Tie-points and fixed-points in N^*

A point x is a (bow) tie-point of a space X if X {set minus} {x} can be partitioned into relatively clopen sets each with x in its closure. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of β N {set minus} N (e.g. [S. Shelah, J. Steprāns, Martin's axiom is consistent with the existence of nowhere trivial automorphisms, Proc. Amer. Math. Soc. 130 (7) (2002) 2097-2106 (electronic). MR 1896046 (2003k:03063), B. Veličković, OCA and automorphisms of P (ω) / fin, Topology Appl. 49 (1) (1993) 1-13]) and in the recent study of (precisely) 2-to-1 maps on β N {set minus} N. In these cases the tie-points have been the unique fixed point of an involution on β N {set minus} N. This paper is motivated by the search for 2-to-1 maps and obtaining tie-points of strikingly differing characteristics.