The algebraic entropy of character automorphisms of a rank 2 free group.
by
Richard J Brown
Johns Hopkins University

We calculate the algebraic entropy of a certain class of polynomial automorphisms of C³, and show that it equals the maximum topological entropy of the action when restricted to its compact, real invariant subvarieties. These polynomial automorphisms arise as actions of the mapping class group of a punctured torus S on the SL(2, C)-character variety of S, which is isomorphic to affine 3-space. The SU(2)-character varieties of S relative to fixed boundary holonomy are real, compact invariant subvarieties. It is known that the topological entropy of the action on these SU(2)-characters is maximized on the relative character variety comprised of reducible characters (those whose boundary holonomy is 2). Here the topological entropy equals the algebraic entropy. In the context of the special linear characters of S, the algebraic entropy depends solely on the topology of S.

Date received: February 16, 2006