We use the set of all periodic points of Hénon-type mappings to develop a theory of the topological and metric properties of their attractors. The topology of a Hénon-type attractor is conveniently represented by a two-dimensional symbol plane, with the allowed and disallowed orbits cleanly separated by the ‘‘pruning front.’’ The pruning front is a function discontinuous on every binary rational number, but for maps with finite dissipation ‖b‖<1, it is well approximated by a few steps, or, in the symbolic dynamics language, by a finite grammar. Thus equipped with the complete list of allowed periodic points, we reconstruct (to resolution of order $b^n$) the physical attractor by piecing together the linearized neighborhoods of all periodic points of cycle length n. We use this representation to compute the singularity spectrum f(α). The description in terms of periodic points works very well in the ‘‘hyperbolic phase,’’ for α larger than some ${\alpha }_c$, where ${\alpha }_c$ is the value of α corresponding to the (conjectured) phase transition.

• Received 23 November 1987

DOI:https://doi.org/10.1103/PhysRevA.38.1503