TOPOLOGICAL ENTROPY OF BUNIMOVICH STADIUM BILLIARDS

The paper proves lim h(F_ℓ|K_ℓ) = log(1+√2) for Bunimovich stadium billiards by (i) constructing finite-type symbolic models Σ̃_N on K_{ℓ,N} for ℓ>2N+2 and computing their entropy via the rome method, yielding a uniform lower bound whose limit is log(1+√2), and (ii) using a 3-state SFT Σ̃ to bound the symbolic entropy from above, giving equality when entropy is defined via the coding; it gives a liminf ≥ log(1+√2) under other definitions . The model reproduces the same lower bound and limit via the same finite-state cutoffs and the same characteristic equation, and then supplies a clean symbolic upper bound using a countable-state shift Σ̃_∞ computed by first-return loops, also yielding log(1+√2). The main mathematical steps agree; the proofs differ in the upper bound. The only caveat is definitional: the model’s upper bound implicitly uses the symbolic/cylinder definition of entropy, mirroring the paper’s assumption in its equality result.