Given $ρ\in(0, 1/3]$, let $μ$ be the Cantor measure satisfying $μ=\frac{1}{2}μf_0^{-1}+\frac{1}{2}μf_1^{-1}$, where $f_i(x)=ρx+i(1-ρ)$ for $i=0, 1$. The support of $μ$ is a Cantor set $C$ generated by the iterated function system $\{f_0, f_1\}$. Continuing the work of Feng et al. (2000) on the pointwise lower and upper densities \[ Θ_*^s(μ, x)=\liminf_{r\to 0}\frac{μ(B(x,r))}{(2r)^s},\qquad Θ^{*s}(μ, x)=\limsup_{r\to 0}\frac{μ(B(x,r))}{(2r)^s}, \] where $s=-\log 2/\logρ$ is the Hausdorff dimension of $C$, we give a complete description of the sets $D_*$ and $D^*$ consisting of all possible values of the lower and upper densities, respectively. We show that both sets contain infinitely many isolated and infinitely many accumulation points, and they have the same Hausdorff dimension as the Cantor set $C$. Furthermore, we compute the Hausdorff dimension of the level sets of the lower and upper densities. Our method consists in formulating an equivalent ``dyadic" version of the problem involving the doubling map on $[0,1)$, which we solve by using known results on the entropy of a certain open dynamical system and the notion of tuning.
We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map.
We consider (not self-similar) Cantor sets defined by a sequence of piecewise linear functions. We prove that the dimension of the harmonic measure on such a set is strictly smaller than its Hausdorff dimension. Some Hausdorff measure estimates for these sets are also provided.
We show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the spectrum of the corresponding Schrödinger operators is Cantor. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to general $\mathrm{SL}(2,\mathbb R)$ cocycles, and obtain that uniform hyperbolic systems form a open and dense set in some one-parameter family.
Given $λ\in(0, 1)$, let $E_λ$ be the self-similar set generated by the iterated function system $\{x/3,(x+λ)/3,(x+2)/3\}$. Then $E_λ$ is a self-similar set with overlaps. We obtain the necessary and sufficient condition for $E_λ$ to be totally self-similar, which is a concept first introduced by Broomhead, Montaldi, and Sidorov in 2004. When $E_λ$ is totally self-similar, all its generating IFSs are investigated, and the size of the set of points having finite triadic codings is determined. Besides, we give some properties of the spectrum of $E_λ$ and show that the spectrum of $E_λ$ vanishes i…
A class of ultrametric Cantor sets $(C, d_{u})$ introduced recently in literature (Raut, S and Datta, D P (2009), Fractals, 17, 45-52) is shown to enjoy some novel properties. The ultrametric $d_{u}$ is defined using the concept of {\em relative infinitesimals} and an {\em inversion} rule. The associated (infinitesimal) valuation which turns out to be both scale and reparametrisation invariant, is identified with the Cantor function associated with a Cantor set $\tilde C$ where the relative infinitesimals are supposed to live in. These ultrametrics are both metrically as well as topologically …
We consider extended CMV matrices with analytic quasi-periodic Verblunsky coefficients with Diophantine frequency vector in the perturbatively small coupling constant regime and prove the analyticity of the tongue boundaries. As a consequence we establish that, generically, all gaps of the spectrum that are allowed by the Gap Labelling Theorem are open and hence the spectrum is a Cantor set. We also prove these results for a related class of almost periodic Verblunsky coefficients and present an application to suitable quantum walk models on the integer lattice.
We construct multidimensional Schrödinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the spectrum in the class of uniformly recurrent Schrödinger operators, namely the coexistence of a half-line and a Cantor-type structure, can be confirmed. Our construction uses Schrödinger operators with separable potentials that decompose into one-dimensional potentials generated by the Fibonacci sequence and relies on the study of such operators via the trace map …
We prove Cantor spectrum and almost-sure Anderson localization for quasiperiodic discrete Schrödinger operators $H = \varepsilonΔ+ V$ with potential $V$ sampled with Diophantine frequency $α$ from an asymmetric, smooth, cosine-like function $v \in C^2(\mathbb{T},[-1,1])$ for sufficiently small interaction $\varepsilon \leq \varepsilon_0(v,α)$. We prove this result via an inductive analysis on scales, whereby we show that locally the Rellich functions of Dirichlet restrictions of $H$ inherit the cosine-like structure of $v$ and are uniformly well-separated.
A solenoidal manifold is the inverse limit space of a tower of proper coverings of a compact manifold. In this work, we introduce new invariants for solenoidal manifolds, their asymptotic Steinitz orders and their prime spectra, and show they are invariants of the homeomorphism type. These invariants are formulated in terms of the monodromy Cantor action associated to a solenoidal manifold. To this end, we continue our study of invariants for minimal equicontinuous Cantor actions. We introduce the three types of prime spectra associated to such actions, and study their invariance properties un…
In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, \[ (H_{b,ϕ} x)_n= x_{n+1} +x_{n-1} + b \cos(2 πn ω+ ϕ)x_n \] on $l^2(\mathbb{Z})$ and its associated eigenvalue equation to deduce that for $b \ne 0,\pm 2$ and $ω$ Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ``Ten Martini Problem'' for these values of $b$ and $ω$. Moreover, we prove that for $|b|\ne 0$ small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open.
We consider products of two Cantor sets, and obtain the optimal estimates in terms of their thickness that guarantee that their product is an interval. This problem is motivated by the fact that the spectrum of the Labyrinth model, which is a two dimensional quasicrystal model, is given by a product of two Cantor sets. We also discuss the connection with the question on the structure of intersections of two Cantor sets, which was considered by many authors previously.
For a compact set $K\subset \mathbb{R}^1$ and a family $\{C_λ\}_{λ\in J}$ of dynamically defined Cantor sets sufficiently close to affine with $\text{dim}_H\, K+\text{dim}_H\, C_λ>1$ for all $λ\in J$, under natural technical conditions we prove that the sum $K+C_λ$ has positive Lebesgue measure for almost all values of the parameter $λ$. As a corollary, we show that generically the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than one.
We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schrödinger operators, the obstruction vanishes and we conclude that uniform hype…
We study the Fourier transforms $\widehatμ(ξ)$ of non-atomic Gibbs measures $μ$ for uniformly expanding maps $T$ of bounded distortions on $[0,1]$ or Cantor sets with strong separation. When $T$ is totally non-linear, then $\widehatμ(ξ) \to 0$ at a polynomial rate as $|ξ| \to \infty$.
To any continuous eigenvalue of a Cantor minimal system $(X,\,T)$, we associate an element of the dimension group $K^0(X,\,T)$ associated to $(X,\,T)$. We introduce and study the concept of irrational miscibility of a dimension group. The main property of these dimension groups is the absence of irrational values in the additive group of continuous spectrum of their realizations by Cantor minimal systems. The strong orbit equivalence (respectively orbit equivalence) class of a Cantor minimal system associated to an irrationally miscible dimension group $(G,\,u)$ (resp. with trivial infinitesim…
This paper is dedicated to the study of two famous subsets of the real line, namely Lagrange spectrum $L$ and Markov spectrum $M$. Our first result, Theorem 2.1, provides a rigorous estimate on the smallest value $t_1$ such that the portion of the Markov spectrum $(-\infty,t_1)\cap M$ has Hausdorff dimension $1$. Our second result, Theorem 3.1, gives a new upper bound on the Hausdorff dimension of the set difference $M\setminus L$. Our method combines new facts about the structure of the classical spectra together with finer estimates on the Hausdorff dimension of Gauss--Cantor sets of continu…
Let $T : Λ\to Λ$ be an expanding map on a Cantor set. For each suitably normalized Hölder continuous potential, we construct a spectral triple from which one may recover the associated Gibbs measure as a noncommutative measure.
Denote by $μ_a$ the distribution of the random sum $(1-a) \sum_{j=0}^\infty ω_j a^j$, where $P(ω_j=0)=P(ω_j=1)=1/2$ and all the choices are independent. For $0<a<1/2$, the measure $μ_a$ is supported on $C_a$, the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length $(1-2a)$, and iterating this process inductively on each of the remaining intervals. We investigate the convolutions $μ_a * (μ_b \circ S_λ^{-1})$, where $S_λ(x)=λx$ is a rescaling map. We prove that if the ratio $\log b/\log a$ is irrational and $λ\neq 0$, then \[ D(μ_a…
We consider digits-deleted sets or Cantor-type sets with $β$-expansions. We calculate the Hausdorff dimension $d$ of these sets and show that $d$ is continuous with respect to $β$. The $d$-dimentional Hausdorff measure of these sets is finite and positive.
These notes from lectures by the author at the CIMPA Research School held in Salta, Argentina, in November 2015 review definitions and properties of various measures of complexity in topological and measure-preserving dynamical systems, especially symbolic dynamical systems.
We prove that the dimension of the harmonic measure of the complementary of a translation-invariant type of Cantor sets as a continuous function of the parameters determining these sets. This results extend a previous one of the author and do not use ergodic theoretic tools, not applicable to our case.
We prove upper and lower bounds for the Lebesgue measure of the set of products $xy$ with $x$ and $y$ in the middle-third Cantor set. Our method is inspired by Athreya, Reznick and Tyson, but a different subdivision of the Cantor set provides a more rapidly converging approximation formula.
In this article we give necessary and sufficient conditions that a complex number must satisfy to be a continuous eigenvalue of a minimal Cantor system. Similarly, for minimal Cantor systems of finite rank, we provide necessary and sufficient conditions for having a measure theoretical eigenvalue. These conditions are established from the combinatorial information of the Bratteli-Vershik representations of such systems. As an application, from any minimal Cantor system, we construct a strong orbit equivalent system without irrational eigenvalues which shares all measure theoretical eigenvalues…
Let $P$ be a Borel probability measure on $\mathbb R^2$ supported by the Cantor dusts generated by a set of $4^u,\ u\geq 1$, contractive similarity mappings satisfying the strong separation condition. For this probability measure, we determine the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$. In addition, it is shown that though the quantization dimension of the measure $P$ is known, the quantization coefficient for $P$ does not exist.
Quantization of a probability distribution is the process of estimating a given probability by a discrete probability that assumes only a finite number of levels in its support. Centroidal Voronoi tessellations (CVT) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. In this paper, we investigate the optimal quantization and the centroidal Voronoi tessellations with $n$ generators for a Borel probability measure $P$ on $\mathbb R$ supported by a dyadic Cantor set generated by two self-similar m…
We study the set $M_\infty(X)$ of all infinite full non-atomic Borel measures on a Cantor space X. For a measure $μ$ from $M_\infty(X)$ we define a defective set $M_μ= \{x \in X : for any clopen set U which contains x we have μ(U) = \infty \}$. We call a measure $μ$ from $M_\infty(X)$ non-defective ($μ\in M_\infty^0(X)$) if $μ(M_μ) = 0$. The paper is devoted to the classification of measures $μ$ from $M_\infty^0(X)$ with respect to a homeomorphism. The notions of goodness and clopen values set $S(μ)$ are defined for a non-defective measure $μ$. We give a criterion when two good non-defective m…
A distortion theory is developed for $S-$unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of $S-$unimodal maps is classified according to a distortion property, called the Markov-property.
We consider a minimal equicontinuous action of a finitely generated group $G$ on a Cantor set $X$ with invariant probability measure $μ$, and stabilizers of points for such an action. We give sufficient conditions under which there exists a subgroup $H$ of $G$ such that the set of points in $X$ whose stabilizers are conjugate to $H$ has full measure. The conditions are that the action is locally quasi-analytic and locally non-degenerate. An action is locally quasi-analytic if its elements have unique extensions on subsets of uniform diameter. The condition that the action is locally non-degene…
We show that under natural technical conditions, the sum of a $C^2$ dynamically defined Cantor set with a compact set in most cases (for almost all parameters) has positive Lebesgue measure, provided that the sum of the Hausdorff dimensions of these sets exceeds one. As an application, we show that for many parameters, the Square Fibonacci Hamiltonian has spectrum of positive Lebesgue measure, while at the same time the density of states measure is purely singular.
In this paper we define a new class of metric spaces, called multi-model Cantor sets. We compute the Hausdorff dimension and show that the Hausdorff measure of a multi-model Cantor set is finite and non-zero. We then show that a bilipschitz map from one multi-model Cantor set to another has constant Radon-Nikodym derivative on some clopen. We use this to obtain an invariant up to bilipschitz homeomorphism.
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 or finitely many such measures (finitely ergodic homeomorphisms). Since every Cantor dynamical system $(X,T)$ can be realized as a Vershik map acting on the path space of a Bratteli diagram, we use combinatorial methods developed in symbolic dynamics and Bratteli diagrams during the last decade to study the simplex of invariant measures.
After the correction of an inaccurate result in the reference, the author uses five different methods, and gets five different inequalities on the Hausdorff measure of the Cartesian product of the middle third Cantor set with itself: $$H^s (C\times C)\leq 1.548563$$ $$H^s (C\times C)\leq 1.504975$$ $$H^s (C\times C)\leq 1.502878$$ $$H^s (C\times C)\leq 1.503263$$ $$H^s (C\times C)\leq 1.502483.$$ The main theory of this paper is the best among them, which is $$H^s (C\times C)\leq 1.502483.$$
We study the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets shrink to a Cantor set of zero Lebesgue measure. We consider both the presence and absence of clustering, which is detected by the Extremal Index, which turns out to be very useful to identify the compatibility between the dynamics and the fractal structure of the limiting Cantor set. The computation of the Extremal Index is connected to the box dimension of the intersection between the Cantor set and i…
We study the set M(X) of full non-atomic Borel (finite or infinite) measures on a non-compact locally compact Cantor set X. For an infinite measure $μ$ in M(X), the set $\mathfrak{M}_μ= \{x \in X : {for any compact open set} U \ni x {we have} μ(U) = \infty \}$ is called defective. We call $μ$ non-defective if $μ(\mathfrak{M}_μ) = 0$. The class $M^0(X) \subset M(X)$ consists of probability measures and infinite non-defective measures. We classify measures $μ$ from $M^0(X)$ with respect to a homeomorphism. The notions of goodness and compact open values set $S(μ)$ are defined. A criterion when t…
Let $N\ge 2$ and $ρ\in(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system \[ \left\{f_i(x)=ρx+\frac{i(1-ρ)}{N-1}: i=0,1,\ldots, N-1\right\}. \] Let $s=\dim_H E$ be the Hausdorff dimension of $E$, and let $μ=\mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $x\in E$, the pointwise lower $s$-density $Θ_*^s(μ,x)$ and upper $s$-density $Θ^{*s}(μ, x)$ of $μ$ at $x$. This extends some early results of Feng et al. (2000). Furthermore, we determine two critical values $a_c$ and $b_c$ …
We show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions that exceeds 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of homogeneous Cantor sets). In our setting the perturbations have more freedom than in the setting of the Palis' conjecture, so our result can be viewed as an affirmative answer to a weaker form of the Palis' conjecture. We also consider self-similar sets with overlaps on the real line (not necessarily homogeneous), and show that one can create an interval by applying arbitrary small pert…
Downarowicz and Maass (2008) proposed topological ranks for all homeomorphic Cantor minimal dynamical systems using properly ordered Bratteli diagrams. In this study, we adopt this definition to the case of all essentially minimal zero-dimensional systems. We consider the cases in which topological ranks are 2 and unique minimal sets are fixed points. Akin and Kolyada (2003), in their study of Li--Yorke sensitivity, showed that if the unique minimal set of an essentially minimal system is a fixed point, then the system must be proximal. However, a finite topological rank implies expansiveness;…
In 1954 Marstrand proved that if K is a subset of R^2 with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we give a combinatorial proof of this theorem when K is the product of regular Cantor sets of class C^{1+a}, a>0, for which the sum of their Hausdorff dimension is greater than 1.
We translate Akin's notion of {\it good} (and related concepts) from measures on Cantor sets to traces on dimension groups, and particularly for invariant measures of minimal homeomorphisms (and their corresponding simple dimension groups), this yields characterizations and examples, which translate back to the original context. Good traces on a simple dimension group are characterized by their kernel having dense image in their annihilating set of affine functions on the trace space; this makes it possible to construct many examples with seemingly paradoxical properties. In order to study the…
The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. For a given $k\geq 2$, let $\{S_j : 1\leq j\leq k\}$ be a set of $k$ contractive similarity mappings such that $S_j(x)=\frac 1 {2k-1} x +\frac{2 (j-1)} {2k-1}$ for all $x\in \mathbb R$, and let $P= \frac 1 k \sum_{j=1}^kP\circ S_j^{-1}$. Then, $P$ is a unique Borel probability measure on $\mathbb R$ such that $P$ has support the Cantor set generated by the similarity mappings $S_j$ for $1\leq j\leq k$.…
Let $C$ be the middle-third Cantor set. Define $C*C=\{x*y:x,y\in C\}$, where $*=+,-,\cdot,÷$ (when $*=÷$, we assume $y\neq0$). Steinhaus \cite{HS} proved in 1917 that \[ C-C=[-1,1], C+C=[0,2]. \] In 2019, Athreya, Reznick and Tyson \cite{Tyson} proved that \[ C÷C=\bigcup_{n=-\infty}^{\infty}\left[ 3^{-n}\dfrac{2}{3},3^{-n}\dfrac {3}{2}\right] . \] In this paper, we give a description of the topological structure and Lebesgue measure of $C\cdot C$. We indeed obtain corresponding results on the uniform $λ$-Cantor sets.
Let $m\in\mathbb N_{\ge 2}$, and let $\mathcal K=\{K_λ: λ\in(0, 1/m]\}$ be a class of Cantor sets, where $K_λ=\{\sum_{i=1}^\infty d_iλ^i: d_i\in\{0,1,\ldots, m-1\}, i\ge 1\}$. We investigate in this paper the likelyhood of a fixed point in the Cantor sets of $\mathcal K$. More precisely, for a fixed point $x\in(0,1)$ we consider the parameter set $Λ(x)=\{λ\in(0,1/m]: x\in K_λ\}$, and show that $Λ(x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, by constructing a sequence of Cantor subsets with large thickness in $Λ(x)$ we prove that the in…
