Topological and metric properties of Hénon-type strange attractors

We use the set of all periodic points of H\'enon-type mappings to develop a theory of the topological and metric properties of their attractors. The topology of a H\'enon-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 <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 (). The description in terms of periodic points works very well in the ``hyperbolic phase, '' for larger than some ₂, where ₂ is the value of corresponding to the (conjectured) phase transition.