Interval Exchange Research Articles

Non-periodic not everywhere dense trajectories in triangular billiards

Building on tools that have been successfully used in the study of rational billiards, such as induced maps and interval exchange transformations, we provide a construction of a one-parameter family of isosceles triangles exhibiting non-periodic trajectories that are not everywhere dense. This provides, by elementary means, a definitive negative answer to a long-standing open question on the density of non-periodic trajectories in triangular billiards.

Metric ultraproducts of groups — Simplicity, perfectness and torsion

We characterise the simplicity of metric ultraproducts of a family of metric groups. We also present several new examples of simple groups, such as metric ultraproducts of finite and infinite symmetric groups, linear groups, and interval exchange transformation groups. Using similar methods, we also examine concepts such as genericity, perfectness, and torsion.

Renormalization of translated cone exchange transformations

In this paper, we investigate a class of non-invertible piecewise isometries on the upper half-plane known as Translated Cone Exchanges. These maps include a simple interval exchange on a boundary we call the baseline. We provide a geometric construction for the first return map to a neighbourhood of the vertex of the middle cone for a large class of parameters, then we show a recurrence in the first return map tied to Diophantine properties of the parameters, and subsequently prove the infinite renormalizability of the first return map for these parameters.

Affine interval exchange maps with a singular conjugacy to an IET

Affine interval exchange maps with a singular conjugacy to an IET

Flow views and infinite interval exchange transformations for recognizable substitutions

A flow view is the graph of a measurable conjugacy Φ between a substitution or S-adic subshift (Σ,σ,μ) and an exchange of infinitely many intervals in ([0,1],F,m), where m is Lebesgue measure. A natural refining sequence of partitions of Σ is transferred to ([0,1],m) using a canonical addressing scheme, a fixed dual substitution S∗, and a shift-invariant probability measure μ. On the flow view, τ∈Σ is shown horizontally at a height of Φ(τ) using colored unit intervals to represent the letters.The infinite interval exchange transformation F is well approximated by exchanges of finitely many intervals, making numeric and graphic methods possible. We prove that in certain cases a choice of dual substitution guarantees that Φ is self-similar. We discuss why the spectral type of Φ∈L2(Σ,μ), is of particular interest. As an example of utility, some spectral results for constant-length substitutions are included.

Interval Exchange Transformations with Vanishing Sah–Arnoux–Fathi Invariant

Interval Exchange Transformations with Vanishing Sah–Arnoux–Fathi Invariant

Languages of general interval exchange transformations

Languages of general interval exchange transformations

Minimality and unique ergodicity of Veech 1969 type interval exchange transformations

Minimality and unique ergodicity of Veech 1969 type interval exchange transformations

Solving the cohomological equation for locally Hamiltonian flows, part I - local obstructions

We study the cohomological equation Xu=f for smooth locally Hamiltonian flows on compact surfaces. The main novelty of the proposed approach is that it is used to study the regularity of the solution u when the flow has saddle loops, which has not been systematically studied before. Then we need to limit the flow to its minimal components. We show the existence and (optimal) regularity of solutions regarding the relations with the associated cohomological equations for interval exchange transformations (IETs). Our main theorems state that the regularity of solutions depends not only on the vanishing of the so-called Forni's distributions (cf. [9,11]), but also on the vanishing of families of new invariant distributions (local obstructions) reflecting the behavior of f around the saddles. Our main results provide some key ingredient for the complete solution to the regularity problem of solutions (in cohomological equations) for a.a. locally Hamiltonian flows (with or without saddle loops) to be shown in [15].The main contribution of this article is to define the aforementioned new families of invariant distributions dσ,jk, Cσ,lk and analyze their effect on the regularity of u and on the regularity of the associated cohomological equations for IETs. To prove this new phenomenon, we further develop local analysis of f near degenerate singularities inspired by tools from [14] and [17]. We develop new tools of handling functions whose higher derivatives have polynomial singularities over IETs.

On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms

Abstract We show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g. piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.

Interval Exchange Transformations Groups: Free Actions and Dynamics of Virtually Abelian Groups

Interval Exchange Transformations Groups: Free Actions and Dynamics of Virtually Abelian Groups

New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows

Abstract We consider smooth locally Hamiltonian flows on compact surfaces of genus g ≥ 1 {g\geq 1} to prove a deviation formula of Birkhoff integrals for smooth observables. Our work generalizes results of [G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 2002, 1, 1–103] and [A. I. Bufetov, Limit theorems for translation flows, Ann. of Math. (2) 179 2014, 2, 431–499] (and a recent result in [K. Frączek and C. Ulcigrai, On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions, preprint 2021]) which prove the existence of the deviation spectrum of Birkhoff integrals for observables whose jets vanish at sufficiently high order around fixed points of the flow. They showed that ergodic integrals display a power deviation spectrum with exactly g positive exponents related to the positive Lyapunov exponents of the cocycle so-called Kontsevich–Zorich, a renormalization cocycle over the Teichmüller flow on a stratum of the moduli space of translation surfaces. Our paper extends the study of the deviation spectrum of ergodic integrals beyond the case of observables whose jets vanish at sufficiently high order around fixed points. We prove the existence of some extra terms in the deviation spectrum related to the non-vanishing of some derivatives of the observable at fixed points. The proof of this new phenomenon is inspired by tools developed in the recent work of the first author and Ulcigrai for locally Hamiltonian flows having only (simple) non-degenerate saddles. In full generality, due to the occurrence of (multiple) degenerate saddles, we introduce new methods of handling functions with polynomial singularities over a full measure set of interval exchange transformations.

Tiling billards on triangle tilings, and interval exchange transformations

AbstractWe consider the dynamics of light rays in triangle tilings where triangles are transparent and adjacent triangles have equal but opposite indices of refraction. We find that the behavior of a trajectory on a triangle tiling is described by an orientation‐reversing three‐interval exchange transformation on the circle, and that the behavior of all the trajectories on a given triangle tiling is described by a polygon exchange transformation. We observe that, for a particular choice of triangle tiling, certain trajectories appear to approach the Rauzy fractal, under rescaling.

Closing lemmas for interval translation maps

A interval translation mapping (or a circle translation mapping) is studied. Such maps can be regarded as interval exchange maps with overlaps. It is known that for any mapping of that type admits a Borel probability invariant non-atomic measure. This measure can be constructed as a weak limit of invariant measures of maps with periodic parameters. Those measures, are just normalized Lebesgue ones on a family of sub-sectors. For such limit measures, in the case of a shift of the arcs of the circle, it is shown that any point of their supports can be made periodic by arbitrarily small change of the parameters of the system without changing the number of segments. For any invariant measure, it is deduced from Poincar´es Recurrence Theorem shows every point can be made periodic by a small change in the parameters of the system, with the number of intervals increasing by two at most.

Relations with a fixed interval exchange transformation

Relations with a fixed interval exchange transformation

Uniform simplicity for subgroups of piecewise continuous bijections of the unit interval

AbstractLet and (resp. ) be the quotient group of the group of all piecewise continuous (resp. piecewise continuous and orientation preserving) bijections of by its normal subgroup consisting in elements with finite support (i.e., which are trivial except at possibly finitely many points). Arnoux's thesis states that and certain groups of interval exchanges are simple, and the proofs of these results are the purpose of the Appendix. We prove the simplicity of the group of locally orientation preserving, piecewise continuous, piecewise affine maps of the unit interval. These results can be improved. Indeed, a group is uniformly simple if there exists a positive integer such that for any , the element can be written as a product of at most conjugates of or . We provide conditions, which guarantee that a subgroup of is uniformly simple. As corollaries, we obtain that , , , , , and some Thompson‐like groups included that the Thompson group are uniformly simple.

Joint ergodicity of piecewise monotone interval maps

For , let µ i be a Borel probability measure on which is equivalent to the Lebesgue measure λ and let be µ i -preserving ergodic transformations. We say that transformations are uniformly jointly ergodic with respect to if for any , limN−M→∞1N−M∑n=MN−1f0(T0 nx)⋅f1(T1 nx)⋯fk(Tk nx)=∏i=0k∫fidμi in L2(λ). We establish convenient criteria for uniform joint ergodicity and obtain numerous applications, most of which deal with interval maps. Here is a description of one such application. Let T G denote the Gauss map, , and, for β > 1, let T β denote the β-transformation defined by . Let T 0 be an ergodic interval exchange transformation. Let be distinct real numbers with and assume that for all . Then for any , limN−M→∞1N−M∑n=MN−1f0(T0nx)⋅f1(Tβ1nx)⋯fk(Tβknx)⋅fk+1(TGnx)=∫f0dλ⋅∏i=1k∫fidμβi⋅∫fk+1dμGin L2(λ). We also study the phenomenon of joint mixing. Among other things we establish joint mixing for skew tent maps and for restrictions of finite Blaschke products to the unit circle.

Rotated odometers

AbstractWe describe the infinite interval exchange transformations, called the rotated odometers, which are obtained as compositions of finite interval exchange transformations and the von Neumann–Kakutani map. We show that with respect to Lebesgue measure on the unit interval, every such transformation is measurably isomorphic to the first return map of a rational parallel flow on a translation surface of finite area with infinite genus and a finite number of ends. We describe the dynamics of rotated odometers by means of Bratteli–Vershik systems, derive several of their topological and ergodic properties, and investigate in detail a range of specific examples of rotated odometers.

Quantitative weak mixing for interval exchange transformations

We establish a dichotomy for the rate of the decay of the Cesàro averages of correlations of sufficiently regular functions for typical interval exchange transformations (IET) which are not rigid rotations (for which weak mixing had been previously established in the works of Katok–Stepin, Veech, and Avila–Forni). We show that the rate of decay is either logarithmic or polynomial, according to whether the IET is of rotation class (i.e., it can be obtained as the induced map of a rigid rotation) or not. In the latter case, we also establish that the spectral measures of Lipschitz functions have local dimension bounded away from zero (by a constant depending only on the number of intervals). In our approach, upper bounds are obtained through estimates of twisted Birkhoff sums of Lipschitz functions, while the logarithmic lower bounds are based on the slow deviation of ergodic averages that govern the relation between rigid rotations and their induced maps.

Rotated odometers and actions on rooted trees

A rotated odometer is an infinite interval exchange transformation (IET) obtained as a composition of the von Neumann–Kakutani map and a finite IET of intervals of equal length. In this paper, we consider rotated odometers for which the finite IET is of i