On subshifts with slow forbidden word growth

In this work, we treat subshifts, defined in terms of an alphabet $\mathcal {A}$ and (usually infinite) forbidden list $\mathcal {F}$ , where the number of n-letter words in $\mathcal {F}$ has ‘slow growth rate’ in n. We show that such subshifts are well behaved in several ways; for instance, they are boundedly supermultiplicative in the sense of Baker and Ghenciu [Dynamical properties of S-gap shifts and other shift spaces. J. Math. Anal. Appl.430(2) (2015), 633–647] and they have unique measures of maximal entropy with the K-property and which satisfy Gibbs bounds on large (measure-theoretically) sets. The main tool in our proofs is a more general result, which states that bounded supermultiplicativity and a sort of measure-theoretic specification property together imply uniqueness of the measure of maximum entropy and our Gibbs bounds. We also show that some well-known classes of subshifts can be treated by our results, including the symbolic codings of $x \mapsto \alpha + \beta x$ (the so-called $\alpha $ - $\beta $ shifts of Hofbauer [Maximal measures for simple piecewise monotonic transformations. Z. Wahrsch. verw. Geb.52(3) (1980), 289–300]) and the bounded density subshifts of Stanley [Bounded density shifts. Ergod. Th. & Dynam. Sys.33(6) (2013), 1891–1928].

Entropy and rotation sets: A toy model approach

Given a continuous dynamical system [Formula: see text] on a compact metric space [Formula: see text] and a continuous potential [Formula: see text], the generalized rotation set is the subset of [Formula: see text] consisting of all integrals of [Formula: see text] with respect to all invariant probability measures. The localized entropy at a point in the rotation set is defined as the supremum of the measure-theoretic entropies over all invariant measures whose integrals produce that point. In this paper, we provide an introduction to the theory of rotation sets and localized entropies. Moreover, we consider a shift map and construct a Lipschitz continuous potential, for which we are able to explicitly compute the geometric shape of the rotation set and its boundary measures. We show that at a particular exposed point on the boundary there are exactly two ergodic localized measures of maximal entropy.

DEGENERATIONS OF COMPLEX DYNAMICAL SYSTEMS

We show that the weak limit of the maximal measures for any degenerating sequence of rational maps on the Riemann sphere ${\hat{{\mathbb{C}}}} $ must be a countable sum of atoms. For a one-parameter family $f_t$ of rational maps, we refine this result by showing that the measures of maximal entropy have a unique limit on $\hat{{\mathbb{C}}}$ as the family degenerates. The family $f_t$ may be viewed as a single rational function on the Berkovich projective line $\mathbf{P}^1_{\mathbb{L}}$ over the completion of the field of formal Puiseux series in $t$, and the limiting measure on $\hat{{\mathbb{C}}}$ is the ‘residual measure’ associated with the equilibrium measure on $\mathbf{P}^1_{\mathbb{L}}$. For the proof, we introduce a new technique for quantizing measures on the Berkovich projective line and demonstrate the uniqueness of solutions to a quantized version of the pullback formula for the equilibrium measure on $\mathbf{P}^1_{\mathbb{L}}$.