Dimensions of surface repellers and attractors of non-linear planar IFSs

What the math gives to ML

The paper provides a non-conformal Bowen formula: under genericity, separation, irreducibility, and expansion assumptions, the Hausdorff and box dimensions of a planar repeller or nonlinear IFS attractor are the unique zero of a sub-additive singular-value pressure. The transferable mechanism is to treat products of neural Jacobians as non-conformal cocycles and use pressure, rather than only the largest singular value, to measure multiscale geometric complexity. A concrete neural application is a multi-branch recurrent or generative architecture whose attractor dimension is controlled by a pressure penalty, with the predicted dimension obtained from an empirical pressure zero-crossing. This is most credible for expanding neural IFSs with explicit branch separation, rather than arbitrary feed-forward networks.