Phase space universality for multimodal maps (Q2576718)

The paper deals with the multimodal map \(f:I\to I\), \(I=[-1,1]\), where \(f\) is a smooth map with a finite number of critical points, all of them local maximum or local minimum, and such that \(f(\partial I) \subset\partial I\). The \(f\) is a multimodal map of type \(n\) if it can be written as a composition of \(n\) unimodal maps: to be more precise, if there exist maps \(f_1,\dots,f_n\) with the following properties \(f=f_n\circ \cdots\circ f_1\); \(f_i:I\to I\) has an unique critical point and \(f_i(\partial I)\subset\partial I\); if \(c_i\) is the critical point of \(f_i\), then \(f_i(c_i)\geq c_{i+1}\bmod n\). The \(n\)-tuple \((f_1,\dots,f_n)\) is a decomposition of \(f\). For each decomposition of \(f\), the author associates an extended map \(F\) defined on \(I_n=\{(x,i):x\in I,1\leq i\leq n\}\) by \(F(x,i)= (f_i(x), i+1\bmod n)\). For the extended map \(F\), the restrictive interval for \(F\) of period \(k\) is defined. Denote by \(N_k\) the period of the restrictive interval. Then we say that \(f\) has \(C\)-bounded combinatorics if \(N_{k+1}/N_k\leq C\). The author investigates the conditions, under which the \(k\)th renormalizations \(R^kf\) and \(R^kg\) have an analytic extension, where \(f\) and \(g\) are multimodal maps having the same combinatorics.