Many of us remember when the Spring Topology Conference became the the Spring Topology and Dynamics conference in part because continua theorists were finding so many things they wanted to work on in dynamics. Classical Interval Dynamics is now a mature field with hundreds of articles and many books. Years ago continua theorists with considerable inspiration from Devaney’s accessible book began extending theorems in interval dynamics like the Sarkovski theorem to chainable continua, that is continua that are the inverse limit of interval functions. A favorite tool of continua theorists, inverse limits, have also been used in dynamical systems since whatever one might call the beginning.

Inspired by Ethan Akin our group has been constructing continua and functions at the same time that have a variety of dynamical properties using what we call Mahavier products, also known as an inverse limit with a set valued function. Akin would probably call what we are doing the dynamics of closed relations. The dynamics are those of shift maps. In other words we extend from the now classic topic of dynamics of shift maps on inverse limits with a single bonding map to continua that cannot be expressed as an inverse limit with a single continuous function on a simpler space like an arc or a tree or a circle, but can be expressed as an inverse limit with a single closed relation. Specifically in this talk we look at various ways to express the Cantor fan and the Lelek fan as a Mahavier products . We obtain transitive homeomorphisms, mixing homeomorphisms, with and without a dense set of periodic orbits and with zero or positive entropy. This is joint work with Iztok Banic, Judy Kennnedy, Chris Mouron, and Goran Erceg.