Hausdorff dimension of attractors for two dimensional Lorenz transformations

2000 ◽  
Vol 116 (1) ◽  
pp. 253-269
Author(s):  
Thomas Steinberger
Keyword(s):  
Hausdorff Dimension ◽  
Two Dimensional

scholarly journals Hausdorff dimension and uniform exponents in dimension two

2018 ◽  
Vol 167 (02) ◽  
pp. 249-284 ◽  
Author(s):  
YANN BUGEAUD ◽  
YITWAH CHEUNG ◽  
NICOLAS CHEVALLIER
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Packing Dimension ◽  
Two Dimensional ◽  
Joint Work ◽  
Unpublished Work

AbstractIn this paper we prove that the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponentμin (1/2, 1) is equal to 2(1 −μ) forμ⩾$\sqrt2/2$, whereas forμ<$\sqrt2/2$it is greater than 2(1 −μ) and at most equal to (3 − 2μ)(1 − μ)/(1 −μ+μ2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) whenμtends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. Moreover, we prove a lower bound for the packing dimension, which appears to be strictly greater than the Hausdorff dimension for μ ⩾ 0.565. . . .

scholarly journals On the notion of the dimension with respect to a dynamical system

1984 ◽  
Vol 4 (3) ◽  
pp. 405-420 ◽  
Author(s):  
Ya. B. Pesin
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Invariant Sets ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Common Features ◽  
Dynamical Properties ◽  
The One ◽  
Hyperbolic Sets

AbstractFor the invariant sets of dynamical systems a new notion of dimension-the so-called dimension with respect to a dynamical system-is introduced. It has some common features with the general topological notion of the dimension, but it also reflects the dynamical properties of the system. In the one-dimensional case it coincides with the Hausdorff dimension. For multi-dimensional hyperbolic sets formulae for the calculation of our dimension are obtained. These results are generalizations of Manning's results obtained by him for the Hausdorff dimension in the two-dimensional case.

scholarly journals The Hausdorff dimension of surfaces in two-dimensional quantum gravity coupled to Ising minimal matter

1997 ◽  
Vol 403 (3-4) ◽  
pp. 197-202 ◽  
Author(s):  
Mark J. Bowick ◽  
Varghese John ◽  
Gudmar Thorleifsson
Keyword(s):  
Hausdorff Dimension ◽  
Quantum Gravity ◽  
Two Dimensional

Identifications and Dimension of the Rauzy Fractal

Fractals ◽  
1997 ◽  
Vol 05 (02) ◽  
pp. 281-294 ◽  
Author(s):  
Víctor F. Sirvent
Keyword(s):  
Hausdorff Dimension ◽  
Fundamental Domain ◽  
Two Dimensional ◽  
Dimensional Torus

In this paper we describe explicitly the identifications on the boundary of the Rauzy fractal that makes it a fundamental domain of the two-dimensional torus, for the action of the lattice Z2 on the plane. Using these identifications, we give a new way to compute the Hausdorff dimension of the Rauzy fractal. Also, we compute the Hausdorff dimension of the realization of the boundary of the Rauzy fractal on the interval.

Diophantine approximation and Hausdorff dimension in Fuchsian groups

1993 ◽  
Vol 113 (2) ◽  
pp. 343-354 ◽  
Author(s):  
S. L. Velani
Keyword(s):  
Hausdorff Dimension ◽  
Unit Circle ◽  
Hyperbolic Space ◽  
Diophantine Approximation ◽  
Fuchsian Groups ◽  
Two Dimensional ◽  
Disc Model ◽  
Image Position ◽  
Straight Lines

The Poincaré disc modelof two-dimensional hyperbolic space supports a metric ρ derived from the differentialGeodesics for the metric ρ are arcs of circles orthogonal to the unit circle S, and straight lines through the origin.

THE HAUSDORFF DIMENSION OF ATTRACTORS APPEARING BY SADDLE-NODE BIFURCATIONS

1991 ◽  
Vol 01 (02) ◽  
pp. 309-325 ◽  
Author(s):  
V. S. AFRAIMOVICH ◽  
M. A. SHERESHEVSKY
Keyword(s):  
Growth Rate ◽  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Strange Attractors ◽  
Asymptotic Formulas ◽  
Two Dimensional ◽  
Saddle Node

We consider the strange attractors which appear as a result of saddle-node vanishing bifurcations in two-dimensional, smooth dynamical systems. Some estimates and asymptotic formulas for the Hausdorff dimension of such attractors are obtained. The estimates demonstrate a dependence of the dimension growth rate after the bifurcation upon the "pre-bifurcational" picture.

SUPER-LIOUVILLE THEORY AS A TWO-DIMENSIONAL, SUPERCONFORMAL SUPERGRAVITY THEORY

1990 ◽  
Vol 05 (02) ◽  
pp. 391-414 ◽  
Author(s):  
JACQUES DISTLER ◽  
ZVONIMIR HLOUSEK ◽  
HIKARU KAWAI
Keyword(s):  
String Theory ◽  
Hausdorff Dimension ◽  
Critical Exponents ◽  
Liouville Theory ◽  
Two Dimensional ◽  
Random Surfaces ◽  
Supergravity Theory ◽  
Arbitrary Topology ◽  
Bosonic Case

In this paper we extend our previous results on the bosonic Liouville theory, to the supersymmetric case. As in the bosonic case, we find that the quantization of the N=1 theory is limited to the region D≤1. We compute the exact critical exponents and the analogue of the Hausdorff dimension of super random surfaces. Our procedure is manifestly covariant and our results hold for the surface of arbitrary topology. We also examine the N=2, O(2) string theory and find that it appears to be well-defined for all D.

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.

On the Hausdorff dimension of Brownian cone points

1985 ◽  
Vol 98 (2) ◽  
pp. 343-353 ◽  
Author(s):  
Steven N. Evans
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Two Dimensional ◽  
Image Position

Let B(t) be a two-dimensional Brownian motion. For 0 < α < 2π, setand, for 0 ≥ β< 2π, let F(α,β) be F(α) rotated through an angle β about the origin.

The Hausdorff dimension of the two-dimensional Edwards' random walk

1989 ◽  
Vol 22 (9) ◽  
pp. 1385-1391 ◽  
Author(s):  
F Koukiou ◽  
J Pasche ◽  
D Petritis
Keyword(s):  
Random Walk ◽  
Hausdorff Dimension ◽  
Two Dimensional
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