Topological Entropy of Nonlinear Time-Varying Systems

The paper defines entropy relative to an initial set K that is compact with nonempty interior, and proves the lower bound h(f,t0,K) ≥ max{χ̌,0} under this standing assumption and via a volume argument; the proof explicitly yields a time-averaged intermediate bound and then upgrades it to χ̌ using liminf properties. The candidate’s counterexample uses K={0}, which violates the paper’s assumption, so it does not refute the theorem. The candidate’s upper-bound argument aligns with the paper’s result. See the paper’s statement of the assumption on K and Theorem 1 with its proof steps on the lower bound and the volume bound used therein .