Daniel J. Thompson

Irregular sets, the -transformation and the almost specification property

Let (X, d) be a compact metric space, f : X → X be a continuous map satisfying a property we call almost specification (which is slightly weaker than the g-almost product property of Pfister and Sullivan), and φ : X → R be a continuous function. We show that the set of points for which the Birkhoff average of φ does not exist (which we call the irregular set) is either empty or has full topological entropy. Every β-shift satisfies almost specification and we show that the irregular set for any β-shift or β-transformation is either empty or has full topological entropy and Hausdorff dimension.

The irregular set for maps with the specification property has full topological pressure

Let (X, d) be a compact metric space, f : X ↦ X be a continuous map with the specification property and ϕ : X ↦ ℝ a continuous function. We consider the set of points for which the Birkhoff average of ϕ does not exist (which we call the irregular set for ϕ) and show that this set is either empty or carries full topological pressure (in the sense of Pesin and Pitskel see, for example [Y.B. Pesin, Dimension Theory in Dimensional Systems: Contemporary Views and Applications, University of Chicago Press, Chicago, 1997]). We formulate various equivalent natural conditions on ϕ that completely describe when the latter situation holds and give examples of interesting systems to which our results apply but were not previously known. As an application, we show that for a suspension flow over a continuous map with specification, the irregular set carries full topological entropy.

A variational principle for topological pressure for certain non-compact sets

Let

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

(X,d)

Equilibrium states beyond specification and the Bowen property

be a compact metric space,

Unique equilibrium states for flows and homeomorphisms with non-uniform structure

f:X \mapsto X

Intrinsic ergodicity via obstruction entropies

be a continuous map with the specification property, and

A thermodynamic definition of topological pressure for non-compact sets

\varphi: X \mapsto \IR

Large deviations for systems with non-uniform structure

be a continuous function. We prove a variational principle for topological pressure (in the sense of Pesin and Pitskel) for non-compact sets of the form \[ \{x \in X : \lim_{n \ra \infty} \frac{1}{n} \sum_{i = 0}^{n-1} \varphi (f^i (x)) = \alpha \}. \] Analogous results were previously known for topological entropy. As an application, we prove multifractal analysis results for the entropy spectrum of a suspension flow over a continuous map with specification and the dimension spectrum of certain non-uniformly expanding interval maps.

Unique equilibrium states for Bonatti–Viana diffeomorphisms

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.