Critical-exponent spectra and rank two inverse realization on biregular trees

Abstract

We study the critical-exponent, or equivalently entropy, spectrum arising from free type-preserving actions on the biregular tree $\mathcal T_{r+1,s+1}$. For an action with a nonempty finite quotient core, the critical exponent agrees with both the volume entropy of the universal cover of the core and the topological entropy of the associated non-backtracking edge shift. The unrestricted spectrum is $[0,\frac12\log(rs)]$, whereas the finitely generated spectrum is a countable dense subset of this interval obtained by taking logarithms of Hashimoto spectral radii. We stratify this finite-state spectrum by the circuit rank of the quotient core. At each fixed rank, finitely many typed kernels parametrize all values. Moreover, every positive entropy value has only finitely many kernel--length realizations, up to type-preserving isomorphism, and every positive accumulation point of a fixed-rank spectrum belongs to a lower-rank stratum. At rank two, the corresponding exponential rates admit a complete inverse classification in terms of three explicit polynomial families, together with a finite exact membership test for algebraic-integer inputs.