There are not many periodic orbits in bunches for iteration of complex quadratic polynomials of one variable

Abstract

It is proved that for every complex quadratic polynomial $f$ with Cremer's fixed point $z_0$ (or periodic orbit) for every $\delta>0$, there is at most one periodic orbit of minimal period $n$ for all $n$ large enough, entirely in the disc (ball) $B(z_0, \exp -\delta n)$ (at most $p$ for a periodic Cremer orbit of period $p$; in fact at most one, by a renormalization remark by C.L.Petersen). Next, it is proved that the number of periodic orbits of period $n$ in a bunch $P_n$, that is for all $x,y\in P_n$, $|f^j(x)- f^j(y)|\le \exp -\delta n$ for all $j=0,...,n-1$, does not exceed $\exp \delta n$. We conclude that the geometric pressure defined with the use of periodic points coincides with the one defined with the use of preimages of an arbitrary typical point. I. Binder, K. Makarov and S. Smirnov (Duke Math. J. 2003) proved this for all polynomials but assuming all periodic orbits were hyperbolic, and asked about general situations. We prove here a positive answer for all quadratic polynomials.