Likely Limit Sets of a Class of $p$-Order Feigenbaum's Maps (Communications in Mathematical Research, 2021)
A continuous map from a closed interval into itself is called a $p$-orderFeigenbaum's map if it is a solution of the Feigenbaum's equation $f^p(λx) = λf(x)$.In this paper, we estimate Hausdorff dimensions of likely limit sets of some $p$-orderFeigenbaum's maps. As an application, it is proved that for any $0 < t < 1$, therealways exists a $p$-order Feigenbaum's map which has a likely limit set with Hausdorffdimension $t$. This generalizes some known results in the special case of $p = 2$.
Keywords: Feigenbaum's equation; Feigenbaum's map; likely limit set; Hausdorff dimension.