Countable Sofic Shifts with a Periodic Direction

As a variant of the equal entropy cover problem, we ask whether all multidimensional sofic shifts with countably many configurations have SFT covers with countably many configurations. We answer this question in the negative by presenting explicit counterexamples. We formulate necessary conditions for a vertically periodic shift space to have a countable SFT cover, and prove that they are sufficient in a natural (but quite restricted) subclass of shift spaces.

Confined-walking mode for an elliptical polarized undulator and its application

Elliptical polarized undulators (EPUs) are broadly used in the soft X-ray energy range. They have the advantage of providing photons with both varied energy and polarization through adjustments to the value of the gap and/or shift magnet arrays in an undulator. Yet these adjustments may create a disturbance on the stability of the electron beam in a storage ring. To correct such a disturbance, it is necessary to establish a feed-forward table of key nodes in the gap-shift-defined two-dimensional parameter space. Such a table can only be scanned during machine-study time. For a free-walking mode, whereby an undulator is allowed to manoeuvre in the whole gap-shift space, all the key nodes need to be scanned at the expense of a large amount of machine-study time. This will greatly delay the employment of a full-polarization capable undulator (especially circularly polarized). By analyzing data-collecting patterns of user experiments, this paper defines a reduced set of key nodes in gap-shift parameter space, with the number of key nodes to be scanned for feed-forwarding scaled down to one-third of the original; and introduces a new walking mode for EPUs: confined-walking mode, whereby the undulator is manoeuvred only within the reduced set of key nodes. Such a mode is firstly realized on the EPUs at the DREAMLINE beamline at Shanghai Synchrotron Radiation Facility (SSRF). Under confined-walking mode, the undulator movements are stable and there is no obvious disturbance to the electron beam with the feed-forward system in operation. Successful experiments have been carried out using the circularly polarized light obtained via the new walking mode. This mode is expected to be applied to future EPUs at SSRF with the increasing requirements for various polarization modes.

Matrix characterization of multidimensional subshifts of finite type

<p>Let X ⊂ A^Zdbe a 2-dimensional subshift of finite type. We prove that any 2-dimensional subshift of finite type can be characterized by a square matrix of infinite dimension. We extend our result to a general d-dimensional case. We prove that the multidimensional shift space is non-empty if and only if the matrix obtained is of positive dimension. In the process, we give an alternative view of the necessary and sufficient conditions obtained for the non-emptiness of the multidimensional shift space. We also give sufficient conditions for the shift space X to exhibit periodic points.</p>

Topological entropy of Markov set-valued functions

We investigate the entropy for a class of upper semi-continuous set-valued functions, called Markov set-valued functions, that are a generalization of single-valued Markov interval functions. It is known that the entropy of a Markov interval function can be found by calculating the entropy of an associated shift of finite type. In this paper, we construct a similar shift of finite type for Markov set-valued functions and use this shift space to find upper and lower bounds on the entropy of the set-valued function.

Phase transitions in long-range Ising models and an optimal condition for factors of -measures

We weaken the assumption of summable variations in a paper by Verbitskiy [On factors of $g$-measures. Indag. Math. (N.S.)22 (2011), 315–329] to a weaker condition, Berbee’s condition, in order for a one-block factor (a single-site renormalization) of the full shift space on finitely many symbols to have a $g$-measure with a continuous $g$-function. But we also prove by means of a counterexample that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is a critical inverse temperature in a one-sided long-range Ising model which is at most eight times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.

TWO-SIDED SHIFT SPACES OVER INFINITE ALPHABETS

Ott, Tomforde and Willis proposed a useful compactification for one-sided shifts over infinite alphabets. Building from their idea, we develop a notion of two-sided shift spaces over infinite alphabets, with an eye towards generalizing a result of Kitchens. As with the one-sided shifts over infinite alphabets, our shift spaces are compact Hausdorff spaces but, in contrast to the one-sided setting, our shift map is continuous everywhere. We show that many of the classical results from symbolic dynamics are still true for our two-sided shift spaces. In particular, while for one-sided shifts the problem about whether or not any$M$-step shift is conjugate to an edge shift space is open, for two-sided shifts we can give a positive answer for this question.

Four-cycle free graphs, height functions, the pivot property and entropy minimality

Fix $d\geq 2$. Given a finite undirected graph ${\mathcal{H}}$ without self-loops and multiple edges, consider the corresponding ‘vertex’ shift, $\text{Hom}(\mathbb{Z}^{d},{\mathcal{H}})$, denoted by $X_{{\mathcal{H}}}$. In this paper, we focus on ${\mathcal{H}}$ which is ‘four-cycle free’. There are two main results of this paper. Firstly, that $X_{{\mathcal{H}}}$ has the pivot property, meaning that, for all distinct configurations $x,y\in X_{{\mathcal{H}}}$, which differ only at a finite number of sites, there is a sequence of configurations $x=x^{1},x^{2},\ldots ,x^{n}=y\in X_{{\mathcal{H}}}$ for which the successive configurations $x^{i},x^{i+1}$ differ exactly at a single site. Secondly, if ${\mathcal{H}}$ is connected ,then $X_{{\mathcal{H}}}$ is entropy minimal, meaning that every shift space strictly contained in $X_{{\mathcal{H}}}$ has strictly smaller entropy. The proofs of these seemingly disparate statements are related by the use of the ‘lifts’ of the configurations in $X_{{\mathcal{H}}}$ to the universal cover of ${\mathcal{H}}$ and the introduction of ‘height functions’ in this context.