##
**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.

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.

Reviewer: Maria Saprykina (Toronto)

### 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 |