Intrinsic ergodicity beyond specification: β-shifts, S-gap shifts, and their factors

Abstract

Abstract. We give sufficient conditions for a shift space (Σ, σ) to be intrinsically ergodic, along with sufficient conditions for every subshift factor of Σ to be intrinsically ergodic. As an application, we show that every subshift factor of a β-shift is intrinsically ergodic, which answers an open question included in Mike Boyle's article "Open problems in symbolic dynamics". We obtain the same result for S-gap shifts, and describe an application of our conditions to more general coded systems. One novelty of our approach is the introduction of a new version of the specification property that is well adapted to the study of symbolic spaces with a non-uniform structure.

Equilibrium states beyond specification and the Bowen property

“…Remark 2.2. When ϕ = 0, the conditions in the main theorem of [7] imply the conditions above except with (W)-specification in place of (S)-specification in (I). The theorem holds true if we assume (W)-specification in (I) but it leads to some additional technicalities in the proof.…”

Bounded density shifts

“…(1) ⇒ (2). Assume that the characteristic sequence does not contain arbitrarily long subwords consisting entirely of 0s.…”

“…Suppose, for a sofic shift f , that we have determined values of p, N and k as in Theorem 6.3 (2). Is it always true that σ N ( − → u ) is periodic with period p for any u ∈ L( f )?…”

“…The alphabet of any β-shift is [0.. β − 1 ]. Further information on β-shifts can be found in [1,2,5,7].…”

The beta-transformation's companion map for Pisot or Salem numbers and their periodic orbits

“…The symbols of the β-expansion of x ∈ [0, 1] are related to iteration of x under the β-transformation T β (x) := βx (mod 1). The dynamics of the β-transformation is encoded as a one sided shift on a space of symbolic sequences (for further details, please refer to [2], [3] or [4]). …”