On double spirals in Fibonacci-like unimodal inverse limit spaces (Q2862924)

The author studies the possibility of asymptotic composants for inverse limits of special unimodal maps with non-periodic recurrent critical points (in particular unimodal maps of Fibonacci-like type).NEWLINENEWLINEHe represents composants of inverse limits with such bonding maps as walks on a tree related to the Hofbauer tree (or tower) of these maps. Although he does not answer the question of whether Fibonacci-like unimodal inverse limit spaces possess asymptotic composants, he demonstrates the existence of (uncountably many) so-called double spirals in these inverse limits, which show that points with different symbolic tails can still be on the same arc in such an inverse limit space.NEWLINENEWLINEThis shows that the converse of a result by Brucks and Diamond does not hold, namely that points in such an inverse limit space with the same tails of their backwards itineraries belong to the same arc component.