Interval Exchange Research Articles

Specular sets

We introduce specular sets. These are subsets of groups which form a natural generalization of free groups. These sets of words are an abstract generalization of the natural codings of interval exchanges and of linear involutions. We consider two important families of sets contained in specular sets: sets of return words and maximal bifix codes. For both families we prove several cardinality results as well as results concerning the subgroup generated by these sets.

Exchange of three intervals: Substitutions and palindromicity

We study purely morphic words coding symmetric non-degenerate three interval exchange transformation which are known to be palindromic, i.e., they contain infinitely many palindromes. We prove that such words are fixed by a conjugate to a morphism of class P, that is, a morphism such that each letter a is mapped to ppa where p and pa are both palindromes. We thus provide a new family of palindromic infinite words satisfying the conjecture of Hof, Knill and Simon. Given a morphism fixing such word, we give a formula to determine the parameters of the underlying three interval exchange and the intercept of the word.

Decoding Rauzy Induction: An Answer to Bufetov's General Question

Given a typical interval exchange transformation, we may naturally associate to it an infinite sequence of matrices through Rauzy induction. These matrices encode visitations of the induced interval exchange transformations within the original. In 2010, W. A. Veech showed that these matrices suffice to recover the original interval exchange transformation, unique up to topological conjugacy, answering a question of A. Bufetov. In this work, we show that interval exchange transformation may be recovered and is unique modulo conjugacy when we instead only know consecutive products of these matrices. This answers another question of A. Bufetov. We also extend this result to any inductive scheme that produces square visitation matrices.

Most interval exchanges have no roots

Let $T$ be an $m$-interval exchange transformation. By the rank of $T$ we mean the dimension of the $\mathbb{Q}$-vector space spanned by the lengths of the exchanged intervals. We prove that if $T$ is minimal and the rank of $T$ is greater than $1+\lfloor m/2 \rfloor$, then $T$ cannot be written as a power of another interval exchange. We also demonstrate that this estimate on the rank cannot be improved.In the case that $T$ is a minimal 3-interval exchange transformation, we prove a stronger result: $T$ cannot be written as a power of another interval exchange if and only if $T$ satisfies Keane's infinite distinct orbit condition. In the course of proving this result, we give a classification (up to conjugacy) of those minimal interval exchange transformations whose discontinuities all belong to a single orbit.

Minimality of interval exchange transformations with restrictions

It is known since 40 years old paper by M. Keane that minimality is a generic (i.e. holding with probability one) property of an irreducible interval exchange transformation. If one puts some integral linear restrictions on the parameters of the interval exchange transformation, then minimality may become an "exotic" property. We conjecture in this paper that this occurs if and only if the linear restrictions contain a Lagrangian subspace of the first homology of the suspension surface. We partially prove it in the "only if" direction and provide a series of examples to support the converse one. We show that the unique ergodicity remains a generic property if the restrictions on the parameters do not contain a Lagrangian subspace (this result is due to Barak Weiss).

Cutting and Shuffling of a Line Segment: Effect of Variation in Cut Location

We present a computational study of the impact of variation in cut location on finite-time mixing of a line segment by cutting and shuffling, which is a one-dimensional piecewise isometry (PWI), also known as an interval exchange transformation (IET). A line segment of unit length is repeatedly cut into subsegments and shuffled according to any one of a variety of permutations. To mimic practical process error, variations drawn from a normal distribution are used to perturb cut locations. Illustrative examples of the mixing behaviors and finite-time measures of mixing are used to analyze the effect of variation in cut location for different permutations of subsegment mixing order. Mixing is significantly improved under irreducible nonrotational permutations when the dynamics show a resonance-like structure without variation. Specifically, the requirement of an irrational subsegment length ratio for good mixing can be relaxed as the underlying periodic dynamics is perturbed by the stochastic variation in cut location. Thus, good mixing can occur even with only four subsegments of roughly the same length for most irreducible nonrotational permutations.

Extensive amenability and an application to interval exchanges

Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group$\text{IET}$of interval exchange transformations that have angular components of rational rank less than or equal to two. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on$\text{IET}$and show that there are subgroups$G<\text{IET}$admitting no finitely supported measure with trivial boundary.

Renormalizable two-parameter piecewise isometries.

We exhibit two distinct renormalization scenarios for two-parameter piecewise isometries, based on 2π/5 rotations of a rhombus and parameter-dependent translations. Both scenarios rely on the recently established renormalizability of a one-parameter triangle map, which takes place if and only if the parameter belongs to the algebraic number field K=Q(5) associated with the rotation matrix. With two parameters, features emerge which have no counterpart in the single-parameter model. In the first scenario, we show that renormalizability is no longer rigid: whereas one of the two parameters is restricted to K, the second parameter can vary continuously over a real interval without destroying self-similarity. The mechanism involves neighbouring atoms which recombine after traversing distinct return paths. We show that this phenomenon also occurs in the simpler context of Rauzy-Veech renormalization of interval exchange transformations, here regarded as parametric piecewise isometries on a real interval. We explore this analogy in some detail. In the second scenario, which involves two-parameter deformations of a three-parameter rhombus map, we exhibit a weak form of rigidity. The phase space splits into several (non-convex) invariant components, on each of which the renormalization still has a free parameter. However, the foliations of the different components are transversal in parameter space; as a result, simultaneous self-similarity of the component maps requires that both of the original parameters belong to the field K.

Invariant measures for piecewise continuous maps

We say that f:[0,1]→[0,1] is a piecewise continuous interval map if there exists a partition 0=x0<x1<⋯<xd<xd+1=1 of [0,1] such that f|(xi−1,xi) is continuous and the lateral limits w0+=limx→0+⁡f(x), wd+1−=limx→1−⁡f(x), wi−=limx→xi−⁡f(x) and wi+=limx→xi+⁡f(x) exist for each i. We prove that every piecewise continuous interval map without connections admits an invariant Borel probability measure. We also prove that every injective piecewise continuous interval map with no connections and no periodic orbits is topologically semiconjugate to an interval exchange transformation.

Infinite interval exchange transformations from shifts

We show that minimal shifts with zero topological entropy are topologically conjugate to interval exchange transformations, which are generally infinite. When these shifts have linear factor complexity (linear block growth), the conjugate interval exchanges are proved to satisfy strong finiteness properties.

Hölder Regularity of the Solutions of the Cohomological Equation for Roth Type Interval Exchange Maps

We prove that the solutions of the cohomological equation for Roth type interval exchange maps are Holder continuous provided that the datum is of class \({C^r}\) with \({r > 1}\) and belongs to a finite-codimension linear subspace.

Renormalization of one-parameter families of piecewise isometries

ABSTRACTWe consider two one-parameter families of piecewise isometries of a rhombus. The rotational component is fixed, and its coefficients belong to a quadratic number field . The translations depend on a parameter s which is allowed to vary in an interval. We investigate renormalizability, and show that recursive constructions of first-return maps on a suitable subdomain eventually produce a scaled-down replica of this domain, but with a renormalized parameter r(s). We treat two quadratic fields: d = 5, 2. In the first case, the renormalization map r is of Lüroth type (a piecewise-affine version of Gauss’ map), whereas in the second case, it is the second iterate of a map f of this type. We show that exact self-similarity corresponds to the eventually periodic points of r (resp. f), and that such parameter values are precisely the elements of the quadratic field that lie in the given interval. The renormalizability proof for is based on a straightforward application of return-map analysis. The octagonal case is far more challenging. The proof is organized by a graph analogous to those used to construct renormalizable interval-exchange transformations. There are 10 distinct renormalization scenarios corresponding to as many closed circuits in the graph. The process of induction along some of these circuits involves intermediate maps undergoing, as the parameter varies, infinitely many bifurcations. Our proofs rely on computer assistance.

On the mixing properties of piecewise expanding maps under composition with permutations, II: Maps of non-constant orientation

For an integer [Formula: see text], let [Formula: see text] be the partition of the unit interval [Formula: see text] into [Formula: see text] equal subintervals, and let [Formula: see text] be the class of piecewise linear maps on [Formula: see text] with constant slope [Formula: see text] on each element of [Formula: see text]. We investigate the effect on mixing properties when [Formula: see text] is composed with the interval exchange map given by a permutation [Formula: see text] interchanging the [Formula: see text] subintervals of [Formula: see text]. This extends the work in a previous paper [N. P. Byott, M. Holland and Y. Zhang, DCDS 33 (2013) 3365–3390], where we considered only the “stretch-and-fold” map [Formula: see text].

Diffusion for chaotic plane sections of 3-periodic surfaces

We study chaotic plane sections of some particular family of triply periodic surfaces. The question about possible behavior of such sections was posed by S. P. Novikov. We prove some estimations on the diffusion rate of these sections using the connection between Novikov's problem and systems of isometries - some natural generalization of interval exchange transformations. Using thermodynamical formalism, we construct an invariant measure for systems of isometries of a special class called the Rauzy gasket, and investigate the main properties of the Lyapunov spectrum of the corresponding suspension flow.

Subgroup of interval exchanges generated by torsion elements and rotations

Denote by G the group of interval exchange transformations (IETs) on the unit interval. Let Gper ⊂ G be the subgroup generated by torsion elements in G (periodic IETs), and let Grot ⊂ G be the subset of 2-IETs (rotations). The elements of the subgroup G1 = hGper,Groti ⊂ G (generated by the sets Gper and Grot) are characterized constructively in terms of their Sah-Arnoux-Fathi (SAF) invariant. The characterization implies that a non-rotation type 3-IET lies in G1 if and only if the lengths of its exchanged intervals are linearly dependent over Q. In particular, G1 ( G. The main tools used in the paper are the SAF invariant and a recent result by Y. Vorobets that Gper coincides with the commutator subgroup of G. 1. A group of IETs Denote by R, Q, N the sets of real, rational and natural numbers. By a standard interval we mean a finite interval of the form X = (a,b) ⊂ R (left closed - right open). We write |X| = b − a for its length. By an IET (interval exchange transformation) we mean a pair (X,f) where X = (a,b) is a standard interval and f is a right continuous bijection f : X → X with a finite set D of discontinuities and such that the translation function γ(x) = f(x)−x is piecewise constant. The map f itself is often referred to as IET, and then X = domain(f) and D = disc(f) denote the domain (also the range) of f and the discontinuity set of f, respectively. Given an IET f : X → X, the set disc(f) partitions X into a finite number of subintervals Xk in such a way that f restricted to each Xk is a translation

On the dual of Rauzy induction

We investigate a certain dual relationship between piecewise rotations of a circle and interval exchange maps. In 2005, Cruz and da Rocha [A generalization of the Gauss map and some classical theorems on continued fractions. Nonlinearity18 (2005), 505–525] introduced a notion of ‘castles’ arising from piecewise rotations of a circle. We extend their idea and introduce a continuum version of castles, which we show to be equivalent to Veech’s zippered rectangles [Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), 201–242]. We show that a fairly natural map defined on castles represents the inverse of the natural extension of the Rauzy map.

Neutral and tree sets of arbitrary characteristic

We study classes of minimal sets defined by restrictions on the possible extensions of the words. These sets generalize the previously studied classes of neutral and tree sets by relaxing the condition imposed on the empty word and measured by an integer called the characteristic of the set. We present several enumeration results holding in these sets of words. These formulae concern return words and bifix codes. They generalize formulae previously known for Sturmian sets or more generally for tree sets. We also give two geometric examples of this class of sets, namely the natural coding of some interval exchange transformations and the natural coding of some linear involutions.

On Cohomological Equations for Suspension Flows over Vershik Automorphisms

In this paper we give sufficient conditions for existence of a solution of cohomological equation for suspension flows over automorphisms of Markov compacta, which were introduced by Vershik and Ito. The main result (Theorem 1) can be regarded as a symbolic analogue of results due to Forni and Marmi, Moussa and Yoccoz for translation flows and interval exchange transformations.

Geometry of Refractions and Reflections Through a Biperiodic Medium

The behavior of light rays obeying Snell's law in a medium made up of two materials with different refractive indices and which are arranged in a periodic chessboard pattern is described. The analysis is in some ways analogous to the study of rational billiards and uses a return map on one surface to prove, among other things, that the number of angles with which any individual ray intersects the lattice is bounded and that if the ratio of refractive indices is large enough, then the dynamics can be described by interval exchange maps.

Whirly 3-interval exchange transformations

Irreducible interval exchange transformations are studied with regard to the whirly property, a condition for a non-trivial spatial factor. A uniformly whirly transformation is defined and is further studied. An equivalent condition is introduced for the whirly transformation. We will prove that almost all 3-interval exchange transformations are whirly, using a combinatorics approach with application of the Rauzy–Veech induction. It is still an open question whether the whirly property is a generic property for $m$-interval exchange transformations for $m\geq 4$.