Mañé-Bochi's theorem
In mathematics , Mañé-Bochi's theorem is a tenet from the theory of dynamic systems .
It says that an area-preserving diffeomorphism of a compact surface is either an Anosov diffeomorphism or can be approximated in the topology by area-preserving diffeomorphisms with Lyapunov exponents that vanish almost everywhere .
Since there are Anosov diffeomorphisms only on the torus , the second alternative always applies to all other surfaces.
The theorem was announced by Ricardo Mañé in his lecture at the International Congress of Mathematicians in 1983 , but the proof was not published until his death in 1995. The first complete proof appeared in 2001 in Jairo Bochi's dissertation .
literature
- R. Mañé: Oseledec's theorem from the generic viewpoint. In: Proc. International Congress of Mathematicians , Vol. 1, 2 (Warsaw, 1983), PWN Publ. Warsaw, 1984, pp. 1269-1276.
- J. Bochi: Genericity of zero Lyapunov exponents. In: Ergod. Th. Dynam. Syst. Volume 22, No. 6, 2002, pp. 1667-1696.
Web links
- M. Viana: (Dis) continuity of Lyapunov exponents (English; PDF; 339 kB)