Related papers: Renormalizing an infinite rational IET
Starting from any pseudo-Anosov map $\varphi$ on a surface of genus $g \geqslant 2$, we construct explicitly a family of Derived from pseudo-Anosov maps $f$ by adapting the construction of Smale's Derived from Anosov maps on the two-torus.…
For $\beta\in(1,2]$ the $\beta$-transformation $T_\beta: [0,1) \to [0,1)$ is defined by $T_\beta ( x) = \beta x \pmod 1$. For $t\in[0, 1)$ let $K_\beta(t)$ be the survivor set of $T_\beta$ with hole $(0,t)$ given by \[K_\beta(t):=\{x\in[0,…
We present shrinking targets results for general systems with the emphasis on applications for IETs (interval exchange transformations) $(J,T)$, $J=[0,1)$. In particular, we prove that if an IET $(J,T)$ is ergodic (relative to the Lebesgue…
We show that the transformation (x_n)_{n\ge 1}\to (1/(1+x_1+...+x_n))_{n\ge 1} of the compact set of sequences (x_n)_{n\ge 1} of numbers from the unit interval [0,1] has a unique fixed point, which is attractive. The fixed point turns out…
We show that there exists an interval exchange and a point so that the orbit of the point equidistributes for a measure that is not ergodic.
It is well known that a purely unrectifiable set cannot support a harmonic measure which is absolutely continuous with respect to the Hausdorff measure of this set. We show that nonetheless there exist elliptic operators on (purely…
We survey recent work done on the values at integer points of irrational inhomogeneous quadratic forms, namely, inhomogeneous analogues of the famous Oppenheim conjecture. We also prove that the set of such forms in two variables whose set…
In this paper we continue to explore infinitely renormalizable H\'enon maps with small Jacobian. It was shown in [CLM] that contrary to the one-dimensional intuition, the Cantor attractor of such a map is non-rigid and the conjugacy with…
Renormalizations can be considered as building blocks of complex dynamical systems. This phenomenon has been widely studied for iterations of polynomials of one complex variable. Concerning non-polynomial hyperbolic rational maps, a recent…
A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira. Recurrent train tracks with a single switch provide a subclass of linear involutions. We call such linear involutions…
Let $I_1=[a_0,a_1),\ldots,I_{k}= [a_{k-1},a_k)$ be a partition of the interval $I=[0,1)$ into $k$ subintervals. Let $f:I\to I$ be a map such that each restriction $f|_{I_i}$ is an increasing Lipschitz contraction. We prove that any $f$…
This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure…
The irrationality exponent of a real number measures how well that number can be approximated by rationals. Real numbers with irrationality exponent strictly greater than $2$ are transcendental numbers, and form a set with rich fractal…
A typical interval exchange transformation has an infinite sequence of matrices associated to it by successive iterations of Rauzy induction. In 2010, W. A. Veech answered a question of A. Bufetov by showing that the interval exchange…
This paper continues the study of the structure of finite intersections of general multiplicative translates $\mathcal{C}(M_1,\ldots,M_n)=\frac{1}{M_1}\Sigma_{3,\bar{2}}\cap\cdots\cap\frac{1}{M_n}\Sigma_{3,\bar{2}}$ for integers $1\leq…
We consider dressed 1-electron states in a translation-invariant model of non-relativistic QED. To start with a well-defined model, the interaction Hamiltonian is cutoff at very large photon energies (ultraviolet cutoff) and regularized at…
In this paper we investigate all-at-once versus reduced regularization of dynamic inverse problems on finite time intervals $(0,T)$. In doing so, we concentrate on iterative methods and nonlinear problems, since they have already been shown…
Naively, the "best" method of renormalization is the one where a momentum cutoff is taken to infinity while maintaining stable results due to a cutoff-dependent adjustment of counterterms. We have applied this renormalization method in the…
Dimension groups are complete invariants of strong orbit equivalence for minimal Cantor systems. This paper studies a natural family of minimal Cantor systems having a finitely generated dimension group, namely the primitive unimodular…
In this paper we study the non-injectivity arising in infinite interval exchange transformations. In particular, we build and analyze an infinite family of infinite interval exchanges semi-conjugated to generalized Thue-Morse subshifts,…