We study how inverse limits on graphs can be embedded in
the plane. For example, it is well known that every
inverse limit on arcs (chainable continuum) can be
embedded in the plane (Bing 1951). Specially, if there is
only one bonding map, then the inverse limit can be
embedded as an attractor of a planar homeomorphism (Barge
and Martin 1990). We are interested in different planar
embeddings of the same space, where planar copies of the
space are considered equivalent if there is a planar
homeomorphism mapping one onto another. For example, in
certain cases one can show that the same space can be
embedded in the plane in uncountably many non-equivalent
ways.
We will show how to embed inverse limits in the plane,
generalizing Bing and Barge-Martin construction.
Furthermore, we will investigate when embedded spaces are
dynamically significant (i.e. which self homeomorphisms
can be extended to a planar homeomorphism) and how to
detect accessible points within them.