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

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.

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 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.

scholarly journals DIOPHANTINE APPROXIMATION ON MANIFOLDS AND LOWER BOUNDS FOR HAUSDORFF DIMENSION

Mathematika ◽  
2017 ◽  
Vol 63 (3) ◽  
pp. 762-779
Author(s):  
Victor Beresnevich ◽  
Lawrence Lee ◽  
Robert C. Vaughan ◽  
Sanju Velani
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds ◽  
Diophantine Approximation

scholarly journals A linear stochastic biharmonic heat equation: hitting probabilities

2022 ◽  
Author(s):  
Adrián Hinojosa-Calleja ◽  
Marta Sanz-Solé
Keyword(s):  
Hausdorff Dimension ◽  
Heat Equation ◽  
Random Field ◽  
Lower Bounds ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Upper And Lower Bounds ◽  
Precise Description ◽  
Field Solution ◽  
Hitting Probabilities

AbstractConsider the linear stochastic biharmonic heat equation on a d–dimen- sional torus ($$d=1,2,3$$ d = 1 , 2 , 3 ), driven by a space-time white noise and with periodic boundary conditions: $$\begin{aligned} \left( \frac{\partial }{\partial t}+(-\varDelta )^2\right) v(t,x)= \sigma \dot{W}(t,x),\ (t,x)\in (0,T]\times {\mathbb {T}}^d, \end{aligned}$$ ∂ ∂ t + ( - Δ ) 2 v ( t , x ) = σ W ˙ ( t , x ) , ( t , x ) ∈ ( 0 , T ] × T d , $$v(0,x)=v_0(x)$$ v ( 0 , x ) = v 0 ( x ) . We find the canonical pseudo-distance corresponding to the random field solution, therefore the precise description of the anisotropies of the process. We see that for $$d=2$$ d = 2 , they include a $$z(\log \tfrac{c}{z})^{1/2}$$ z ( log c z ) 1 / 2 term. Consider D independent copies of the random field solution to (0.1). Applying the criteria proved in Hinojosa-Calleja and Sanz-Solé (Stoch PDE Anal Comp 2021. 10.1007/s40072-021-00190-1), we establish upper and lower bounds for the probabilities that the path process hits bounded Borel sets.This yields results on the polarity of sets and on the Hausdorff dimension of the path process.

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