Geometric representation of interval exchange maps over algebraic number fields

We consider the restriction of interval exchange transformations to algebraic number fields, which leads to maps on lattices. We characterize renormalizability arithmetically, and study its relationships with a geometrical quantity that we call the drift vector. We exhibit some examples of renormalizable interval exchange maps with zero and non-zero drift vector, and carry out some investigations of their properties. In particular, we look for evidence of the finite decomposition property: each lattice is the union of finitely many orbits.Comment: 34 pages, 8 postscript figure

Geometrical Models for Substitutions

International audienceWe consider a substitution associated with the Arnoux-Yoccoz interval exchange transformation (IET) related to the tribonacci substitution. We construct the so-called stepped lines associated with the fixed points of the substitution in the abelianization (symbolic) space. We analyze various projections of the stepped line, recovering the Rauzy fractal, a Peano curve related to work in [Arnoux 88], another Peano curve related to the work of [McMullen 09] and [Lowenstein et al. 07], and also the interval exchange transformation itself

Cross sections for geodesic flows and \alpha-continued fractions

We adjust Arnoux's coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the \alpha-continued fractions, for each $\alpha$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and their recently introduced \alpha-variants.Comment: 20 pages, 2 figure

An algorithm to identify automorphisms which arise from self-induced interval exchange transformations

We give an algorithm to determine if the dynamical system generated by a positive automorphism of the free group can also be generated by a self-induced interval exchange transformation. The algorithm effectively yields the interval exchange transformation in case of success.Comment: 26 pages, 8 figures. v2: the article has been reorganized to make for a more linear read. A few paragraphs have been added for clarit

Optical Coatings as Mirrors for Optical Diagnostics

The aim of this work was to provide a comprehensive insight concerning coated films which might be used for first mirrors in ITER. The influence of the mirror crystallite size has been addressed as well as the coating techniques to provide nanocrystalline films. Tests of coated mirrors both in laboratories and in tokamaks are reviewed. For the tokamak tests a wide angle camera system has been installed in JET-ILW which is composed of a mirror box with 3 stainless steel mirrors coated with rhodium viewing the torus through a conically shaped aperture. The system delivered the required image quality for plasma monitoring and wall protection. No or insignificant degradation of the optical transmittance has been observed during the experimental campaign in 2014 with about 3000 plasma pulses in different magnetic field configurations

Multi-machine scaling of the main SOL parallel heat flux width in tokamak limiter plasmas

As in many of today’s tokamaks, plasma start-up in ITER will be performed in limiter configuration on either the inner or outer midplane first wall (FW). The massive, beryllium armored ITER FW panels are toroidally shaped to protect panel-to-panel misalignments, increasing the deposited power flux density compared with a purely cylindrical surface. The chosen shaping should thus be optimized for a given radial profile of parallel heat flux, q in the scrape-off layer (SOL) to ensure optimal power spreading. For plasmas limited on the outer wall in tokamaks, this profile is commonly observed to decay exponentially as q q = − exp ( / r λ ) 0 q omp , or, for inner wall limiter plasmas with the double exponential decay comprising a sharp near-SOL feature and a broader main SOL width, λq omp. The initial choice of λq omp , which is critical in ensuring that current ramp-up or down will be possible as planned in the ITER scenario design, was made on the basis of an extremely restricted L-mode divertor dataset, using infra-red thermography measurements on the outer divertor target to extrapolate to a heat flux width at the main plasma midplane. This unsatisfactory situation has now been significantly improved by a dedicated multi-machine ohmic and L-mode limiter plasma study, conducted under the auspices of the International Tokamak Physics Activity, involving 11 tokamaks covering a wide parameter range with R = = 0.4–2.8 m, 1 B I 0 p .2–7.5 T, = 9–2500 kA. Measurements of λq omp in the database are made exclusively on all devices using a variety of fast reciprocating Langmuir probes entering the plasma at a variety of poloidal locations, but with the majority being on the low field side. Statistical analysis of the database reveals nine reasonable engineering and dimensionless scalings. All yield, however, similar predicted values of λq omp mapped to the outside midplane. The engineering scaling with the highest statistical significance, λ = ( / ( )) ( / /κ) − − q 10 P V W m a R omp tot 3 0.38 1.3 , dependent on input power density, aspect ratio and elongation, yields λq omp = [7, 4, 5] cm for Ip = [2.5, 5.0, 7.5] MA, the three reference limiter plasma currents specified in the ITER heat and nuclear load specifications. Mapped to the inboard midplane, the worst case (7.5 MA) corresponds to λq ~ 57 1 ± 4 imp mm, thus consolidating the 50mm width used to optimize the FW panel toroidal shape.EURATOM 633053Czech Science Foundation GA CR P205/12/2327, GA15-10723S, MSMT LM2011021US Department of Energy DE-FG02- 07ER54917, DE-AC02-09CH11466, DE-FC02-04ER5469

Escape orbits and Ergodicity in Infinite Step Billiards

In a previous paper we defined a class of non-compact polygonal billiards, the infinite step billiards: to a given decreasing sequence of non-negative numbers $\{p_{n}$, there corresponds a table \Bi := \bigcup_{n\in\N} [n,n+1] \times [0,p_{n}]. In this article, first we generalize the main result of the previous paper to a wider class of examples. That is, a.s. there is a unique escape orbit which belongs to the alpha and omega-limit of every other trajectory. Then, following a recent work of Troubetzkoy, we prove that generically these systems are ergodic for almost all initial velocities, and the entropy with respect to a wide class of ergodic measures is zero.Comment: 27 pages, 8 figure