An Entire Function Extension of the Accelerated Collatz Map

Ideas / Approaches
1

I would like to share a simple entire function that interpolates
the accelerated Collatz (Syracuse) map on the integers.

The function is defined by:

image
image848×142 4.71 KB

For integer values, this reproduces exactly the Syracuse map.

It can also be written as a convex combination of two affine maps:

image
image782×362 8.05 KB

I am interested in feedback on whether this type of analytic extension
has been studied in similar form, and whether it may be useful in the
context of complex dynamical systems.

2

Nice!

For your question, a classic reference for ideas about real-valued Collatz is

M. Chamberland, A continuous extension of the 3x + 1 problem to the real line, Dynamics of Continuous, Discrete, and Impulsive Systems, 2 (1996) 495-509.

Comparing with that should be interesting.

3

Thanks for the reference, this is very helpful.