Adding machines, kneading maps, and endpoints

Given a unimodal map f, let I = [ c 2 , c 1 ] denote the core and set E = { ( x 0 , x 1 , … ) ∈ ( I , f ) | x i ∈ ω ( c , f ) for all i ∈ N } . It is known that there exist strange adding machines embedded in symmetric tent maps f such that the collection of endpoints of ( I , f ) is a proper subset of E and such that lim k → ∞ Q ( k ) ≠ ∞ , where Q ( k ) is the kneading map. the partition structure of an adding machine to provide a sufficient condition for x to be an endpoint of ( I , f ) in the case of an embedded adding machine. We then show there exist strange adding machines embedded in symmetric tent maps for which the collection of endpoints of ( I , f ) is precisely E . Examples of this behavior are provided where lim k → ∞ Q ( k ) does and does not equal infinity, and in the case where lim k → ∞ Q ( k ) = ∞ , the collection of endpoints of ( I , f ) is always E .