scholarly journals Lower bounds for the Hausdorff dimension of the geometric Lorenz attractor: The homoclinic case

2008 ◽  
Vol 22 (3) ◽  
pp. 699-709 ◽  
Author(s):  
Cristina Lizana ◽  
◽  
Leonardo Mora
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds ◽  
Lorenz Attractor ◽  
Geometric Lorenz Attractor

scholarly journals Mass Transference Principle: From Balls to Arbitrary Shapes

2020 ◽  
Author(s):  
Henna Koivusalo ◽  
Michał Rams
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds ◽  
Arbitrary Shape ◽  
Strong Result ◽  
Transference Principle ◽  
Arbitrary Shapes

Abstract The mass transference principle, proved by Beresnevich and Velani in 2006, is a strong result that gives lower bounds for the Hausdorff dimension of limsup sets of balls. We present a version for limsup sets of open sets of arbitrary shape.

A degenerate singularity generating geometric Lorenz attractors

1995 ◽  
Vol 15 (5) ◽  
pp. 833-856 ◽  
Author(s):  
Freddy Dumortier ◽  
Hiroshi Kokubu ◽  
Hiroe Oka
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Lorenz Attractor ◽  
Degenerate Singularity ◽  
Field Singularity ◽  
Lorenz Attractors ◽  
Geometric Lorenz Attractor

AbstractA degenerate vector field singularity in R3 can generate a geometric Lorenz attractor in an arbitrarily small unfolding of it. This enables us to detect Lorenz-like chaos in some families of vector fields, merely by performing normal form calculations of order 3.

scholarly journals Lorenz knots are prime

1984 ◽  
Vol 4 (1) ◽  
pp. 147-163 ◽  
Author(s):  
R. F. Williams
Keyword(s):  
Differential Equation ◽  
Periodic Orbits ◽  
Lorenz Attractor ◽  
The Real ◽  
Geometric Lorenz Attractor

AbstractLorenz knots are the periodic orbits of a certain geometrically defined differential equation in ℝ3. This is called the ‘geometric Lorenz attractor’ as it is only conjecturally the real Lorenz attractor. These knots have been studied by the author and Joan Birman via a ‘knot-holder’, i.e. a certain branched two-manifold H. To show such knots are prime we suppose the contrary which implies the existence of a splitting sphere, S2. The technique of the proof is to study the intersection S2∩H. A novelty here is that S2∩H is likewise branched.

Lower bounds for the Hausdorff dimension of attractors

1995 ◽  
Vol 7 (3) ◽  
pp. 457-469 ◽  
Author(s):  
Michael Y. Li ◽  
James S. Muldowney
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds

scholarly journals On the C 1 robust transitivity of the geometric Lorenz attractor

2017 ◽  
Vol 262 (12) ◽  
pp. 5928-5938
Author(s):  
J. Carmona ◽  
D. Carrasco-Olivera ◽  
B. San Martín
Keyword(s):  
Lorenz Attractor ◽  
Geometric Lorenz Attractor

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.

scholarly journals Dynamics of Newton's functions of Barna's polynomials

2002 ◽  
Vol 29 (2) ◽  
pp. 79-84
Author(s):  
Piyapong Niamsup
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Real Roots ◽  
Exceptional Sets ◽  
Real Polynomials

We define Barna's polynomials as real polynomials with all real roots of which at least four are distinct. In this paper, we study the dynamics of Newton's functions of such polynomials. We also give the upper and lower bounds of the Hausdorff dimension of exceptional sets of these Newton's functions.

INVARIANT SUBSETS OF VOLUME HYPERBOLIC SETS

2005 ◽  
Vol 07 (06) ◽  
pp. 839-848
Author(s):  
I. W. AGUILAR ◽  
E. H. APAZA ◽  
C. A. MORALES
Keyword(s):  
Three Dimensional ◽  
Closed Surface ◽  
Invariant Set ◽  
Lorenz Attractor ◽  
Invariant Subset ◽  
Dominated Splitting ◽  
Three Dimensional Flow ◽  
Hyperbolic Set ◽  
Hyperbolic Sets ◽  
Geometric Lorenz Attractor

A volume hyperbolic set is a compact invariant set with a dominated splitting whose external bundles uniformly contract and expand the volume respectively [1]. Examples of volume hyperbolic sets for diffeomorphisms or flows are the hyperbolic sets, the geometric Lorenz attractor [3] and the singular horseshoe [6]. We shall prove that no invariant subset of a volume hyperbolic set of a three-dimensional flow is homeomorphic to a closed surface.

Lower bounds of the Hausdorff dimension for the images of Feller processes

2015 ◽  
Vol 97 ◽  
pp. 222-228 ◽  
Author(s):  
V. Knopova ◽  
R.L. Schilling ◽  
J. Wang
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds
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