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 with Cremer's fixed point (or periodic orbit) for every , there is at most one periodic orbit of minimal period for all large enough, entirely in the disc (ball) (and at most for a Cremer orbit of period ; in fact, at most one, by a renormalisation remark by C.L. Petersen). Next, it is proved that the number of periodic orbits of period in a bunch , that is, for all , for all , does not exceed . We conclude that the geometric pressure defined with the use of periodic points coincides with the one defined with the use of pre‐images of an arbitrary typical point. I. Binder, N. 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.