On the number of parameters $c$ for which the point $x=0$ is a superstable periodic point of $f_c(x) = 1 - cx^2$

TL;DR

This paper investigates the distribution and growth rate of parameters c for which the quadratic map exhibits superstable periodic points at x=0, revealing exponential growth in such parameters within a specific interval.

Contribution

It introduces a recursive method to identify these parameters and establishes a lower bound on their exponential growth rate for a class of quadratic-like maps.

Findings

The number of such parameters grows exponentially with n.

A recursive depiction of parameter appearance in [0, 2] is provided.

The growth rate limit infimum is at least log 2.

Abstract

Let $f_c(x) = 1 - cx^2$ be a one-parameter family of real continuous maps with parameter $c \ge 0$. For every positive integer $n$, let $N_n$ denote the number of parameters $c$ such that the point $x = 0$ is a (superstable) periodic point of $f_c(x)$ whose least period divides $n$ (in particular, $f_c^n(0) = 0$). In this note, we find a recursive way to depict how {\it some} of these parameters $c$ appear in the interval $[0, 2]$ and show that $\liminf_{n \to \infty} (\log N_n)/n \ge \log 2$ and this result is generalized to a class of one-parameter families of continuous real-valued maps that includes the family $f_c(x) = 1 - cx^2$.