Irrational Rotation Research Articles

On the distribution of Sudler products and Birkhoff sums for the irrational rotation

On the distribution of Sudler products and Birkhoff sums for the irrational rotation

Construction of smooth isomorphic and finite-to-one extensions of irrational rotations which are not almost automorphic

Construction of smooth isomorphic and finite-to-one extensions of irrational rotations which are not almost automorphic

Subshifts of finite symbolic rank

Abstract The definition of subshifts of finite symbolic rank is motivated by the finite rank measure-preserving transformations which have been extensively studied in ergodic theory. In this paper, we study subshifts of finite symbolic rank as essentially minimal Cantor systems. We show that minimal subshifts of finite symbolic rank have finite topological rank, and conversely, every minimal Cantor system of finite topological rank is either an odometer or conjugate to a minimal subshift of finite symbolic rank. We characterize the class of all minimal Cantor systems conjugate to a rank- $1$ subshift and show that it is dense but not generic in the Polish space of all minimal Cantor systems. Within some different Polish coding spaces of subshifts, we also show that the rank-1 subshifts are dense but not generic. Finally, we study topological factors of minimal subshifts of finite symbolic rank. We show that every infinite odometer and every irrational rotation is the maximal equicontinuous factor of a minimal subshift of symbolic rank $2$ , and that a subshift factor of a minimal subshift of finite symbolic rank has finite symbolic rank.

Limit law of rescaled hitting time functions for quadratic irrational rotations

Limit law of rescaled hitting time functions for quadratic irrational rotations

Exponential convergence of the weighted Birkhoff average

In this paper, we consider the polynomial and exponential convergence rates of the weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively.

Non-Abelian factors for actions of [formula omitted] and other non-C⁎-simple groups

Let Γ be a countable group and (X,Γ) a compact topological dynamical system. We study the question of the existence of an intermediate C⁎-subalgebra ACr⁎(Γ)<A<C(X)⋊rΓ, which is not of the form A=C(Y)⋊rΓ, corresponding to a factor map (X,Γ)→(Y,Γ). Here Cr⁎(Γ) is the reduced C⁎-algebra of Γ and C(X)⋊rΓ is the reduced C⁎-crossed-product of (X,Γ). Our main results are: (1) For Γ which is not C⁎-simple, when (X,Γ) admits a Γ-invariant probability measure, then such a sub-algebra always exists. (2) For Γ=Z and (X,Γ) an irrational rotation of the circle X=R/Z, we give a full description of all these non-crossed-product subalgebras.

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.

Uniqueness of Denjoy minimal sets for twist maps with zero entropy *

For monotone twist maps with zero topological entropy, we show that the set of recurrent points with irrational rotation number α can be described by a single orientation preserving circle homeomorphism and hence there is either an invariant circle, or a unique Denjoy minimal set, of rotation number α.

Inclusions of $C^*$-algebras arising from fixed-point algebras

We examine inclusions of C^* -algebras of the form A^H \subseteq A \rtimes_{r} G , where G and H are groups acting on a unital simple C^* -algebra A by outer automorphisms and H is finite. It follows from a theorem of Izumi that A^H \subseteq A is C^* -irreducible, in the sense that all intermediate C^* -algebras are simple. We show that A^H \subseteq A \rtimes_{r} G is C^* -irreducible for all G and H as above if and only if G and H have trivial intersection in the outer automorphisms of A , and we give a Galois type classification of all intermediate C^* -algebras in the case when H is abelian and the two actions of G and H on A commute. We illustrate these results with examples of outer group actions on the irrational rotation C^* -algebras. We exhibit, among other examples, C^* -irreducible inclusions of AF-algebras that have intermediate C^* -algebras that are not AF-algebras; in fact, the irrational rotation C^* -algebra appears as an intermediate C^* -algebra.

On Graphical Shapes of χ2 Statistics of Leading Digits of Irrational Rotations

We discuss the outline of the shapes of graphs of χ 2 statistics for distributions of leading digits of irrational rotations under some conditions on mth convergent. We give some estimates of important coefficients Lk’s, which determine the graphical shapes of χ2 statistics. This means that the denominator qm of mth convergent and the large partial quotient am+1 determine the outline of shapes of graphs, when we observe values of χ 2 statistics with step qm.

Relatively pointwise recurrence on local dendrites

Let X be a local dendrite and let f : X → X be a continuous map. Denote by P ( f ) and R ( f ) the sets of periodic and recurrent points of f, respectively. We say that f is relatively pointwise recurrent if R ( f ) ¯ = X . We show that, if X is almost meshed (i.e the union of the interior of all free arcs is dense in X), then f is relatively pointwise recurrent if and only if either f is topologically conjugate to an irrational rotation on the circle or it has a dense set of periodic points.

On the metric upper density of Birkhoff sums for irrational rotations

This article examines the value distribution of for almost every α where is ranging over a long interval and f is a 1-periodic function with discontinuities or logarithmic singularities at rational numbers. We show that for N in a set of positive upper density, the order of is of Khintchine-type, unless the logarithmic singularity is symmetric. Additionally, we show the asymptotic sharpness of the Denjoy–Koksma inequality for such f, with applications in the theory of numerical integration. Our method also leads to a generalized form of the classical Borel–Bernstein Theorem that allows very general modularity conditions.

Hitting functions for mixed partitions

Let $T_{\rho}$ be an irrational rotation on a unit circle $S^{1}\simeq [0,1)$. Consider the sequence $\{\mathcal{P}_{n}\}$ of increasing partitions on $S^{1}$. Define the hitting times $N_{n}(\mathcal{P}_n;x,y):= \inf\{j\geq 1\mid T^{j}_{\rho}(y)\in P_{n}(x)\}$, where $P_{n}(x)$ is an element of $\mathcal{P}_{n}$ containing $x$. D. Kim and B. Seo in [9] proved that the rescaled hitting times $K_n(\mathcal{Q}_n;x,y):= \frac{\log N_n(\mathcal{Q}_n;x,y)}{n}$ a.e. (with respect to the Lebesgue measure) converge to $\log2$, where the sequence of partitions $\{\mathcal{Q}_n\}$ is associated with chaotic map $f_{2}(x):=2x \bmod 1$. The map $f_{2}(x)$ has positive entropy $\log2$. A natural question is what if the sequence of partitions $\{\mathcal{P}_n\}$ is associated with a map with zero entropy. In present work we study the behavior of $K_n(\tau_n;x,y)$ with the sequence of mixed partitions $\{\tau_{n}\}$ such that $ \mathcal{P}_{n}\cap [0,\frac{1}{2}]$ is associated with map $f_{2}$ and $\mathcal{D}_{n}\cap [\frac{1}{2},1]$ is associated with irrational rotation $T_{\rho}$. It is proved that $K_n(\tau_n;x,y)$ a.e. converges to a piecewise constant function with two values. Also, it is shown that there are some irrational rotations that exhibit different behavior.

Spectral characteristics of Schrödinger operators generated by product systems

Motivated by the question of what spectral properties of dynamically defined Schrödinger operators may be preserved under periodic perturbations, we study ergodic Schrödinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. The scenario given by a periodic background potential corresponds to a separable structure in which the sampling function is the sum of two pieces, each of which depends only on a single factor of the product system. However, in each case that we study, our methods apply more generally to sampling functions that allow non-trivial dependencies between the product factors. In the case in which one factor is a Boshernitzan subshift, we prove that either the resulting operators are periodic or the resulting spectra must be Cantor sets. The main ingredient is a suitable stability result for the Boshernitzan condition under taking products. We also discuss the stability of purely singular continuous spectrum, which, given the zero-measure spectrum result, amounts to stability results for eigenvalue exclusion. In particular, we examine situations in which the existing criteria for the exclusion of eigenvalues are stable under periodic perturbations. As a highlight of this, we show that any simple Toeplitz subshift over a binary alphabet exhibits uniform absence of eigenvalues on the hull for any periodic perturbation whose period is commensurate with the coding sequence. This is new, even in the case in which the periodic background vanishes entirely. In the case of a full shift, we give an effective criterion to compute exactly the spectrum of a random Anderson model perturbed by a potential of period two, and we further show that the naive generalization of this criterion does not hold for period three. Next, we consider quasi-periodic potentials with potentials generated by trigonometric polynomials with periodic background. We show that the quasiperiodic cocycle induced by passing to blocks of period length is subcritical when the coupling constant is small and supercritical when the coupling constant is large. Thus, the spectral type is absolutely continuous for small coupling and pure point (for a.e. frequency and phase) when the coupling is large.

New Kronecker–Weyl type equidistribution results and Diophantine approximation

An interesting result of Veech more than 50 years ago is a parity, or mod 2, version of the Kronecker–Weyl equidistribution theorem concerning the irrational rotation sequence {qalpha }, q=0,1,2,3,ldots If alpha is badly approximable and bin (0,1) satisfies bne {malpha } for any min {mathbb {Z}}, then the parity of cardinalities of the sets {1leqslant qleqslant N,{:},{qalpha }in [0,b)} as Nrightarrow infty is evenly distributed. We first answer a question of Veech and establish a stronger form of the mod n analog of his result. Furthermore, for irrational alpha and b={malpha } for some min {mathbb {N}}, we give a simple yet precise characterization of those cases that give rise to even distribution. We also obtain time-quantitative description of some very striking violations of uniformity—this part is particularly number theoretic in nature, and involves Ostrowski representations of positive integers and alpha -expansions of real numbers. The Veech discrete 2-circle problem can also be visualized as a problem that concerns 1-direction geodesic flow on a surface obtained by modifying the surface comprising two side-by-side squares by the inclusion of symmetric barriers and gates on the vertical edges, with appropriate modification of the vertical edge identifications. We establish a far-reaching generalization of this case to ones that concern 1-direction geodesic flow on surfaces obtained by modifying a finite square tiled translation surface in analogous but not necessarily symmetric ways.

The structures of pointwise recurrent quasi-graph maps

We show that a continuous map f from a quasi-graph X to itself is pointwise recurrent if and only if one of the following two statements holds: (1) X is a simple closed curve and f is topologically conjugate to an irrational rotation on the unit circle S1; (2) f is a periodic homeomorphism.

Non-rigid rank-one infinite measures on the circle

For a class of irrational numbers, depending on their Diophantine properties, we construct explicit rank-one transformations that are totally ergodic and not weakly mixing and classify when the measure is finite or infinite. In the finite case they are isomorphic to irrational rotations, giving explicit rank-one cutting and spacer parameters for these irrational rotations. In the infinite case we use the constructions to provide examples of non-weakly mixing infinite measure-preserving ergodic transformations which do not have any nontrivial probability preserving factors with discrete spectrum, thereby contributing to the program of Aaronson and Nadkarni and a question of Glasner and Weiss. We also obtain nonsingular versions of these examples for each Krieger ration set.

On $$SL(2,\mathbb{R})$$-Cocycles over Irrational Rotations with Secondary Collisions

We consider a skew product $F_{A} = (\sigma_{\omega}, A)$ over irrational rotation $\sigma_{\omega}(x) = x + \omega$ of a circle $\mathbb{T}^{1}$. It is supposed that the transformation $A: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ being a $C^{1}$-map has the form $A(x) = R(\varphi(x)) Z(\lambda(x))$, where $R(\varphi)$ is a rotation in $\mathbb{R}^{2}$ over the angle $\varphi$ and $Z(\lambda)= diag\{\lambda, \lambda^{-1}\}$ is a diagonal matrix. Assuming that $\lambda(x) \ge \lambda_{0} > 1$ with a sufficiently large constant $\lambda_{0}$ and the function $\varphi$ be such that $\cos \varphi(x)$ possesses only simple zeroes, we study hyperbolic properties of the cocycle generated by $F_{A}$. We apply the critical set method to show that, under some additional requirements on the derivative of the function $\varphi$, the secondary collisions compensate weakening of the hyperbolicity due to primary collisions and the cocycle generated by $F_{A}$ becomes hyperbolic in contrary to the case when secondary collisions can be partially eliminated.

A Balian–Low type theorem for Gabor Riesz sequences of arbitrary density

Gabor systems are used in fields ranging from audio processing to digital communication. Such a Gabor system (g,varLambda ) consists of all time-frequency shifts pi (lambda ) g of a window function g in L^2({mathbb {R}}) along a lattice varLambda subset {mathbb {R}}^2. We focus on Gabor systems that are also Riesz sequences, meaning that one can stably reconstruct the coefficients c = (c_lambda )_{lambda in varLambda } from the function sum _{lambda in varLambda } c_lambda , pi (lambda ) g. In digital communication, a function of this form is used to transmit the digital sequence c. It is desirable for g to be well localized in time and frequency, since the transmitted signal will then be almost compactly supported in time and frequency if the sequence c has finite support. In this paper, we study what additional structural properties the signal space mathcal {G}(g,varLambda ), i.e., the span of the Gabor system, satisfies in addition to being a closed subspace of L^2({mathbb {R}}). The most well-known result in this direction—the Balian–Low theorem—states that if g is well localized in time and frequency and if (g,varLambda ) is a Riesz sequence, then mathcal {G}(g,varLambda ) is necessarily a proper subspace of L^2({mathbb {R}}). We prove a generalization of this result related to the invariance of mathcal {G}(g,varLambda ) under time-frequency shifts. Precisely, we show that if (g,varLambda ) is a Riesz sequence with g being well localized in time and frequency (precisely, g should belong to the so-called Feichtinger algebra), then pi (mu ) mathcal {G}(g,varLambda ) subset mathcal {G}(g,varLambda ) holds if and only if mu in varLambda . For lattices of rational density, this was already known, with the proof based on Zak transform techniques. These methods do not generalize to arbitrary lattices, however. Instead, our proof for lattices of irrational density relies on combining methods from time-frequency analysis with properties of a special C^*-algebra, the so-called irrational rotation algebra.

On the $$\chi ^2$$ statistics of leading digits of irrational rotations with a large second or third partial quotient

On the $$\chi ^2$$ statistics of leading digits of irrational rotations with a large second or third partial quotient