DYNAMICS ON HUBBARD TREES Lluis Alseda and Nuria Fagella Lluis Alseda Departament de Matematiques Universitat Autonoma de Barcelona Edifici C 08193 Bellaterra, Barcelona Spain e-mail: alseda@mat.uab.es Nuria Fagella Departament de Matematica Aplicada i Analisi Universitat de Barcelona Gran Via 585 08007 Barcelona Spain e-mail: fagella@maia.ub.es ABSTRACT It is well known that the Hubbard tree of a postcritically finite complex polynomial contains all the combinatorial information of the polynomial. In fact, an abstract Hubbard tree as defined in \cite{pure1} uniquely determines the polynomial up to affine conjugation. In this paper we study how much of the ``dynamical information'' is captured by the Hubbard tree of a quadratic Misiurewicz polynomial. More precisely, we study dynamical features such as entropy, transitivity or periodic structure of the polynomial restricted to the Hubbard tree, and compare them with the properties of the polynomial on its Julia set. Our results show that there is a strong connection between the renormalization properties of the polynomial and the mentioned dynamical features of the polynomial on its Hubbard tree. As a consequence of this relation we obtain criteria to check if a quadratic Misiurewicz polynomial is renormalizable by means of its Hubbard tree.