Inverse limits with a single set valued function on [0, 1]

We are a long way from a characterization of continua that are the inverse limit with a single upper semi-continuous set valued function on [0, 1]. Recently we have shown that an arc is the only finite graph which is the inverse limit with a single upper semi-continuous set valued function on [0, 1]. We can generate many interesting and difficult questions by asking for a fixed graph G, if a continuum is the inverse limit with single valued bonding maps on G, and it is also the inverse limit with a single upper semi-continuous set valued function on [0, 1], then must it be chainable? In this talk we will discuss some new techniques for exploiting the dynamics of shift maps on inverse limits with a single set valued function and use them to examine cyclically connected and circle-like continua that are the inverse limit with a single upper semi-continuous set valued function on [0, 1].