35 research outputs found
Ahlfors-Regular Curves In Metric Spaces
We discuss 1-Ahlfors-regular connected sets in a general metric space and
prove that such sets are `flat' on most scales and in most locations. Our
result is quantitative, and when combined with work of I. Hahlomaa, gives a
characterization of 1-Ahlfors regular subsets of 1-Ahlfors-regular curves in
metric spaces. Our result is a generalization to the metric space setting of
the Analyst's (Geometric) Traveling Salesman theorems of P. Jones, K. Okikiolu,
and G. David and S. Semmes, and it can be stated in terms of average Menger
curvature.Comment: 25 pages, No figures. revisions after referee comments. to appear in
Annales Academae Scientiarum Fennicae Mathematic
Multiscale analysis of 1-rectifiable measures II: characterizations
A measure is 1-rectifiable if there is a countable union of finite length
curves whose complement has zero measure. We characterize 1-rectifiable Radon
measures in -dimensional Euclidean space for all in terms of
positivity of the lower density and finiteness of a geometric square function,
which loosely speaking, records in an gauge the extent to which
admits approximate tangent lines, or has rapidly growing density ratios, along
its support. In contrast with the classical theorems of Besicovitch, Morse and
Randolph, and Moore, we do not assume an a priori relationship between
and 1-dimensional Hausdorff measure. We also characterize purely
1-unrectifiable Radon measures, i.e. locally finite measures that give measure
zero to every finite length curve. Characterizations of this form were
originally conjectured to exist by P. Jones. Along the way, we develop an
variant of P. Jones' traveling salesman construction, which is of independent
interest.Comment: 47 pages, 4 figures (v3: added/updated figures, new Remarks 2.1, 4.6,
5.8, minor improvements, final version
An upper bound for the length of a Traveling Salesman path in the Heisenberg group
We show that a sufficient condition for a subset in the Heisenberg group
(endowed with the Carnot-Carath\'{e}odory metric) to be contained in a
rectifiable curve is that it satisfies a modified analogue of Peter Jones's
geometric lemma. Our estimates improve on those of \cite{FFP}, by replacing the
power of the Jones--number with any power . This complements
(in an open ended way) our work \cite{Li-Schul-beta-leq-length}, where we
showed that such an estimate was necessary, but with .Comment: 19 pages. No figures; V2 several (inconsequential) errors corrected.
V3 minor changes. Accepted to Revista Matem\'atica Iberoamerican
The Analyst's traveling salesman theorem in graph inverse limits
We prove a version of Peter Jones' Analyst's traveling salesman theorem in a
class of highly non-Euclidean metric spaces introduced by Laakso and
generalized by Cheeger-Kleiner. These spaces are constructed as inverse limits
of metric graphs, and include examples which are doubling and have a Poincare
inequality. We show that a set in one of these spaces is contained in a
rectifiable curve if and only if it is quantitatively "flat" at most locations
and scales, where flatness is measured with respect to so-called monotone
geodesics. This provides a first examination of quantitative rectifiability
within these spaces.Comment: 38 page