Let
be a family of unimodal maps with topological entropies
, and
be their natural
extensions, where
.
Subject to some regularity conditions, which are satisfied by tent maps and quadratic maps,
we give a complete description of the prime ends of the Barge–Martin embeddings of
into the sphere. We
also construct a family
of sphere homeomorphisms with the property that each
is a
factor of
,
by a semiconjugacy for which all fibers except one contain at most three points,
and for which the exceptional fiber carries no topological entropy; that is,
unimodal natural extensions are virtually sphere homeomorphisms. In the case
where is the tent
family, we show that
is a
generalized pseudo-Anosov map for the dense set of parameters for which
is postcritically
finite, so that
is the completion of the unimodal generalized pseudo-Anosov family introduced by de
Carvalho and Hall (Geom. Topol. 8 (2004) 1127–1188).
Keywords
natural extensions, inverse limits, unimodal maps, prime
ends, sphere homeomorphisms