ON CLASSIFICATION OF TENT MAPS INVERSE LIMITS: A COUNTEREXAMPLE

Generalized tent functions are functions from [0, 1] to [0, 1] whose graphs are unions of two straight line segments, one from (0, 0) to (a, b), and the other one from (a, b) to (1, 0), where (a, b) is any point in [0, 1] x [0, 1]. The point (a, b) is called the top point of the graph of such function. I. Banic, M. Crepnjak, M. Merhar and U. Milutinovie recently described a family F = {C-t vertical bar t is an element of [1, infinity)} of curves in (0, 1) x [0, 1] and showed that for each positive integer n, the following holds true. If (a, b), (c, d) is an element of C-n, are top points of two generalized tent functions, then the corresponding inverse limits are homeomorphic. They also discuss if the same holds true for any t is an element of [1, infinity). More precisely, let (a, b) and (c, d) be top points of two generalized tent functions. Are then the corresponding inverse limits homeomorphic, if (a, b), (c, d) is an element of C-t? They pose this question as an open problem. In this paper we construct a curve D in (0, 1) x [0, 1] satisfying the following properties: (1) D is not an element of F and (2) if (a, b), (c, d) is an element of D, then the corresponding inverse limits are homeomorphic. Using this curve and the Ingram conjecture we answer the above question in negative.