We work within the one-parameter family of symmetric tent maps, where the slope is the parameter. Given two such tent maps $f_a$, $f_b$ with periodic critical points, we show that the inverse limit spaces ${(}_a,f_a)$ and ${(}_b,g_b)$ are not homeomorphic when a ≠ b. To obtain our result, we define topological substructures of a composant, called “wrapping points” and “gaps”, and identify properties of these substructures preserved under a homeomorphism.

It is known that every homeomorphism of the plane which admits an invariant non-separating continuum has a fixed point in the continuum. In this paper we show that any branched covering map of the plane of degree d, |d| ≤ 2, which has an invariant, non-separating continuum Y, either has a fixed point in Y, or is such that Y contains a minimal (in the sense of inclusion among invariant continua), fully invariant, non-separating subcontinuum X. In the latter case, f has to be of degree -2 and X has...

Let f be a unimodal map in the logistic or symmetric tent family whose restriction to the omega limit set of the turning point is topologically conjugate to an adding machine. A combinatoric characterization is provided for endpoints of the inverse limit space (I,f), where I denotes the core of the map.

This paper surveys some recent results concerning inverse limits of tent maps. The survey concentrates on Ingram’s Conjecture. Some motivation is given for the study of such inverse limits.