We present a scheme for constructing examples of new types of continua and dynamical systems using generalized inverse limits. We start with two compact metric spaces X and Y, a surjective, continuous map T from Y to itself, and a continuous surjection either from Y onto X or from X onto Y.

- (1) If k is a continuous surjection from Y onto X, then f=kTk^(-1) is a set valued map from X to the closed subsets of X, and we can consider the generalized inverse limit formed with this one bonding map, as well as the induced shift map, which is a continuous map from the inverse limit onto itself, and the relationship of the generalized inverse limit with the standard inverse limit generated by T on Y.
- (2) If h is a continuous surjection from X onto Y, then f=h^(-1) Th is a set valued map from X to the closed subsets of X, and likewise we can consider the generalized inverse limit generated by this bonding map, along with the dynamics of the shift map on this inverse limit, and the relationship of the generalized inverse limit with the standard inverse limit generated by T on Y.

Professor Judy Kennedy received her PhD from Auburn University in 1975. Most of her professional career was spent in the Department of Math Sciences at the University of Delaware in Newark, Delaware, but in 2007 she moved to the Department of Mathematics at Lamar University in Beaumont, Texas. She has published nearly 80 papers in areas of continuum theory and dynamical systems (with a number of co-authors). She has given talks in 22 countries so far; and she loves doing and teaching mathematics. Her current research interest is the investigation of topology and dynamics of inverse limits with set-valued functions.