Injectivity and surjectivity of Collatz functions

The paper studies conditions for injectivity and surjectivity of generalized Collatz functions. Necessary and sufficient conditions for each property, involving the complete set of parameters, are derived. The injectivity condition is formulated as a property of gcd matrices and the surjectivity condition is shown to be related to covering congruences. Furthermore, it is proved that injectivity implies Sigma(i=0)(d-1) 1/m(i) less than or equal to1 while surjectivity implies Sigma(i=0)(d-1) 1/m(i) greater than or equal to1. Next, some implications of injectivity and surjectivity are discussed. In particular, it is shown that any injective Collatz function which is not surjective has divergent trajectories. (C) 2004 Elsevier B.V. All rights reserved.