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

In this note, we establish a connection between the dynamical degree, or algebraic entropy of a certain class of polynomial automorphisms of $\mathbb{R}^3$, and the maximum topological entropy of the action when restricted to compact invariant subvarieties. Indeed, when there is no cancellation of leading 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 $\operatorname{SU}(2)$-character varieties of $S$ embedded in $\mathbb{R}^3$. 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 calculate the algebraic entropy of the induced polynomial automorphisms on the character varieties and show that it too solely depends on the topology of $S$.