scholarly journals ON BADLY APPROXIMABLE COMPLEX NUMBERS

2010 ◽  
Vol 52 (2) ◽  
pp. 349-355 ◽  
Author(s):  
R. ESDAHL-SCHOU ◽  
S. KRISTENSEN
Keyword(s):  
Hausdorff Dimension ◽  
Complex Numbers ◽  
Maximal Dimension ◽  
Ring Of Integers ◽  
Fractal Set ◽  
Badly Approximable

AbstractWe show that the set of complex numbers which are badly approximable by ratios of elements of the ring of integers in $\(\mathbb{Q}(\sqrt{-D})\)$, where D ∈ {1, 2, 3, 7, 11, 19, 43, 67, 163} has maximal Hausdorff dimension. In addition, the intersection of these sets is shown to have maximal dimension. The results remain true when the sets in question are intersected with a suitably regular fractal set.

scholarly journals Badly approximable points on self-affine sponges and the lower Assouad dimension

2017 ◽  
Vol 39 (3) ◽  
pp. 638-657 ◽  
Author(s):  
TUSHAR DAS ◽  
LIOR FISHMAN ◽  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Assouad Dimension ◽  
Badly Approximable Points ◽  
Bounded Below ◽  
Badly Approximable ◽  
Sierpinski Sponges

We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. For self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford–McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

DYNAMICS AND BIFURCATIONS OF A FAMILY OF RATIONAL MAPS WITH PARABOLIC FIXED POINTS

2011 ◽  
Vol 21 (11) ◽  
pp. 3323-3339
Author(s):  
RIKA HAGIHARA ◽  
JANE HAWKINS
Keyword(s):  
Hausdorff Dimension ◽  
Fixed Points ◽  
Riemann Sphere ◽  
Julia Set ◽  
Critical Orbit ◽  
Rational Maps ◽  
Complex Numbers ◽  
Period 2 ◽  
The Family ◽  
Dynamical Properties

We study a family of rational maps of the Riemann sphere with the property that each map has two fixed points with multiplier -1; moreover, each map has no period 2 orbits. The family we analyze is Ra(z) = (z3 - z)/(-z2 + az + 1), where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).

Relative Galois module structure of rings of integers and elliptic functions

1983 ◽  
Vol 94 (3) ◽  
pp. 389-397 ◽  
Author(s):  
M. J. Taylor
Keyword(s):  
Number Field ◽  
Finite Extension ◽  
Elliptic Functions ◽  
Complex Numbers ◽  
Module Structure ◽  
Galois Module ◽  
Integral Ideal ◽  
Ring Of Integers ◽  
Rings Of Integers ◽  
Imaginary Number

Let K be a quadratic imaginary number field with discriminant less than −4. For N either a number field or a finite extension of the p-adic field p, we let N denote the ring of integers of N. Moreover, if N is a number field then we write for the integral closure of [½] in N. For an integral ideal & of K we denote the ray classfield of K with conductor & by K(&). Once and for all we fix a choice of embedding of K into the complex numbers .

scholarly journals Good’s theorem for Hurwitz continued fractions

2020 ◽  
Vol 16 (07) ◽  
pp. 1433-1447
Author(s):  
Gerardo Gonzalez Robert
Keyword(s):  
Hausdorff Dimension ◽  
Complex Plane ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Analogous Result ◽  
Complex Numbers ◽  
Real Numbers ◽  
Dimension Formula

Good’s Theorem for regular continued fraction states that the set of real numbers [Formula: see text] such that [Formula: see text] has Hausdorff dimension [Formula: see text]. We show an analogous result for the complex plane and Hurwitz Continued Fractions: the set of complex numbers whose Hurwitz Continued fraction [Formula: see text] satisfies [Formula: see text] has Hausdorff dimension [Formula: see text], half of the ambient space’s dimension.

Variation of topological pressure and dimension: from polynomials to complex Hénon maps

2013 ◽  
Vol 34 (3) ◽  
pp. 1018-1036
Author(s):  
CHRISTIAN WOLF
Keyword(s):  
Hausdorff Dimension ◽  
Julia Set ◽  
Topological Pressure ◽  
Dimension Theory ◽  
Maximal Dimension ◽  
Pressure Function ◽  
Small Perturbations ◽  
Henon Maps ◽  
Hénon Maps ◽  
Regularity Results

AbstractWe study the topological pressure and dimension theory of complex Hénon maps which are small perturbations of one-dimensional polynomials. In particular, we derive regularity results for the generalized pressure function in a neighborhood of the degenerate map (i.e. the polynomial). This unifies results concerning the regularity of the pressure function for polynomials by Ruelle and for complex Hénon maps by Verjovsky and Wu. We then apply this regularity to show that the Hausdorff dimension of the Julia set is a continuous non-differentiable function in a neighborhood of the polynomial. Furthermore, we establish uniqueness of the measure of maximal dimension and show that the Hausdorff dimension of the Julia set of a complex Hénon map is discontinuous at the boundary of the hyperbolicity locus.

scholarly journals Dimension estimates for sets of uniformly badly approximable systems of linear forms

2015 ◽  
Vol 11 (07) ◽  
pp. 2037-2054 ◽  
Author(s):  
Ryan Broderick ◽  
Dmitry Kleinbock
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds ◽  
Higher Dimensions ◽  
Upper And Lower Bounds ◽  
Homogeneous Dynamics ◽  
Linear Forms ◽  
One Dimensional ◽  
Approximation Constant ◽  
The One ◽  
Badly Approximable

The set of badly approximable m × n matrices is known to have Hausdorff dimension mn. Each such matrix comes with its own approximation constant c, and one can ask for the dimension of the set of badly approximable matrices with approximation constant greater than or equal to some fixed c. In the one-dimensional case, a very precise answer to this question is known. In this note, we obtain upper and lower bounds in higher dimensions. The lower bounds are established via the technique of Schmidt games, while for the upper bound we use homogeneous dynamics methods, namely exponential mixing of flows on the space of lattices.

Random fractals and their intersection with winning sets

2021 ◽  
pp. 1-30
Author(s):  
YIFTACH DAYAN
Keyword(s):  
Hausdorff Dimension ◽  
General Result ◽  
Countable Collection ◽  
Percolation Process ◽  
Fractal Percolation ◽  
Random Fractals ◽  
Affine Hyperplane ◽  
Badly Approximable

Abstract We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realisation of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$ ), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$ , $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$ , where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$ . We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$ . We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.

HAUSDORFF DIMENSION OF CERTAIN RANDOM SELF-AFFINE FRACTALS

2011 ◽  
Vol 11 (04) ◽  
pp. 627-642 ◽  
Author(s):  
NUNO LUZIA
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Finite Number ◽  
The Self ◽  
Affine Transformations ◽  
Fractal Set ◽  
Bernoulli Measure ◽  
Sierpinski Carpets

In this work we are interested in the self-affine fractals studied by Gatzouras and Lalley [5] and by the author [11] who generalize the famous general Sierpinski carpets studied by Bedford [1] and McMullen [13]. We give a formula for the Hausdorff dimension of sets which are randomly generated using a finite number of self-affine transformations each generating a fractal set as mentioned before. The choice of the transformation is random according to a Bernoulli measure. The formula is given in terms of the variational principle for the dimension.

scholarly journals Schmidt’s game, fractals, and orbits of toral endomorphisms

2010 ◽  
Vol 31 (4) ◽  
pp. 1095-1107 ◽  
Author(s):  
RYAN BRODERICK ◽  
LIOR FISHMAN ◽  
DMITRY KLEINBOCK
Keyword(s):  
Hausdorff Dimension ◽  
Recent Result ◽  
Lacunary Sequence ◽  
Alternative Proof ◽  
Integer Matrix ◽  
Image Position ◽  
Schmidt’S Game ◽  
Badly Approximable

AbstractGiven an integer matrix M∈GLn(ℝ) and a point y∈ℝn/ℤn, consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y∈ℚn/ℤn, the set $ \tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M∈GLn(ℝ)∩Mn×n(ℤ) and y∈ℝn/ℤn, and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m×n matrices. Furthermore, we show that sets of the form $ \tilde E(M,y)$ and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝn. As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint, arXiv:0912.2445] on badly approximable systems of affine forms.

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