The real teapot
Lluis Alseda, Jozef Bobok, Michał Misiurewicz and Lubomir Snoha
Abstract
In his last paper, William Thurston defined the Master Teapot as the
closure of the set of pairs (z,s), where s is the slope
of a tent map Ts with the turning point periodic,
and the complex number z is a Galois conjugate of s. In
this case 1/z is a zero of the kneading determinant
of Ts. We remove the restriction that the turning
point is periodic, and sometimes look beyond tent maps. However, we
restrict our attention to zeros x=1/z in the real
interval (0,1). By the results of Milnor and Thurston, the kneading
determinant has such a zero if and only if the map has positive
topological entropy. We show that the first (smallest) zero is simple,
but among other zeros there may be multiple ones. We describe a class
of unimodal maps, so-called R-even ones, whose kneading determinant
has only one zero in (0,1). In contrast with this, we show that
generic mixing tent maps have kneading determinants with infinitely
many zeros in (0,1). We prove that the second zero in (0,1) of the
kneading determinant of a unimodal map, provided it exists, is always
larger than or equal to the third root of ½ and if the
kneading sequence begins with $RLNR,
where N>1, then the best lower bound for the second zero is in
fact N+1st root of ½. We also investigate (partially
numerically) the shape of the Real Teapot, consisting of the
pairs (s,x), where x in (0,1) is a zero of the kneading
determinant of Ts, and s is in (1,2].