Hausdorff Dimension Dim Research Articles

Dimension spectra of lines1

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp ( L ) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim ( a , b ) is equal to the effective packing dimension Dim ( a , b ), then sp ( L ) contains a unit interval. We also show that, if the dimension dim ( a , b ) is at least one, then sp ( L ) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

Hausdorff dimension estimates for restricted families of projections in [formula omitted

This paper is concerned with restricted families of projections in R3. Let K⊂R3 be a Borel set with Hausdorff dimension dim⁡K=s>1. If G is a smooth and sufficiently well-curved one-dimensional family of two-dimensional subspaces, the main result states that there exists σ(s)>1 such that dim⁡πV(K)≥σ(s) for almost all V∈G. A similar result is obtained for some specific families of one-dimensional subspaces.

Symbolic Dynamics: Entropy = Dimension = Complexity

Let d be a positive integer. Let G be the additive monoid ? d or the additive group ? d . Let A be a finite set of symbols. The shift action of G on A G is given by S g (x)(h) = x(g+h) for all g, h ? G and all x ? A G . A G-subshift is defined to be a nonempty closed set X ⊆ A G such that S g (x)?X for all g ? G and all x ? X. Given a G-subshift X, the topological entropy ent(X) is defined as usual (Ruelle Trans. Am. Math. Soc. 187, 237---251, 1973). The standard metric on A G is defined by ?(x, y) = 2 ? | F n | $2^{-|F_{n}|}$ where n is as large as possible such that x?F n = y?F n . Here F n = {0, 1, ? , n} d if G = ? d , and F n = {?n, ? , ?1, 0, 1, ? , n} d if G = ? d . For any X ⊆ A G the Hausdorff dimension dim(X) and the effective Hausdorff dimension effdim(X) are defined as usual (Hausdorff Math. Ann. 79, 157---179 1919; Reimann 2004; Reimann and Stephan 2005) with respect to the standard metric. It is well known that effdim(X) = sup x?X lim inf n K(x?F n )/|F n | where K denotes Kolmogorov complexity (Downey and Hirschfeldt 2010). If X is a G-subshift, we prove that ent(X) = dim(X) = effdim(X), and ent(X) ? limsup n K(x?F n )/|F n | for all x ? X, and ent(X) = lim n K(x?F n )/|F n | for some x ? X.

Sharp nonremovability examples for Hölder continuous quasiregular mappings in the plane

Let α ∈ 2 (0, 1), K ≥ 1, and \( d = 2\frac{{1 + \alpha K}} {{1 + K}} \). Given a compact set E ⊂ ℂ, it is known that if \( \mathcal{H}^d (E) = 0 \), then E is removable for α-Hölder continuous K-quasiregular mappings in the plane. The sharpness of the index d is shown with the construction, for any t > d, of a set E of Hausdorff dimension dim(E) = t which is not removable. In this paper, we improve this result and construct compact nonremovable sets E such that \( 0 < \mathcal{H}^d (E) < \infty \). For the proof, we give a precise planar K-quasiconformal mapping whose Hölder exponent is strictly bigger than \( \frac{1} {K} \) and which exhibits extremal distortion properties.

Constructive Dimension and Turing Degrees

This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dim H(S) and constructive packing dimension dim P(S) is Turing equivalent to a sequence R with dim H(R)≥(dim H(S)/dim P(S))−ε, for arbitrary ε>0. Furthermore, if dim P(S)>0, then dim P(R)≥1−ε. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension.A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dim H(S)/dim P(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previously-known zero-one law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dim H(S)=dim P(S)) such that dim H(S)>0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1.Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truth-table and bounded Turing reductions differ in their ability to extract dimension.

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin &amp; Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

ON THE MENDÈS FRANCE FRACTAL DIMENSION OF STRANGE ATTRACTORS: THE HÉNON CASE

We consider the Hénon attractor ℋ as a curve limit of continuous planar curves H(n), as n → ∞. We describe a set of tools for studying the Hausdorff dimension dim H of a certain family of such curves, and we adapt these tools to the particular case of the Hénon attractor, estimating its Hausdorff dimension dim H (ℋ) to be about 1.258, a number smaller than the usual estimates for the box-counting dimension of the attractor. We interpret this discrepancy.

Degrees of freedom of a time series

We give a formal proof that if f is a smooth dynamics on a d-dimensional smooth manifold and μ is an ergodic and exact dimensional measure with Hausdorff dimension dim μ>d−1, then the number d of degrees of freedom of the dynamics can be recovered from the observation of an orbit. We implement, with this purpose, an algorithm based on the analysis of the microstructure of μ. We show how a correct estimation of d permits the computation of the Liapunov spectrum with a high accuracy avoiding the issue of the spurious exponents.

Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions

Erdős (1939, 1940) studied the distribution νλ of the random series P∞ 0 ±λn, and showed that νλ is singular for infinitely many λ ∈ (1/2, 1), and absolutely continuous for a.e. λ in a small interval (1 − δ, 1). Solomyak (1995) proved a conjecture made by Garsia (1962) that νλ is absolutely continuous for a.e. λ ∈ (1/2, 1). In order to sharpen this result, we have developed a general method that can be used to estimate the Hausdorff dimension of exceptional parameters in several contexts. In particular, we prove: • For any λ0 > 1/2, the set of λ ∈ [λ0, 1) such that νλ is singular has Hausdorff dimension strictly less than 1. • For any Borel set A ⊂ Rd with Hausdorff dimension dim A > (d + 1)/2, there are points x ∈ A such that the pinned distance set {|x− y| : y ∈ A} has positive Lebesgue measure. Moreover, the set of x where this fails has Hausdorff dimension at most d + 1− dim A. • Let Kλ denote the middle-α Cantor set for α = 1 − 2λ and let K ⊂ R be any compact set. Peres and Solomyak (1998) showed that for a.e. λ ∈ (λ0, 1/2) such that dim K + dimKλ > 1, the sum K + Kλ has positive length; we show that the set of exceptional λ in this statement has Hausdorff dimension at most 2− dim K − dimKλ0 . • For any Borel set E ⊂ Rd with dim E > 2, almost all orthogonal projections of E onto lines through the origin have nonempty interior, and the exceptional set of lines where this fails has dimension at most d + 1− dim E. • If μ is a Borel probability measure on Rd with correlation dimension greater than m + 2γ, then for a “prevalent” set of C1 maps f : Rd → Rm (in the sense described by Hunt, Sauer and Yorke (1992)), the image of μ under f has a density with at least γ fractional derivatives in L2(Rm).