  The classical core entropy for a post critically finite (PCF) polynomial f
with deg(f)>1 is defined to be the topological entropy of f restricted to its
Hubbard tree. We fully generalize this notion by a new quantity, called the
(general) core entropy, which is well defined whenever f has a connected Julia
set. If f is PCF, the core entropy equals the classical version. If two
polynomials are J-equivalent they share the same core entropy. If f is
renormalizable there is a direct connection between the core entropy of f and
that corresponding to the small Julia set. We also analyze the map that sends
every parameter c in the Mandelbrot set to the core entropy of the polynomial
z^2+c. In particular, the level set of this entropy map at log2 is of full
harmonic measure and the Mandelbrot set is locally connected at each point in
this level set.
