New York Journal of Mathematics
Volume 30 (2024), 783-827


Sushmanth J. Akkarapakam and Patrick Morton

Periodic points of algebraic functions related to a continued fraction of Ramanujan

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Published: June 3, 2024.
Keywords: Periodic points, algebraic function, 2-adic field, extended ring class fields, Ramanujan continued fraction.
Subject [2020]: 14H05, 37F05, 11R37, 11D88, 11R29.

A continued fraction v(τ) of Ramanujan is evaluated at certain arguments in the field K = Q(\sqrt{-d}), with -d = 1 (mod 8), in which the ideal (2) = P2 P'2 is a product of two prime ideals. These values of v(τ) are shown to generate the inertia field of P2 or P'2 in an extended ring class field over the field K. The conjugates over Q of these same values, together with 0, -1 ± \sqrt{2}, are shown to form the exact set of periodic points of a fixed algebraic function F(x), independent of d. These are analogues of similar results for the Rogers-Ramanujan continued fraction.



Author information

Sushmanth J. Akkarapakam
Department of Mathematics
University of Missouri at Columbia
208 Math Sci Building, 810 Rollins St.
Columbia, MO 65211, USA


Patrick Morton
Department of Mathematical Sciences
Indiana University-Purdue University at Indianapolis (IUPUI)
402 N. Blackford St., LD 270
Indianapolis, IN 46202, USA