Related papers: Renormalizing an infinite rational IET
We establish a dichotomy for the rate of the decay of the Ces\`aro averages of correlations of sufficiently regular functions for typical interval exchange transformations (IET) which are not rigid rotations (for which weak mixing had been…
For each irrational $\alpha\in[0,1)$ we construct a continuous function $f\: [0,1)\to \R$ such that the corresponding cylindrical transformation $[0,1)\times\R \ni (x,t) \mapsto (x+\alpha, t+ f(x)) \in [0,1)\times\R$ is transitive and the…
It has been shown in (Gaidashev et al, 2010) and (Gaidashev et al, 2011) that infinitely renormalizable area-preserving maps admit invariant Cantor sets with a maximal Lyapunov exponent equal to zero. Furthermore, the dynamics on these…
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…
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…
The classical Cantor's intersection theorem states that in a complete metric space $X$, intersection of every decreasing sequence of nonempty closed bounded subsets, with diameter approaches zero, has exactly one point. In this article, we…
It is shown that Schroedinger 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…
Period doubling H\'enon renormalization of strongly dissipative maps is generalized in arbitrary finite dimension. In particular, a small perturbation of toy model maps with dominated splitting has invariant $C^r$ surfaces embedded in…
Denote by $\mu_a$ the distribution of the random sum $(1-a) \sum_{j=0}^\infty \omega_j a^j$, where $P(\omega_j=0)=P(\omega_j=1)=1/2$ and all the choices are independent. For $0<a<1/2$, the measure $\mu_a$ is supported on $C_a$, the central…
We present a computational study of finite-time mixing of a line segment by cutting and shuffling. A family of one-dimensional interval exchange transformations is constructed as a model system in which to study these types of mixing…
Interval exchange transformations are typically uniquely ergodic maps and therefore have uniformly distributed orbits. Their degree of uniformity can be measured in terms of the star-discrepancy. Few examples of interval exchange…
We prove that any over-twist pattern is conjugate to an interval exchange transformation with bounded number of segments of isometry, restricted on one of its cycles. The bound is independent of the period and over-rotation number of the…
A map $f{:}\,[0,1)\to [0,1)$ is a {\it piecewise contraction of $n$ intervals} ($n$-PC) if there exist $0<\lambda<1$ and a partition of $[0,1)$ into intervals $I_1,\ldots,I_n$ such that $f\vert_{I_i}$ is $\lambda$-Lipschitz for every $1\le…
We consider the infrared problem in a model of a freely propagating, nonrelativistic charged particle of mass 1 in interaction with the quantized electromagnetic field. The hamiltonian of the system is regularized by an infrared cutoff…
We investigate, both analytically and numerically, the kinetic and stochastic counterpart of the triadic Cantor set. The generator that divides an interval either into three equal pieces or into three pieces randomly and remove the middle…
We study the topology and the Hausdorff dimension of a random Cantor set with overlaps, generated by an iterated function system with scaling ratio equal to the Golden Mean. The results extend known formulas to a case where the Open Set…
We introduce and study bi-Lipschitz-invariant dimensions that range between the box and Assouad dimensions. The quasi-Assouad dimensions and $\theta$-spectrum are other special examples of these intermediate dimensions. These dimensions are…
We introduce a definition of admissibility for subintervals in interval exchange transformations. Using this notion, we prove a property of the natural codings of interval exchange transformations, namely that any derived set of a regular…
A sharp bound on the number of invariant components of an interval exchange transformation is provided. More precisely, it is proved that the number of periodic components n_per and the number of minimal components n_min of an interval…
In this note, we investigate the coboundaries of interval exchange transformations of 3 intervals (3-IETs). More precisely, we show that a differentiable function with absolutely continuous derivative with bounded variation, whose integral…