Comparison of several generating partitions of the Hénon map

Abstract

The existence and uniqueness of generating partitions in non-hyperbolic dynamical systems is usually studied in a simple, exemplary system, namely the Henon map at standard parameter values a = 1.4 and b = 0.3. We compare its standard partition with three other binary partitions, which are quite different from the standard partition but also appear to be generating. One of these partitions passes twice through the same orbit of homoclinic tangencies, providing a counterexample to a recent conjecture by Jaeger and Kantz ( J. Phys. A: Math. Gen. 30 L567). Introducing some simple rules to manipulate symbolic sequences, we show how to translate symbolic sequences produced by one partition into sequences produced by the other partitions. This proves that all these partitions are as good approximations to generating partitions as the standard partition. We also construct an infinite number of binary partitions, which are all quite similar to the standard partition, derive their translation rules, and prove the same equivalence. It is not known for sure whether any of these partitions is indeed generating. But if one of them is generating, then they all are.