## Institute for Mathematical Sciences

## Preprint ims05-04

** Anca Radulescu**
* The Connected Isentropes Conjecture in a Space of Quartic Polynomials*

Abstract: 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.

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