Oliver Jenkinson

Rotation, entropy, and equilibrium states

For a dynamical system (X, T) and function f : X Rd we consider the corresponding generalised rotation set. This is the convex subset of Rd consisting of all integrals of f with respect to T-invariant probability measures. We study the entropy H(e) of rotation vectors e, and relate this to the directional entropy lHl(L) of Geller & Misiurewicz. For (X, T) a mixing subshift of finite type, and f of summable variation, we prove that if the rotation set is strictly convex then the functions -H and H are in fact one and the same. For those rotation sets which are not strictly convex we prove that 1H((e) and H(e) can differ only at non-exposed boundary points e.

Frequency Locking on the Boundary of the Barycentre Set

We consider the doubling map T : z Z2 of the circle. For each T-invariant probability measure μ we define its barycentre b(μ) = ∫S1 Z dμ(z), which describes its average weight around the circle. We study the set Ω of all such barycentres, a compact convex set with nonempty interior. Its boundary ∂Ω has a countable dense set of points of nondifferentiability, the worst possible regularity for the boundary of a convex set. We explain this behaviour in terms of the frequency locking of rotation numbers for a certain class of invariant measures, each supported on the closure of a Sturmian orbit.

Characterisations of balanced words via orderings

Three new characterisations of balanced words are presented. Each of these characterisations is based on the ordering of a shift orbit, either lexicographically or with respect to the norm |ċ|1 (which counts the number of occurrences of the symbol 1).

Geometric barycentres of invariant measures for circle maps

For a continuous circle map T , define the barycentre of any T -invariant probability measure \mu to be b (\mu)=\int_{S^1} z \, d \mu( z ). The set \Omega of all such barycentres is a compact convex subset of \mathbb{C}. If T is conjugate to a rational rotation via a Mobius map, we prove \Omega is a disc. For every piecewise-onto expanding map we prove that the barycentre set has non-empty interior. In this case, each interior point is the barycentre of many invariant measures, but we prove that amongst these there is a unique one which maximizes entropy, and that this measure belongs to a distinguished two-parameter family of equilibrium states. This family induces a real-analytic radial foliation of int(\Omega), centred around the barycentre of the global measure of maximal entropy, where each ray is the barycentre locus of some one-parameter section of the family. We explicitly compute these rays for two examples. While developing this framework we also answer a conjecture of Z. Coelho [ 6 ] regarding limits of sequences of equilibrium states.

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim(EA) is between 0 and 1. It is shown that the set intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim(EA), and employ it to investigate numerically the way in which intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbanski, that is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim(E{1,2}), improving on the one given in [18].

Optimization and majorization of invariant measures

The set of ×2-invariant measures can be equipped with the partial order of majorization, describing relative dispersion. The minimal elements for this order are precisely the Sturmian measures of Morse and Hedlund. This yields new characterisations of Sturmian measures, and has applications to the ergodic optimization of convex functions.

On the Ruelle eigenvalue sequence

For certain real analytic data, we show that the eigenvalue sequence of the associated transfer operator L is insensitive to the holomorphic function space on which L acts. Explicit bounds on this eigenvalue sequence are established.

Ergodic optimization for noncompact dynamical systems

The purpose of this note is to initiate the study of ergodic optimization for general topological dynamical systems T:X→ X, where the topological space X need not be compact. Given , four possible notions of largest ergodic average are defined; for compact metrisable X these notions coincide, while for general Polish spaces X they are related by inequalities, each of which may be strict. We seek conditions on f which guarantee the existence of a normal form, in order to characterize its maximizing measures in terms of their support. For compact metrisable X it suffices to find a fixed point form. For general Polish X this is not the case, but an extra condition on f, essential compactness, is shown to imply the existence of a normal form. When T:X→ X is the full shift on a countable alphabet, essential compactness yields an easily checkable criterion for the existence of a normal form.

BALANCED WORDS AND MAJORIZATION

When expressed in terms of base-2 expanions, balanced words are majorized by other words of the same slope. Consequently they have smaller standard deviation and larger geometric mean than all words with given arithmetic mean (or slope), they can be expressed as a doubly stochastic average of any such word, and they can be derived from any such word by a finite number of transfers.

Lyapunov optimizing measures for C 1 expanding maps of the circle

For a generic C-1 expanding map of the circle, the Lyapunov maximizing measure is unique and fully supported, and has zero entropy.