What's the name of this fractal? 
I created it with some simple chaos game restrictions:


*

*You can always move towards the point you moved to on the last step

*When on the top left corner, you can go to the right corners

*When on the top right corner, you can go to the left corners

*When on the bottom corners, you can only go to the corner that is opposite


Your figure can be constructed by a graph directed iterated function system.  An iterated function system typically has no restrictions on which transforms may follow each other.  A graph-directed IFS has restrictions like the one you have imposed: there is a directed graph in which the transformations correspond to edges.  In your example the nodes of this directed graph would correspond to the corners of your figure, like this:

For a reference, with fractal dimension calculations, see:
"Hausdorff dimension in graph directed constructions" R. Daniel Mauldin and S. C. Williams (Trans. Amer. Math. Soc. 309 (1988), 811-829) http://www.ams.org/journals/tran/1988-309-02/S0002-9947-1988-0961615-4/

We introduce the notion of geometric constructions in $R^m$ governed by a directed graph $G$ and by similarity ratios which are labelled with the edges of this graph.

