  This note is a shortened version of my dissertation thesis, defended at Stony
Brook University in December 2004. It illustrates how dynamic complexity of a
system evolves under deformations. The objects I considered are quartic
polynomial maps of the interval that are compositions of two logistic maps. In
the parameter space $P^{Q}$ of such maps, I considered the algebraic curves
corresponding to the parameters for which critical orbits are periodic, and I
called such curves left and right bones. Using quasiconformal surgery methods
and rigidity I showed that the bones are simple smooth arcs that join two
boundary points. I also analyzed in detail, using kneading theory, how the
combinatorics of the maps evolves along the bones. The behavior of the
topological entropy function of the polynomials in my family is closely related
to the structure of the bone-skeleton. The main conclusion of the paper is that
the entropy level-sets in the parameter space that was studied are connected.

Dynamical Systems