AI 中文总结
研究双正则树上自由保型离散作用，通过商复杂度分层临界指数谱，无限制谱为$[0,\frac{1}{2}\log(rs)]$，有限生成谱可数稠密，秩二时可通过特定多项式族进行完全有效逆分类。
AI 中文摘要
对于双正则树$\mathcal T_{r + 1,s + 1}$上的自由保型离散作用，我们通过商复杂度对临界指数谱进行分层。无限制谱是整个区间$[0,\frac{1}{2}\log(rs)]$，而有限生成谱是可数且稠密的，由有限型核的桥本半径编码。在固定秩时，有限多个型核参数化所有值，且每个非零聚点属于更低秩的层。秩为二时，通过八字形、θ和哑铃多项式族可进行完全有效的逆分类。
英文摘要
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.
Comments29 pages, 4 figures
