ON TOPOLOGICAL ENTROPY OF PIECEWISE SMOOTH VECTOR FIELDS

The candidate solution reproduces the paper’s construction: define the trajectory space and time-one map, introduce the next-escape time τ and induced map P on a set of concatenated escape–return arcs, prove τ and P are continuous, code by returns to obtain a semiconjugacy onto a full shift (hence infinite entropy for P), and then pass this entropy to the flow’s time-one map via a commuting diagram or a constant rescaling lemma. This matches Theorem 6.2 and Lemma 6.1 in the paper, along with the continuity/coding results (Propositions 6.1–6.3) and the factor/iterate entropy facts (Propositions 3.1 and 3.3) . The model’s only substantive gap is an unnecessary compactness claim for the trajectory subset to bound τ away from zero; the paper avoids this by either restricting to a region with uniform bounds or appealing to the rescaling lemma.