Non-Archimedean indifferent components of rational functions that are not disks

Abstract

Consider a rational function of degree d ≥ 2 d\geq 2 acting on the Berkovich projective line of a complete and algebraically closed non-archimedean field. Rivera-Letelier has asked if a rational function in this setting can have infinitely many cycles of indifferent components that are not disks, and if not, if there exists a bound in terms of the degree of the function. In this work we show that for d = 2 d=2 the bound is d − 1 d-1 . By imposing an extra condition on the residue field and a connectivity requirement on the cycles of indifferent components that are not disks, the bound d − 1 d-1 is also achieved when d ≥ 3 d\geq 3 . To ensure that the bound is realized, we describe how to construct a rational function of degree d ≥ 2 d\geq 2 with exactly d − 1 d-1 cycles of indifferent components that are not disks.