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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.

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
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