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In this note, we establish a connection between the dynamical de- gree, or algebraic entropy of a certain class of polynomial automorphisms of R3, and the maximum topological entropy of the action when restricted to compact invariant subvarieties. Indeed, when there is no cancellation of lead- ing terms in the successive iterates of the polynomial automorphism, the two quantities are equal. In general, however, the algebraic entropy overestimates the topological entropy. These polynomial automorphisms arise as extensions of mapping class actions of a punctured torus S on the relative SU(2)-character varieties of S embedded inR3. It is known that the topological entropy of these mapping class actions is maximized on the relative character variety comprised of reducible characters (those whose boundary holonomy is 2). Here we cal- culate the algebraic entropy of the induced polynomial automorphisms on the character varieties and show that it too solely depends on the topology of S.

The algebraic entropy of the special linear character automorphisms of a free group on two generators