有限核穷竭与编码移位中严格序列压力逼近的失效
Finite Cores Exhaustion and the Failure of Strict Sequential-Pressure Approximation in Coded Shifts
AI总结：
本文证明编码移位中有限核压力收敛于序列压力，给出计算全局压力的算法，并构造反例反驳相关猜想。
AI中文摘要：
设 $X=X(\mathcal{G})$ 为一个编码移位，其生成集唯一地表示其拼接集。我们证明，对于每个连续势 $\varphi$，有限生成子移位（有限核）的压力收敛到序列压力 $P_{\mathrm{seq}}(\varphi,\mathcal{G})$，即不变测度在拼接集上赋予全质量时自由能的上确界。因此，完全序列压力是有限核恢复全局压力的确切条件。证明是对Burr、Das、Wolf和Yang的诱导论证的改进。给定生成元的任意枚举和语言的按长度排序的枚举，我们获得一个算法，该算法计算每个具有完全序列压力的可计算势的全局压力。平衡态的集合是递归紧的，并且唯一平衡态可由相同数据计算。对于零势，$h_{\mathrm{con}}\geq h_{\mathrm{res}}$ 因此蕴含拓扑熵和任何唯一最大熵测度的可计算性。我们还构造了全二元移位的一个唯一表示的呈现，其中零势具有完全序列压力，但不是满足 $P_{\mathrm{seq}}>P_{\mathrm{res}}$ 的势的一致极限，从而反驳了Burr、Das、Wolf和Yang的一个猜想。
英文摘要：
Let $X=X(\mathcal{G})$ be a coded shift whose generating set uniquely represents its concatenation set. We prove that, for every continuous potential $φ$, the pressures of the finite-generator subshifts (the finite cores) converge to the sequential pressure $P_{\mathrm{seq}}(φ,\mathcal{G})$, the supremum of free energy over invariant measures giving full mass to the concatenation set. Thus full sequential pressure is the exact condition for the finite cores to recover global pressure. The proof is a refinement of the inducing argument of Burr, Das, Wolf, and Yang. Given an arbitrary enumeration of the generators and a length-ordered enumeration of the language, we obtain an algorithm which computes the global pressure of every computable potential with full sequential pressure. The set of equilibrium states is recursively compact, and a unique equilibrium state is computable from the same data. For the zero potential, $h_{\mathrm{con}}\geq h_{\mathrm{res}}$ therefore implies computability of topological entropy and of any unique measure of maximal entropy. We also construct a uniquely represented presentation of the full binary shift for which the zero potential has full sequential pressure but is not a uniform limit of potentials satisfying $P_{\mathrm{seq}}>P_{\mathrm{res}}$, disproving a conjecture of Burr, Das, Wolf, and Yang.
