Finite Cores Exhaustion and the Failure of Strict Sequential-Pressure Approximation in Coded Shifts

RESUMO

Let $X=X(\mathcal{G})$ be a coded shift whose generating set uniquely represents its concatenation set. We prove that, for every continuous potential $\varphi$, the pressures of the finite-generator subshifts (the finite cores) converge to the sequential pressure $P_{\mathrm{seq}}(\varphi,\mathcal{G})$, 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.