Non-ergodic interval exchange transformations

We construct interval exchange transformations on four intervals satisfying a strong irrationality condition and having exactly two ergodic invariant probability measures. This shows that although Kronecker’s theorem remains true for interval exchange transformations, the Weyl equidistribution theorem is false even under the strongest irrationality assumptions.     

Languages of k-interval exchange transformations

This paper gives a complete characterization of those sequencesof subword complexity (k – 1)n + 1that are natural codings of orbits of k-interval exchange transformations,thereby answering an old question of Rauzy.     

Local complexity functions of interval exchange transformations

We studied numerically complexity functions for interval exchange transformations. We have shown that they grow linearly in time as well as the ?-complexity function. Moreover, we found out that they depend also linearly on ? where ? is the Lebesgue measure of a set of initial points. This allows us to hypothesize that the dimension of the measure related to the ?-complexity function could be determined by studying the dependence of local complexity functions on ?.     

An entropy estimate for infinite interval exchange transformations

We determine upper bound estimates for minimal entropy growth rates in measure-preserving systems by utilizing the universal representability of such systems by means of interval exchange transformations. Using these estimates, we also establish several criteria for the identification of systems with vanishing entropy.     

Cocycles over interval exchange transformations and multivalued Hamiltonian flows

We consider interval exchange transformations of periodic type and construct different classes of ergodic cocycles of dimension?1 over this special class of IETs. Then using Poincaré sections we apply this construction to obtain the recurrence and ergodicity for some smooth flows on non-compact manifolds which are extensions of multivalued Hamiltonian flows on compact surfaces.     

Almost periodic Schrödinger operators along interval exchange transformations

It is shown that Schrödinger operators, with potentials along the shift embedding of irreducible interval exchange transformations in a dense set, have pure singular continuous spectrum for Lebesgue almost all points of the interval. Such potentials are natural generalizations of the Sturmian case.     

Interval exchange transformations and some special flows are not mixing

An interval exchange transformation (I.E.T.) is a map of an interval into itself which is one-to-one and continuous except for a finite set of points and preserves Lebesgue measure. We prove that any I.E.T. is not mixing with respect to any Borel invariant measure. The same is true for any special flow constructed by any I.E.T. and any “roof” function of bounded variation. As an application of the last result we deduce that in any polygon with the angles commensurable with π the billiard flow is not mixing on two-dimensional invariant manifolds. The author is partially supported by grant NSF MCS 78-15278.     

Cantor singular continuous spectrum for operators along interval exchange transformations

It is shown that Schrödinger operators, with potentials along the shift embedding of Lebesgue almost every interval exchange transformations, have Cantor spectrum of measure zero and pure singular continuous for Lebesgue almost all points of the interval.

     

On the commutator group of the group of interval exchange transformations

We study the group of interval exchange transformations and obtain several characterizations of its commutator group. In particular, it turns out that the commutator group is generated by elements of order 2.     

Witness Rectangle Graphs

In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x, y in P are adjacent whenever the open isothetic rectangle with x and y as opposite corners contains at least one point in W. WRGs are representative of a larger family of witness proximity graphs introduced in two previous papers. We study graph-theoretic properties of WRGs. We prove that any WRG has at most two non-trivial connected components. We bound the diameter of the non-trivial connected components of a WRG in both the one-component and two-component cases. In the latter case, we prove that a graph is representable as a WRG if and only if each component is a connected co-interval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs. In addition, we prove that a WRG with no isolated vertices has domination number at most four. Moreover, we show that any combinatorial graph can be drawn as a WRG using a combination of positive and negative witnesses. Finally, we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n points.     

The Golden Rectangle

Fibonacci(Leonardo Pisano) was an outstanding mathematician and scholar whose interests lay particularly in solving practical problems.In Chapter 2,you came across the Fibonacci series.The series has a number of interesting properties.If we label each number(term) in the series using t1,     

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.     

Structure of three-interval exchange transformations II: a combinatorial description of the tranjectories

Astract  We describe an algorithm for generating the symbolic sequences which code the orbits of points under an interval exchange transformation on three intervals. The algorithm has two components. The first is an arithmetic division algorithm applied to the lengths of the intervals. This arithmetic construction was originally introduced by the authors in an earlier paper and may be viewed as a two-dimensional generalization of the regular continued fraction. The second component is a combinatorial algorithm which generates the bispecial factors of the associated symbolic subshift as a function of the arithmetic expansion. As a consequence, we obtain a complete characterization of those sequences of block complexity 2n+1 which are natural codings of orbits of three-interval exchange transformations, thereby answering an old question of Rauzy. Partially supported by NSF grant INT-9726708.     

An extension of Lagrange’s theorem to interval exchange transformations over quadratic fields

LetT be an interval exchange transformation onN intervals whose lengths lie in a quadratic number field. Let {T _n } _n=1 ^∞ be any sequence of interval exchange transformations such thatT ₁ =T andT _n is the first return map induced byT _n-1 on one of its exchanged intervals I_n-1. We prove that {T _n } _n=1 ^∞ contains finitely many transformations up to rescaling. If the interval I_n is chosen according to a consistent pattern of induction, e.g., the first interval is chosen, then there existk,n ₀ ∈ ℤ⁺, λ ∈R ⁺ such that for alln ≥n ₀,I _n = λI _n+k andT _n ,T _n+k are the same up to rescaling. Rephrased arithmetically, this says that a certain family of vectorial division algorithms, applied to quadratic vector spaces, yields sequences of remainders that are eventually periodic. WhenN = 2 the assertion reduces to Lagrange’s classical theorem that the simple continued fraction expansion of a quadratic irrational is eventually periodic. We also discuss the case of periodic induced sequences. These results have applications to topology. In particular, every projective measured foliation on Thurston’s boundary to Teichmüller space that is minimal and metrically ‘quadratic’ is fixed by a hyperbolic element of the modular group. Moreover, if the foliation is orientable, it covers (via a branched covering) an irrational foliation of the two-torus. We also obtain a new proof, for quadratic irrationals, of Boshernitzan’s result that a minimal rank 2 interval exchange transformation is uniquely ergodic. The first author was supported in part by NSF-DMS-9224667. The second author was supported in part by an NSF-NATO fellowship, held at the Université Paris-Sud, Orsay.     

Covering a Rectangle With Equal Circles

Recently, Tarnai and Gáspár [22] used mechanically inspired computer simulations to construct thin coverings of a square with up to ten equal circles. We generalise the problem to rectangles and determine the thinnest coverings of a general rectangle with up to five equal circles. Partial results are presented for coverings with seven circles.