scholarly journals The sum of Rademacher functions and Hausdorff dimension

1990 ◽  
Vol 108 (1) ◽  
pp. 97-103 ◽  
Author(s):  
Tian-You Hu ◽  
Ka-Sing Lau
Keyword(s):  
Hausdorff Dimension ◽  
Rademacher Functions ◽  
Dimension 2 ◽  
Weierstrass Functions

AbstractFor 0 < α < 1, let for 0 ≤ x < 1, where is the sequence of Rademacher functions. We give a class of fα so that their graphs have Hausdorff dimension 2 − α. The result is closely related to the corresponding unsolved question for the Weierstrass functions.

scholarly journals On the Hausdorff dimension of the Julia set of a regularly growing entire function

2010 ◽  
Vol 148 (3) ◽  
pp. 531-551 ◽  
Author(s):  
WALTER BERGWEILER ◽  
BOGUSŁAWA KARPIŃSKA
Keyword(s):  
Entire Function ◽  
Hausdorff Dimension ◽  
Julia Set ◽  
Transcendental Entire Function ◽  
Dimension 2

AbstractWe show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.

scholarly journals On the Hausdorff Dimension of CAT(κ) Surfaces

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
David Constantine ◽  
Jean-François Lafont
Keyword(s):  
Hausdorff Dimension ◽  
Geodesic Flow ◽  
Short Proof ◽  
Closed Surface ◽  
Upper And Lower Bounds ◽  
Two Dimensional ◽  
Dimensional Hausdorff Measure ◽  
Dimension 2 ◽  
Uniformity Condition ◽  
Metric Balls

AbstractWe prove that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally,we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(−1) manifolds.

SUPER BROWNIAN MOTION IS A FRACTAL MEASURE FOR WHICH THE MULTIFRACTAL FORMALISM IS INVALID

Fractals ◽  
1995 ◽  
Vol 03 (04) ◽  
pp. 737-746 ◽  
Author(s):  
S. JAMES TAYLOR
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Positive Constant ◽  
Packing Measure ◽  
Borel Set ◽  
Multifractal Formalism ◽  
Fractal Measure ◽  
Measure Function ◽  
Super Brownian Motion ◽  
Dimension 2

Whenever Xt is the measure value of super Brownian motion in Rd(d≥3), and [Formula: see text]St is the topological support of Xt it is known7 that there is a positive constant c, depending only on d, such that for every Borel set A, [Formula: see text] There is no such exact measure function for packing measure, but it follows from the precise results in Ref. 16 that the packing dimension as well as the Hausdorff dimension of St is 2. This means that Xt is dimension regular with exact dimension 2. We describe some of the key ideas, written up in Ref. 24, which show that a.s. [Formula: see text] while [Formula: see text] is not empty for 2<β<4. Further Aβ is not dimension regular since dim Aβ=[Formula: see text]. For this reason the multifractal formalism used in Ref. 11 or Ref. 9 is invalid because their function τ(q) cannot exist.

Hausdorff dimension 2 for Julia sets of quadratic polynomials

2001 ◽  
Vol 237 (3) ◽  
pp. 571-583 ◽  
Author(s):  
Stefan-M. Heinemann ◽  
Bernd O. Stratmann
Keyword(s):  
Hausdorff Dimension ◽  
Julia Sets ◽  
Quadratic Polynomials ◽  
Dimension 2
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