A partial classification of inverse limit spaces of tent maps with periodic critical points (Q697611)

This paper deals with ``tent'' maps \(f_a:[0,1]\rightarrow[0,1]\) where \(a\in[\sqrt 2,2]\) and \(f_a(x)= \)min\(\{ax,a(1-x)\}\). Such \(f_a\) gives rise to the inverse sequence, each of whose bonding maps is \(f_a\) and whose limit we shall denote by \(X_a\). If \(a\neq b\), then one may ask if \(X_a\) is homeomorphic to \(X_b\), but only partial results exist in answer to this question. A map \(f:[a,b]\rightarrow[a,b]\) is called unimodal if there is a unique point \(c\in[a,b]\), called the turning point, such that \(f\) is increasing on \([a,c]\) and \(f\) is decreasing on \([c,b]\). Observe that \(c=\frac{1}{2}\) is a turning point for each of the maps \(f_a\). It is assumed in this work that for some \(n\in\mathbb N\), \(f_a^n(c)=c\), and the first such \(n\) is called the period of the tent map \(f_a\). From this the notion of the kneading sequence \(\{b_0,\dots,b_n\}\) of \(f_a\) is defined, where \(b_n=C\), and for \(0\leq i<n\), \(b_i=R\) if \(f_a^i(c)>c\) and \(b_i=L\) otherwise. The kneading sequence is said to have even or odd parity depending respectively on the parity of the number of \(R\)'s. The author's main result is the following: \textbf{Main Theorem} Let \(f_a\) and \(f_b\) be tent maps with kneading sequences of the same finite period. If \(X_a\) is homeomorphic to \(X_b\), then one of the following holds: (1) The kneading sequences of \(f_a\) and \(f_b\) have the same parity and have the same number of \(R\)'s. (2) The kneading sequences of \(f_a\) and \(f_b\) have different parity and the number of \(R\)'s in that of \(f_a\) is the same as the number of \(L\)'s in that of \(f_b\).