A piecewise linear homeomorphism of the circle which is periodic under renormalization

The paper proves that the PL circle homeomorphism f with three affine branches has rot(f)=√2−1 by introducing a renormalization f↦f* based on a first-return construction (in backward time), showing m_f=m_{f*}=2 and f**=f, and invoking rot(f)=1/(m_f+rot(f*)) so that x=1/(2+x) implies x=√2−1. Theorem 4 states the claim and the algorithm (Proposition 1) gives the key identity, with an explicit computation of f* confirming the 2-cycle under renormalization and m-values equal to 2 . The candidate solution uses a forward-time inducing scheme on L=[0,3/8), computes the first return map F and its rescaling, observes the same {2,3} return-time pattern at every level, and concludes that every continued-fraction partial quotient equals 2, hence rot(f)=[0;2,2,2,…]=√2−1. This is the same renormalization/continued-fraction mechanism as in the paper, differing only by using forward returns (vs the paper’s backward-return star operator). Minor presentational imprecision aside (e.g., the claim of “same combinatorial order” under rescaling), the core computations and logic align with the paper’s argument.