Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. (English) Zbl 1034.37003

Forni, Giovanni

Ann. Math. (2) 155, No. 1, 1-103 (2002).

This paper is quite long and it is very difficult to summarize all of this. But main purpose of this paper is to prove Theorem {0.1} below.
Let \(M\) be a compact orientable surface of genus \(g\geq2\). Let \((\Sigma, i)\) be a divisor on \(M\), \(\Sigma=\{p_1,\ldots,p_\sigma\}\subset M\), \(i=(i_1,\ldots,i_\sigma)\in Z^\sigma\), \(i_k<0\) and \(\sum_ki_k=2-2g\), \({\mathcal F}^i(M,\Sigma)\) be the space of orientable measured foliations, \({\mathcal E}^i_\omega(M,\Sigma)\) be the space of all smooth \(\omega\)-preserving vector fields on \(M\backslash\Sigma\) such that \(\eta_X=i_X\omega\) is smooth on \(M\) and the orbit foliation \({\mathcal F}_X=\{\eta_X=0\}\in{\mathcal F}^i(M,\Sigma)\).
Theorem 0.1. For almost all \(X\in{\mathcal D}^i_\omega(M,\Sigma)\), the flow \(\Phi^\tau_X\) has a deviation spectrum in the following sense. There exists a finite set of exponents \[ \lambda_1'(X)=1> \lambda_2'(X)>\cdots> \lambda_s'(X)>0 \] and splitting of the space \({\mathcal T}^1_X(M)\) of \(X\)-invariant distributions of order 1 \[ {\mathcal T}^1_X(M)={\mathcal T}^1_X(\lambda_1')\oplus\cdots\oplus {\mathcal T}^1_X(\lambda_s') \] such that the following holds. Let \(f\in H^1_0(M\backslash\Sigma)\) be a weakly differentiable function with \(L^2\) partial derivatives, supported in \(M\backslash\Sigma\), such that \[ D^X(f)=0,\quad\text{for all }\quad D_X\in{\mathcal T}^1_X(\lambda_1')\oplus\cdots \oplus{\mathcal T}^1_X(\lambda_i'). \] If \(i<s\), then for all \(p\in M\) with regular forward trajectory, \[ \limsup_{T\to+\infty} {\log\int_0^{\mathcal T}f(\Phi_X(p,\tau)\,d\tau\over\log{\mathcal T}} \leq\lambda_{i+1}'(X) \] and there exists \(D^X_{i+1}\in{\mathcal T}_X(\lambda_{i+1}')\backslash\{0\}\) such that, if \(D_{i+1}^X(f)\neq0\), then equality holds for almost all \(p\in M\).
If \(i=s\), then for all \(p\in M\) with regular forward trajectory, \[ \limsup_{T\to+\infty} {\log\int_0^{\mathcal T}f(\Phi_X(p,\tau)\,d\tau\over\log{\mathcal T}}=0. \] The multiplicities \(m_i(X)=\dim{\mathcal T}_X(\lambda_i')\) satisfy the conditions \(\sum_im_i(X)=g\) and \(m_1(X)=1\), since \({\mathcal T}_X(\lambda_1')=R\cdot\omega\). The deviation spectrum with multiplicities \(\{(\lambda_i'(X),m_i(X))\}\) is locally constant on \({\mathcal E}^i_\omega(M,\Sigma)\).

Reviewer: Makoto Mori (Tokyo)

MSC:

37A25	Ergodicity, mixing, rates of mixing
32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
37C85	Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37C40	Smooth ergodic theory, invariant measures for smooth dynamical systems
37D25	Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37E35	Flows on surfaces
53D25	Geodesic flows in symplectic geometry and contact geometry

Keywords:

ergodicity; geodesic flow; Teichmüller flow; topological dynamics; foliation; flows on orientable surfaces; invariant measure