THE STRUCTURE OF POINTWISE RECURRENT EXPANSIVE HOMEOMORPHISMS

The paper proves Theorem 1.2 by first reducing to dim(X)=0 and then using a canonical coding to obtain a conjugacy with a subshift; under positive recurrence and in dimension 0, each orbit-closure is minimal, yielding semisimplicity. The core of the dimension argument is to assume dim(X)>0, use a hyperbolic (Lyapunov) metric for expansive maps (Reddy) to set up Assumption 1 (or pass to an iterate), invoke Mañé’s connectedness lemma to find a local stable plaque that meets the boundary, refine via a boundary-bumping/recurrence selection to obtain an almost periodic point with a nontrivial connected stable plaque, and then construct a nested sequence of continua producing a point that is neither positively nor negatively recurrent—a contradiction to pointwise recurrence. This structure is explicit in Section 3 (Lemma 3.1, Assumption 1, Lemma 3.5) and Section 4 of the paper, including the final nonrecurrent-point construction and the reduction to coding and Proposition 2.3 for semisimplicity . The candidate solution follows the same plan: hyperbolic metric/iterate; a Mañé-type connected-set lemma; boundary bumping and recurrence to get a nontrivial stable plaque at an almost periodic point; a nested-continua contradiction yielding a nonrecurrent point; zero-dimensionality; symbolic coding; and, with positive recurrence, semisimplicity via Katznelson–Weiss. One minor overstatement is that Step 2 claims a stronger version of Mañé’s lemma (“for every x and every η>0”) than is used or stated in the paper (which asserts existence at some point/scale); however, the remainder of the argument only needs the weaker, standard existence statement and proceeds identically. Overall, both proofs align closely, and the paper’s steps are sound and complete for the stated results.