Real dynamics of a 3-power extension of the 3x+1 function

This paper investigates the real dynamics of a generalization of the 3x + 1 function using powers of three. We will see that any cycle of positive integers is attractive for this generalization and the cycle has an expansion factor given by Terras' coefficient function. We will see the function has a negative Schwarzian derivative for x greater than or equal to 0 and will be able to identify invariant intervals and approximately locate the fixed points and critical points. The special simplicity of dynamics around the cycle (1, 2) means there is a natural generalization of total stopping time for this function. We conjecture that the odd critical points of this generalization are well behaved. In particular, they lie in the immediate basin of total stopping time surrounding each odd integer.

Title	Real dynamics of a 3-power extension of the 3x+1 function
Creator	Dumont, J. P.
Creator	Reiter, Clifford A.
Publisher	Dynamics of Continuous, Discrete and Impulsive Systems A
Academic Department	Mathematics
Division	Natural Sciences
Organization	Lafayette College
Date Issued	December 2003
Date Available	2023-04-17
Type	Article
Language	English
Keyword	Syracuse problem
	3n+1
	Collatz problem
	hailstone function
	real dynamics
Bibliographic Citation	Dumont, J. P. and C. A. Reiter (2003 Dec) "Real dynamics of a 3-power extension of the 3x+1 function." Dynamics of Continuous, Discrete and Impulsive Systems A 10 (6): 875-893.
Standard Identifier	Handle 10385/qz20sv02j
Permalink	http://hdl.handle.net/10385/qz20sv02j
Rights Statement	In Copyright
Rights Holders	Watam Press