The emergence of 4-cycles in polynomial maps over the extended integers

Let $f(x) in mathbb{Z}[x]$; for each integer $alpha$ it is interesting to consider the number of iterates $n_{alpha}$, if possible, needed to satisfy $f^{n_{alpha}}(alpha) = alpha$. The sets ${alpha, f(alpha), ldots, f^{n_{alpha} - 1}(alpha), alpha}$ generated by the iterates of $f$ are called cycles. For $mathbb{Z}[x]$ it is known that cycles of length 1 and 2 occur, and no others. While much is known for extensions to number fields, we concentrate on extending $mathbb{Z}$ by adjoining reciprocals of primes. Let $mathbb{Z}[1/p_1, ldots, 1/p_n]$ denote $mathbb{Z}$ extended by adding in the reciprocals of the $n$ primes $p_1, ldots, p_n$ and all their products and powers with each other and the elements of $mathbb{Z}$. Interestingly, cycles of length 4, called 4-cycles, emerge for polynomials in $mathbb{Z}left[1/p_1, ldots, 1/p_nright][x]$ under the appropriate conditions. The problem of finding criteria under which 4-cycles emerge is equivalent to determining how often a sum of four terms is zero, where the terms are $pm 1$ times a product of elements from the list of $n$ primes. We investigate conditions on sets of primes under which 4-cycles emerge. We characterize when 4-cycles emerge if the set has one or two primes, and (assuming a generalization of the ABC conjecture) find conditions on sets of primes guaranteed not to cause 4-cycles to emerge.