A Method for Estimating Entropy of Real Birational Maps with Constrained Critical Orbits

The entropy of a discrete dynamical system is a rough gauge of its complexity. We wish to find a bound for the entropy of a system. We particularly concern ourselves with the specific family of maps $f:\overline{\R} \ imes \overline{\R} \ ightarrow \overline{\R} \ imes \overline{\R}$ defined by $f(x,y) = \left(y\?rac{x+a}{x-1},x+a-1\ ight)$, where $a \in \R$ is a parameter subject to conditions specified later. This family first appeared in various statistical physics papers and was considered mathematically by Bedford and Diller. Our method is combinatorial, using invariant and critical curves for $f$ to define a partition of $\overline{\R} \ imes \overline{\R}$ that is convenient for coding orbits under $f$. We may expand these codings to hold for points without well-defined orbits. These more general codings allow us to construct a set with easier-to-calculate entropy. This new set's entropy is a lower bound for the entropy of $f$.