Full-Dimensional One-Sided Trapping for Smooth Area-Preserving Plane Maps

A point of an invertible plane dynamical system is one-sided trapped if its forward semiorbit is bounded and its backward semiorbit is unbounded. We construct a single smooth area-preserving diffeomorphism of the plane whose trapped set has Hausdorff dimension two. The map has the explicit form F(x,y)=(y,h(y)-x), with h given by a locally finite sum of smooth affine coding blocks. Its trapped set contains compact sets of dimensions 2-3/(j+3) for every positive integer j, and their backward orbits escape geometrically through a common channel. Thus the discrete-time maximum is two in every C^m class, including C-infinity. In contrast, for every complete jointly smooth autonomous area-preserving flow on the plane, a nonempty trapped set has dimension one. A self-contained critical-value argument proves this contrast. Finite-regularity continuous-time maxima are not determined. This is an unrefereed preprint; no independent expert review, formal verification or absolute priority certification is claimed.