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The connected Isentropes conjecture in a space of quartic polynomials
This note is a shortened version of my dissertation paper, 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 evolve 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.