Search results for "Hausdorff measure"

Menger curvature and rectifiability in metric spaces

2008

We show that for any metric space $X$ the condition \[ \int_X\int_X\int_X c(z_1,z_2,z_3)^2\, d\Hm z_1\, d\Hm z_2\, d\Hm z_3 < \infty, \] where $c(z_1,z_2,z_3)$ is the Menger curvature of the triple $(z_1,z_2,z_3)$, guarantees that $X$ is rectifiable.

Planar Sobolev homeomorphisms and Hausdorff dimension distortion

2011

We investigate how planar Sobolev-Orlicz homeomorphisms map sets of Hausdorff dimension less than two. With the correct gauge functions the generalized Hausdorff measures of the image sets are shown to be zero.

Dimension gap under Sobolev mappings

2015

Abstract We prove an essentially sharp estimate in terms of generalized Hausdorff measures for the images of boundaries of Holder domains under continuous Sobolev mappings, satisfying suitable Orlicz–Sobolev conditions. This estimate marks a dimension gap, which was first observed in [2] for conformal mappings.

Separation properties of (n, m)-IFS attractors

2017

Abstract The separation properties of self similar sets are discussed in this article. An open set condition for the (n, m)- iterated function system is introduced and the concepts of self similarity, similarity dimension and Hausdorff dimension of the attractor generated by an (n, m) - iterated function system are studied. It is proved that the similarity dimension and the Hausdorff dimension of the attractor of an (n, m) - iterated function system are equal under this open set condition. Further a necessary and sufficient condition for a set to satisfy the open set condition is established.

On boundaries of attractors in dynamical systems

2021

Abstract Fractal geometry is one of the beautiful and challenging branches of mathematics. Self similarity is an important property, exhibited by most of the fractals. Several forms of self similarity have been discussed in the literature. Iterated Function System (IFS) is a mathematical scheme to generate fractals. There are several variants of IFSs such as condensation IFS, countable IFS, etc. In this paper, certain properties of self similar sets, using the concept of boundary are discussed. The notion of boundaries like similarity boundary and dynamical boundary are extended to condensation IFSs. The relationships and measure theoretic properties of boundaries in dynamical systems are a…

2020

Abstract This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a C ∞ -hypersurface S without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure H 12 . As a consequence, we show that any Lipschitz map defined on a subset of a Carnot group of Hausdorf…

Weakly controlled Moran constructions and iterated functions systems in metric spaces

2011

We study the Hausdorff measures of limit sets of weakly controlled Moran constructions in metric spaces. The separation of the construction pieces is closely related to the Hausdorff measure of the corresponding limit set. In particular, we investigate different separation conditions for semiconformal iterated function systems. Our work generalizes well known results on self-similar sets in metric spaces as well as results on controlled Moran constructions in Euclidean spaces.

Tangential behavior of functions and conical densities of Hausdorff measures.

2005

We construct a $C^1$-function $f\colon [0,1]\to \mathbb{R}$ such that for almost all $x\in(0,1)$, there is $r>0$ for which $f(y)>f(x)+f'(x)(y-x)$ when $y\in(x,x+r)$ and $f(y)< f(x)+f'(x)(y-x)$ when $y\in(x-r,x)$. The existence of such functions is related to a problem concerning conical density properties of Hausdorff measures on $\mathbb{R}^n$. We also discuss the tangential behavior of typical $C^1$-functions, using an improvement of Jarnik's theorem on essential derived numbers

Notions of Dirichlet problem for functions of least gradient in metric measure spaces

2019

We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a (1, 1)-Poincaré inequality. Since one of the two notions is not amenable to the direct method of the calculus of variations, we construct, based on an approach of Juutinen and Mazón-Rossi–De León, solutions by considering the Dirichlet problem for p-harmonic functions, p>1, and letting p→1. Tools developed and used in this paper include the inner perimeter measure of a domain. Peer reviewed

Sharp estimate on the inner distance in planar domains

2020

We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlev\'e length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painlev\'e length bound $\kappa(E) \le\pi \mathcal{H}^1(E)$ is sharp.