We discuss fractal structures of invariant sets. To develop a general
- in particular in a higherdimensional setting - dimension theory is a
notoriously difficult problem. One natural approach to study
higherdimensional dynamical systems (and perhaps eventually their
dimension theory) is to consider gradually more complex (e.g.
higher-dimensional) settings. Following this approach, we consider
sets which can be considered as graphs over hyperbolic sets in 2
dimensions. Moreover, we restrict our considerations to sets which are
invariant under diffeomorphisms which simultaneously have a
partially-hyperbolic and a hyperbolic structure. We describe the
(critical: either Lipschitz or at all scales Hölder continuous)
regularity of such graphs and draw conclusions about their
box-counting dimension. A key ingredient for the dimension arguments
will be the presence of a blender-like horseshoe.
The results grew from discussions with M. Gröger and T. Oertel-Jäger (Jena).