Explicit Dynamical Systems on the Sierpiński Curve

The paper constructs a measure on the inverse limit S∞ so that (S∞,H∞,μ) is measure-theoretically isomorphic to the quotient sphere system (S0,H0,ν), by projecting away the blown-up circles, and then invokes standard factor/isomorphism entropy results (Prop. 2.4.1–2.4.2) to conclude h(H∞)=h(H0) and hence to transfer the base entropies. For (T2,FA) and (Xg,Tλ), the finite number of branch points allows immediate equality; for (Cn,Bn), the paper proves equality via a factor from (Σn,σn) and Ledrappier–Walters to handle the fiber term and then shows every r>0 is realized (Prop. 4.0.1) . The candidate’s solution uses the same backbone: almost-everywhere bijectivity across blow-ups/inverse limit to get measure isomorphism, and finite-to-one quotient to preserve entropy; it computes base entropies by standard results and realizes all r>0 via Bernoulli weights. Differences are primarily in citations and presentation (Abramov–Rokhlin/Pesin vs. Ledrappier–Walters), but the logical steps match the paper’s argument and yield the same conclusions, namely Theorem C .