have a symmetry property: $$\lambda_j(f,x)=-\lambda_{2q-j+1}(f,x)$$. Therefore, $$\dim (E_x^+)=\dim (E_x^-)$$, and $$\dim (E_x^0)$$ is even.
Theorem 3. If the map $$f\in \text{Sympl}^1_\mu \mapsto {LE}_q(f)$$ is continuous at $$f=f_0$$, then for $$\mu$$-almost every $$x\in M$$ either dim$$E^0_x\geq 2$$, or the splitting $$E^+_x \oplus E^-_x$$ is hyperbolic along the orbit of $$x$$.
As in the volume-preserving case, the function $$f \mapsto {LE}_q(f)$$ is continuous on a residual subset $$R_1$$ of $$\text{Sympl}^1_\mu$$. On a slightly smaller residual set $$R_2$$ the following is true,
Theorem 4. There exists a residual set $$R_2\subset\text{Sympl}^1_\mu$$ such that every $$f\in R_2$$ either is Anosov or has at least two zero Lyapunov exponents at almost every point.
For $$d=2$$ this theorem was announced by R. Mañé [Proc. Int. Congr. Math., Warszawa 1983, Vol. 2, 1269–1276 (1984; Zbl 0584.58007); François Ledrappier (ed.) et al., 1st international conference on dynamical systems, Montevideo, Uruguay, 1995 – a tribute to Ricardo Mañé. Proceedings. Harlow: Longman. Pitman Res. Notes Math. Ser. 362, 110–119 (1996; Zbl 0870.58083)] and proved by J. Bochi [Ergodic Theory Dyn. Syst. 22, 1667–1696 (2002; Zbl 1023.37006)].
The results above extend to linear cocycles, which are defined as follows. Let $$M$$ be a compact Hausdorff space, $$\mu$$ a Borel probability measure, and $$f:M\to M$$ a homeomorphism that preserves $$\mu$$. Given a continuous map $$A:M\to \text{GL}(d,\mathbb R)$$, one associates a linear cocycle
$F_A:M\times \mathbb R^d \to M\times \mathbb R^d, \quad F_A(x,v)=(f(x), A(x)v).$
All the main results, along with the context, numerous examples and open problems are very well explained in the introduction.

### MSC:

 37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010) 37A05 Dynamical aspects of measure-preserving transformations 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations 37D30 Partially hyperbolic systems and dominated splittings 37C20 Generic properties, structural stability of dynamical systems

### Citations:

Zbl 0584.58007; Zbl 0870.58083; Zbl 1023.37006
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