Abstract:

An enrichment of a degree $d$ rational map $f$ is an accumulation point of the double sequence $(f_n^N)_{n,N}$ for some uniformly convergent sequence $(f_n)_n$ of rational functions of degree $d$ converging to $f$. When $f$ has a $p$-periodic Siegel disc $Delta$ with a Brjuno rotation number $theta$, I will explain that every isomorphism,commuting with $f^p$, between two $f^p$-invariant sub-discs of the cycle generated by $Delta$ is an enrichment of $f$. For this the control of the size of linearisation domains of the maps $f_n$ is crucial and the quadratic family plays here a central role. A surprising fact is that for some sequences $(f_n)_n$ converging to $f$, the double sequence $(f_n^N)_{n,N}$ is normal on a $f^p$-invariant subdisc of $Delta$ but every subsequence diverges on a bigger subdisc. If time permits I will give a geometric interpretation of this result in the family of quadratic polynomials $P_{lambda} : zmapstolambda z+z^2$ ($lambdainC^*$).