To Define the Core Entropy for All Polynomials Having a Connected Julia Set

Abstract

The classical core entropy h core (f ) for post critically finite (PCF) polynomials f with degree d ≥ 2 is defined to be the topological entropy of f restricted to its Hubbard tree. We fully generalize this notion by a new quantity E core (f ), called the (general) core entropy of f , which is well defined for all polynomials f with degreefor all parameters c running through the Mandelbrot set M. Contents 1. Introduction 2. E core (f ) and the usc Decomposition D(f ) for f 3. E core (f ) and the Classical Core Entropy h core (f ) 4. E core (f ) and the Maximal Dendrite Factor of (J, f ) 5. E core (f ) and the Core Entropy for Small Julia Sets 6. E core (f ) and the Huasdorff Dimension of B * (f ) 7. The Function h(c) = E core (f c ) with f c (z) = z 2 + c and Thurston's Algorithm 8. The Tuning Lemma and the Level Set h −1 (0) 9. The Monotonicity Lemma and Continuity Points of h(c) 10. Three Remarks References