ZERO-DIMENSIONAL AND SYMBOLIC EXTENSIONS OF TOPOLOGICAL FLOWS

The paper proves Theorem 2.1: every topological semi-flow admits a zero-dimensional principal extension, with the roof function chosen constant 1, via a two-step construction: (i) build a zero-dimensional principal extension for any suspension semi-flow using Downarowicz–Huczek and Abramov’s formula (Proposition 2.3), and (ii) show the time-1 suspension (X1,(φ1)1) is a principal extension of (X,Φ), then compose to obtain the desired zero-dimensional principal extension with roof 1 . The candidate solution follows the same backbone: reduce to φ1, apply Downarowicz–Huczek to get a zero-dimensional principal extension, suspend with roof 1, and define π([z,s])=φs(πZ(z)); principality is checked by identifying μ with ν×Leb and using that hψ1(ν×Leb)=hS(ν) together with the principal property of πZ. This is essentially the composition used in the paper, just presented in a single pass. Minor technicalities (e.g., using the affinity of μ↦hφ1(μ) and the fact φs-conjugacy preserves hφ1) are standard and align with the paper’s use of Abramov’s formula and entropy monotonicity, so both proofs are correct and substantially the same.