  In this paper we extend the notion of an $\alpha$-family of maps to discrete
systems defined by simple difference equations with the fractional Caputo
difference operator. The equations considered are equivalent to maps with
falling factorial-law memory which is asymptotically power-law memory. We
introduce the fractional difference Universal, Standard, and Logistic
$\alpha$-Families of Maps and propose to use them to study general properties
of discrete nonlinear systems with asymptotically power-law memory.
