Mathematics – Dynamical Systems
Scientific paper
2009-03-09
Mathematics
Dynamical Systems
To clarify all the details and simply the proof, I made many changes to the latter version, however the proof follows the same
Scientific paper
For every $C^2$-small perturbation $B$, we prove that the map $(x,y)\mapsto (x^2+a+y,0)+B(x,y,a)$ preserves a physical, SRB probability, for a positive Lebesgue measure set of parameters $a$. When the perturbation $B$ is zero, this is the Jakobson Theorem; when the perturbation is a small constant times $(0,1)$, this is the celebrated Benedicks-Carleson Theorem. In particular, a new proof of the last Theorem is given, by basically mixing analytical ideas of Benedicks-Carleson and the combinatorial formalism of Yoccoz puzzle with new geometrical and arithmetic ingredients. These proofs are enlarged to the $C^2$-topology. Also the dynamics can be an endomorphism.
Berger Pierre
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