Establish an analytical theory of the arithmetic dependence of the dynamical exponent

Establish an analytical framework that connects the substitution hierarchy and associated arithmetic properties of generalized Fibonacci irrational modulations to the dynamical critical exponent z through spectral self-similarity, trace-map renormalization, or the multifractal properties of critical eigenstates.

Background

The central numerical finding is that generalized Fibonacci irrational modulations preserve the correlation-length exponent ν≈1 while continuously changing the dynamical exponent z. For the m=1 family, the authors observe an approximately linear relationship between z and the inflation factor associated with the generalized Fibonacci substitution matrix.

Although the numerical results indicate a connection between the dynamical exponent and the arithmetic or inflation structure of the irrational modulation, the mechanism producing this dependence is not derived analytically. The paper identifies the missing theory as one relating substitution hierarchies, spectral self-similarity, trace-map renormalization, or critical-state multifractality to the observed dynamical scaling.

References

Although the numerical evidence strongly supports such a connection, the precise theoretical relation between the dynamical exponent and the arithmetic properties of the irrational modulation remains an open problem. In particular, it would be highly desirable to establish an analytical framework connecting the substitution hierarchy, spectral self-similarity, trace-map renormalization, or multifractal properties of critical eigenstates to the observed arithmetic dependence of $z$.

— Arithmetic Tuning of Dynamical Critical Exponents in Quasiperiodic Localization Transitions  (2608.23990 - Yi et al., 25 Aug 2026) in Section 5, “Discussion and Summary”