Periodic cycles for an extension of generalized 3x + 1 functions

The Collatz conjecture is an open problem involving the function. The function belongs to a class of generalized functions of relatively prime type. This paper focuses on exploring periodic cycles for an extension of a generalized function of relatively prime type. By extending its domain to , the result shows that every integer periodic point is isolated in the usual topology on . Moreover, every positive integer periodic cycle for the extension is attracting if the generalized function is satisfied by parameters under some conditions.