Cantor-type Sets Research Articles

Pluripolar hulls and fine holomorphy

Examples by Poletsky and the author and by Zwonek show the existence of nowhere extendable holomorphic functions with the property that the pluripolar hull of their graphs is much larger than the graph of the respective functions, and contains multiple sheets. We will explain this phenomenon by fine analytic continuation of the function over part of a Cantor-type set involved. This gives more information on the hull, and allows for weakening and effectiveness of the conditions in the original examples.

Daubechies' time-frequency localization operator on Cantor type sets II

We study a version of the fractal uncertainty principle in the joint time-frequency representation. Namely, we consider Daubechies' localization operator projecting onto spherically symmetric n-iterate Cantor sets with an arbitrary base M>1 and alphabet A. We derive an upper bound asymptote up to a multiplicative constant for the operator norm in terms of the base M and the alphabet size |A| of the Cantor set. For any fixed base and alphabet size, we show that there are Cantor sets such that the asymptote is optimal. In particular, the asymptote is precise for mid-third Cantor set, which was studied in part I [19]. Nonetheless, this does not extend to every Cantor set as we provide examples where the optimal asymptote is not achieved.

Algebraic genericity and special properties within sequence spaces and series

In this note we study the lineability and algebrability problems for certain classes of sequence spaces and series. In particular, we extend some algebrability results to the context of p-adic fields. We also prove that the subset of nonconvergent sequences of ℓ∞(ℝ) whose set of accumulation points is not a Cantor-type set is strongly 𝔠-algebrable.

Logarithmic dimension and bases in Whitney spaces

We give a formula for the logarithmic dimension of the generalized Cantor-type set $K$. In the case when the logarithmic dimension of $K$ is smaller than $1$, we construct a Faber basis in the space of Whitney functions $\mathcal{E}(K)$.

Quasi-equivalence of bases in some Whitney spaces

AbstractIf the logarithmic dimension of a Cantor-type set K is smaller than $1$ , then the Whitney space $\mathcal {E}(K)$ possesses an interpolating Faber basis. For any generalized Cantor-type set K, a basis in $\mathcal {E}(K)$ can be presented by means of functions that are polynomials locally. This gives a plenty of bases in each space $\mathcal {E}(K)$ . We show that these bases are quasi-equivalent.

Trace and Density Results on Regular Trees

The boundary of a regular tree can be viewed as a Cantor-type set. We equip our tree with a weighted distance and a weighted measure via the Euclidean arc-length and consider the associated first-order Sobolev spaces. We give characterizations for the existence of traces and for the density of compactly supported functions.

Large sets with small injective projections

Let $\ell_1,\ell_2,\dots$ be a countable collection of lines in ${\mathbb R}^d$. For any $t \in [0,1]$ we construct a compact set $\Gamma\subset{\mathbb R}^d$ with Hausdorff dimension $d-1+t$ which projects injectively into each $\ell_i$, such that the image of each projection has dimension $t$. This immediately implies the existence of homeomorphisms between certain Cantor-type sets whose graphs have large dimensions. As an application, we construct a collection $E$ of disjoint, non-parallel $k$-planes in $\mathbb{R}^d$, for $d \geq k+2$, whose union is a small subset of $\mathbb{R}^d$, either in Hausdorff dimension or Lebesgue measure, while $E$ itself has large dimension. As a second application, for any countable collection of vertical lines $w_i$ in the plane we construct a collection of nonvertical lines $H$, so that $F$, the union of lines in $H$, has positive Lebesgue measure, but each point of each line $w_i$ intersects at most one $h\in H$ and, for each $w_i$, the Hausdorff dimension of $F\cap w_i$ is zero.

Restriction of Laplace–Beltrami Eigenfunctions to Arbitrary Sets on Manifolds

Abstract Given a compact Riemannian manifold $(M, g)$ without boundary, we estimate the Lebesgue norm of Laplace–Beltrami eigenfunctions when restricted to a wide variety of subsets $\Gamma $ of $M$. The sets $\Gamma $ that we consider are Borel measurable, Lebesguenull but otherwise arbitrary with positive Hausdorff dimension. Our estimates are based on Frostman-type ball growth conditions for measures supported on $\Gamma $. For large Lebesgue exponents $p$, these estimates provide a natural generalization of $L^p$ bounds for eigenfunctions restricted to submanifolds, previously obtained in [ 8, 18, 19, 32]. Under an additional measure-theoretic assumption on $\Gamma $, the estimates are shown to be sharp in this range. As evidence of the genericity of the sharp estimates, we provide a large family of random, Cantor-type sets that are not submanifolds, where the above-mentioned sharp bounds hold almost surely.

A general renormalization procedure on the one-dimensional lattice and decay of correlations

We present a general form of renormalization operator [Formula: see text] acting on potentials [Formula: see text]. We exhibit the analytical expression of the fixed point potential [Formula: see text] for such operator [Formula: see text]. This potential can be expressed in a natural way in terms of a certain integral over the Hausdorff probability on a Cantor type set on the interval [0,1]. This result generalizes a previous one by Baraviera, Leplaideur and Lopes where the fixed point potential [Formula: see text] was of Hofbauer type. For the potentials of Hofbauer type (a well-known case of phase transition) the decay is like [Formula: see text], [Formula: see text]. Among other things we present the estimation of the decay of correlation of the equilibrium probability associated to the fixed potential [Formula: see text] of our general renormalization procedure. In some cases we get polynomial decay like [Formula: see text], [Formula: see text], and in others a decay faster than [Formula: see text], when [Formula: see text]. The potentials [Formula: see text] we consider here are elements of the so-called family of Walters’ potentials on [Formula: see text] which generalizes a family of potentials considered initially by Hofbauer. For these potentials some explicit expressions for the eigenfunctions are known. In a final section we also show that given any choice [Formula: see text] of real numbers varying with [Formula: see text] there exists a potential [Formula: see text] on the class defined by Walters which has a invariant probability with such numbers as the coefficients of correlation (for a certain explicit observable function).

Self‐similar sets, simple augmented trees and their Lipschitz equivalence

Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graph on the IFS has been established recently. In the paper, we introduce a notion of simple augmented tree which is a Gromov hyperbolic graph. By generalizing a combinatorial device of rearrangeable matrix, we show that there exists a near-isometry between the simple augmented tree and the symbolic space of the IFS, so that their hyperbolic boundaries are Lipschitz equivalent. We then apply this to consider the Lipschitz equivalence of self-similar sets with or without assuming the open set condition. Moreover, we also provide a criterion for a self-similar set to be a Cantor-type set which completely answers an open question raised in \cite{LaLu13}. Our study extends the previous works.

Random walks and induced Dirichlet forms on self-similar sets

Let K be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea [25], it has been proved that on the symbolic space X of K a natural augmented tree structure E exists; it is hyperbolic, and the hyperbolic boundary ∂HX with the Gromov metric is Hölder equivalent to K. In this paper we consider certain reversible random walks with return ratio 0<λ<1 on (X,E). We show that the Martin boundary M can be identified with ∂HX and K. With this setup and a device of Silverstein [41], we obtain precise estimates of the Martin kernel and the Naïm kernel in terms of the Gromov product. Moreover, the Naïm kernel turns out to be a jump kernel satisfying the estimate Θ(ξ,η)≍|ξ−η|−(α+β), where α is the Hausdorff dimension of K and β depends on λ. For suitable β, the kernel defines a regular non-local Dirichlet form on K. This extends the results of Kigami [27] concerning random walks on certain trees with Cantor-type sets as boundaries (see also [5]).

MORE ABOUT CANTOR LIKE SETS IN HYPERBOLIC NUMBERS

In this paper, we discuss the construction of new Cantor like sets in the hyperbolic plane. Also, we study the arithmetic sum of two of these Cantor like sets, as well as of those previously introduced in the literature. An hyperbolization, in the sense of Gromov, of the commutative ring of hyperbolic numbers is also given. Finally, we present the construction of a Cantor-type set as hyperbolic boundary.

Rate of convergence: the packing and centered Hausdorff measures of totally disconnected self-similar sets

In this paper we obtain the rates of convergence of the algorithms given in [13] and [14] for an automatic computation of the centered Hausdorff and packing measures of a totally disconnected self-similar set. We evaluate these rates empirically through the numerical analysis of three standard classes of self-similar sets, namely, the families of Cantor type sets in the real line and the plane and the class of Sierpinski gaskets. For these three classes and for small contraction ratios, sharp bounds for the exact values of the corresponding measures are obtained and it is shown how these bounds automatically yield estimates of the corresponding measures, accurate in some cases to as many as 14 decimal places. In particular, the algorithms accurately recover the exact values of the measures in all cases in which these values are known by geometrical arguments. Positive results, which confirm some conjectural values given in [13] and [14] for the measures, are also obtained for an intermediate range of larger contraction ratios. We give an argument showing that, for this range of contraction ratios, the problem is inherently computational in the sense that any theoretical proof, such as those mentioned above, might be impossible, so that in these cases, our method is the only available approach. For contraction ratios close to those of the connected case our computational method becomes intractably time consuming, so the computation of the exact values of the packing and centered Hausdorff measures in the general case, with the open set condition, remains a challenging problem.

Orthogonal polynomials for the weakly equilibrium Cantor sets

Let K ( γ ) K(\gamma ) be the weakly equilibrium Cantor-type set introduced by the second author in an earlier work. It is proven that the monic orthogonal polynomials Q 2 s Q_{2^s} with respect to the equilibrium measure of K ( γ ) K(\gamma ) coincide with the Chebyshev polynomials of the set. Procedures are suggested to find Q n Q_{n} of all degrees and the corresponding Jacobi parameters. It is shown that the sequence of the Widom factors is bounded below.

Fatness and thinness of some general Cantor sets for doubling measures

In this paper, we consider some classes of general Cantor sets satisfying (αk)- regular condition and obtain necessary and sufficient conditions to characterize the fatness and thinness of the sets for doubling measures under suitable conditions. It is applied to classify some kinds of Cantor-type sets.

Lipschitz equivalence of self-similar sets and hyperbolic boundaries II

In [13], two of the authors gave a study of Lipschitz equivalence of self-similar sets through the augmented trees, a class of hyperbolic graphs introduced by Kaimanovich in [9] and developed by Lau and Wang [10]. In this paper, we continue such investigation. We remove a major assumption in the main theorem in [13] by using a new notion of quasi-rearrangeable matrix, and show that the hyperbolic boundary of any simple augmented tree is Lipschitz equivalent to a Cantor-type set. We then apply this result to consider the Lipschitz equivalence of certain totally disconnected self-similar sets as well as their unions.

Geometric analysis on Cantor sets and trees

Abstract Using uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit smoothness exponent. Further, we study quasisymmetries between the boundaries of two trees, and show that they have rough quasiisometric extensions to the trees. Conversely, we show that every rough quasiisometry between two trees extends as a quasisymmetry between their boundaries. In both directions we give sharp estimates for the involved constants. We use this to obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor type sets.

Widom Factors

Given a non-polar compact set K,we define the n-th Widom factor Wn(K) as the ratio of the sup-norm of the n-th Chebyshev polynomial on K to the n-th degree of its logarithmic capacity. By G. Szegő, the sequence \((W_{n}(K))_{n=1}^{\infty }\) has subexponential growth. Our aim is to consider compact sets with maximal growth of the Widom factors. We show that for each sequence \((M_{n})_{n=1}^{\infty }\) of subexponential growth there is a Cantor-type set whose Widom’s factors exceed Mn. We also present a set K with highly irregular behavior of the Widom factors.

Two measures on Cantor sets

We give an example of Cantor-type set for which its equilibrium measure and the corresponding Hausdorff measure are mutually absolutely continuous. Also we show that these two measures are regular in the Stahl–Totik sense.

One-dimensional Schrödinger operators with [formula omitted]-interactions on Cantor-type sets

We introduce a novel approach for defining a δ′-interaction on a subset of the real line of Lebesgue measure zero which is based on Sturm–Liouville differential expression with measure coefficients. This enables us to establish basic spectral properties (e.g., self-adjointness, lower semiboundedness and spectral asymptotics) of Hamiltonians with δ′-interactions concentrated on sets of complicated structures.