Search results for "Jordan curve"
showing 6 items of 6 documents
A proof of Carleson's 𝜀2-conjecture
2021
In this paper we provide a proof of the Carleson 𝜀2-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson 𝜀2-square function. peerReviewed
Invariant Jordan curves of Sierpinski carpet rational maps
2015
In this paper, we prove that if $R\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpi\'nski carpet, then there is an integer $n_0$, such that, for any $n\ge n_0$, there exists an $R^n$-invariant Jordan curve $\Gamma$ containing the postcritical set of $R$.
A proof of Carleson's $\varepsilon^2$-conjecture
2019
In this paper we provide a proof of the Carleson $\varepsilon^2$-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson $\varepsilon^2$-square function.
A continuous circle of pseudo-arcs filling up the annulus
1999
We prove an early announcement by Knaster on a decomposition of the plane. Then we establish an announcement by Anderson saying that the plane annulus admits a continlous decomposition into pseudo-arcs such that the quotient space is a simple closed curve. This provides a new plane curve, "a selectible circle of pseudo-aics", and answers some questions of Lewis.
Homogeneous Suslinian Continua
2011
AbstractA continuumis said to be Suslinian if it does not contain uncountably many mutually exclusive non-degenerate subcontinua. Fitzpatrick and Lelek have shown that a metric Suslinian continuum X has the property that the set of points at which X is connected im kleinen is dense in X. We extend their result to Hausdorff Suslinian continua and obtain a number of corollaries. In particular, we prove that a homogeneous, non-degenerate, Suslinian continuum is a simple closed curve and that each separable, non-degenerate, homogenous, Suslinian continuum is metrizable.
Curves as measured foliation on noncompact surfaces
1993
In the present work, that regards the Thurston's theory, we prove that, if we choose a closed curve, how we wish, on a noncompact surface, it is always possible to construct a particular masured foliation that has the choosed curve like a leaf; we also prove this foliation has a remarkable property that makes very easy to mesure all homotopy classes of closed curves of our surface. To prove this statement we need some Propositions and some Lemma that we also demonstre.