bibkey: bowenlanford1970shift authors: Rufus Bowen and Oscar E. Lanford III year: 1970 title: Zeta functions of restrictions of the shift transformation doi: null url: https://people.math.harvard.edu/~knill/history/lanford/papers/BowenLanford.pdf claim: Finite forbidden-word shifts have periodic-point counts given by traces of a finite transition matrix and dynamical zeta equal to its inverse characteristic determinant. strata_touched: [] license: citation-only triage: anchor

Zeta functions of restrictions of the shift transformation

The primary scan, printed pp.43–45, defines the two-sided shift, its periodic-point zeta and the transition matrix for a finite forbidden-word set. Section 2, Lemmas 1–2 and Theorem 1, gives

$$N_m=\mathrm{tr}(T^m),\exp \left(\sum _{m\ge 1}\frac{N_m{z}^m}{m}\right)=\prod _j(1-{\lambda }_{j}z{)}^{-1}=\det (I-zT{)}^{-1},$$

where eigenvalues are counted with multiplicity. The local power series has its own convergence domain; rational continuation outside that domain is not a convergent sum of periodic configurations.

The example on printed pp.44–45 forbids 00 and uses $T=\left(\begin{array}{cc}0& 1\\ 1& 1\end{array}\right)$. Exchanging the two symbols gives the forbidden-11 matrix $Q=\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)$. Both have characteristic polynomial $X^2-X-1$ and $\det (I-zT)=1-z-z^2$. The scan’s final example denominator on p.45 prints $1-z+z^2$, which conflicts with its displayed matrix and eigenvalues. The general theorem and the determinant calculation supply the minus sign; the conflicting printed example formula is not used.

The result is consumed inside the labelled-seam to periodic-point bridge in §5 of the seams continuation. The matrix-periodic configurations belong to a specified transition graph, not automatically to the original ordered substitution tree. The paper does not supply a physical cycle, a Riemann zeta identification or RH.