Bounded Orbits of Quadratic Collatz-type Recursions

TL;DR

This paper characterizes bounded orbits in two quadratic Collatz-type mappings, showing conditions for boundedness and the absence of cycles in one case, and the equivalence of boundedness and reaching a cycle in the other.

Contribution

It provides a complete characterization of bounded orbits for two specific quadratic Collatz-type functions, including cycle analysis and orbit behavior.

Findings

In one map, bounded orbits reach a cycle, which can have any length.

In the other map, all bounded orbits eventually reach zero, with no cycles present.

The study advances understanding of orbit structures in quadratic Collatz-type problems.

Abstract

We characterize all bounded orbits of two similar Collatz-type quadratic mappings of the set of non-negative integers. In one case, where cycles of all possible lengths may occur, an orbit is bounded if and only if it reaches a cycle. For the other map we prove that every bounded orbit must reach 0 (in particular, there are no cycles).