Multi-dimensioned intertwined basin boundaries and the kicked double rotor

Using two examples, a four-dimensional kicked double rotor and a simple noninvertible one-dimensional map, we show that basin boundary dimensions can be different regions of phase space. For example, they can be fractal or not fractal depending on the region. In addition, we show that these regions of different dimension can be intertwined on arbitrarily fine scale. We conjecture, based on these examples, that a basin boundary typically can have at most a finite number of possible dimension values.