Intrinsic ergodicity beyond specification: \({\beta}\)-shifts, \(S\)-gap shifts, and their factors (Q1936816)

A novel form of specification well adapted to studying shift dynamical systems is introduced here, and used to extend our understanding of the prevalance of intrinsic ergodicity (the property of having a single measure of maximal entropy; it is automatic for shift systems that there will be at least one measure of maximal entropy) in several ways. For both \(\beta\)-shifts and \(S\)-gap shifts it is shown that every subshift factor is intrinsically ergodic. In addition it is shown that the measure of maximal entropy is obtained as a weak* limit of uniform atomic measures supported on the set of periodic points of period at most \(n\) as \(n\to\infty\). For coded systems (which subsume both \(\beta\)-shifts and \(S\)-gap shifts) a combinatorial condition is found that guarantees intrinsic ergodicity.