Escape-rate monotonicity for prime words with respect to cylinder measure

Prove that for a subshift of finite type equipped with an arbitrary Markov measure μ_P, and for any two prime words u and v of the same length, if the cylinder measures satisfy μ_P(C_u) < μ_P(C_v), then the corresponding escape rates satisfy ρ(C_u) < ρ(C_v).

Background

The paper studies escape rates into holes in subshifts of finite type under general Markov measures, providing explicit formulas via spectral radii and generating functions. In simple settings, such as two-symbol shifts or product measures, certain patterns are known (e.g., equal escape rates for prime words with equal measures), but these patterns do not extend in general to larger alphabets or arbitrary Markov measures.

Motivated by counterexamples showing that previously observed relationships can fail, the authors propose a conjecture restricted to prime words. The conjecture posits a natural monotonicity: among prime words of equal length, larger cylinder measure should correspond to larger escape rate. This aims to identify a robust ordering principle for escape rates beyond special cases.