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 \ensuremath{\Vert}b\ensuremath{\Vert}<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(\ensuremath{\alpha}). The description in terms of periodic points works very well in the ``hyperbolic phase,'' for \ensuremath{\alpha} larger than some ${\ensuremath{\alpha}}_{c}$, where ${\ensuremath{\alpha}}_{c}$ is the value of \ensuremath{\alpha} corresponding to the (conjectured) phase transition.