There are not many periodic orbits in bunches for iteration of complex quadratic polynomials of one variable
Feliks Przytycki
TL;DR
This work addresses how geometric pressures defined via periodic points relate to pressures defined via preimages for quadratic polynomials with Cremer phenomena. The authors combine distortion estimates near Cremer points with external-ray dynamics and Milnor's orbit portraits to bound nearby periodic orbits and verify Hypothesis H for all quadratics. Consequently, for $t\ge 0$, the variational, tree, and periodic pressures coincide, extending pressure-equivalence results beyond hyperbolic settings. The findings bridge dynamical and thermodynamic formalisms in non-hyperbolic complex dynamics and solidify the role of external rays and orbit portraits in counting near-periodic behavior.
Abstract
It is proved that for every complex quadratic polynomial $f$ with Cremer's fixed point $z_0$ (or periodic orbit) for every $δ>0$, there is at most one periodic orbit of minimal period $n$ for all $n$ large enough, entirely in the disc (ball) $B(z_0, \exp -δn)$ (at most $p$ for a periodic Cremer orbit of period $p$). Next, it is proved that the number of periodic orbits of period $n$ in a bunch $P_n$, that is for all $x,y\in P_n$, $|f^j(x)- f^j(y)|\le \exp -δn$ for all $j=0,...,n-1$, does not exceed $\exp δn$. 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.