Papers by Christopher Bishop
arXiv (Cornell University), Jul 15, 2020
For any δ > 0 we construct an entire function f with three singular values whose Julia set has Ha... more For any δ > 0 we construct an entire function f with three singular values whose Julia set has Hausdorff dimension at most 1 + δ. Stallard proved that the dimension must be strictly larger than 1 whenever f has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known.
arXiv (Cornell University), Jan 7, 2023
We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing... more We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function f on a compact set K, the critical points of our approximants may be taken to lie in any given domain containing K, and all the critical values in any given neighborhood of the polynomially convex hull of f (K).
This is a mathematically rigorous introduction to fractals which emphasizes examples and fundamen... more This is a mathematically rigorous introduction to fractals which emphasizes examples and fundamental ideas. Building up from basic techniques of geometric measure theory and probability, central topics such as Hausdorff dimension, self-similar sets and Brownian motion are introduced, as are more specialized topics, including Kakeya sets, capacity, percolation on trees and the traveling salesman theorem. The broad range of techniques presented enables key ideas to be highlighted, without the distraction of excessive technicalities. The authors incorporate some novel proofs which are simpler than those available elsewhere. Where possible, chapters are designed to be read independently so the book can be used to teach a variety of courses, with the clear structure offering students an accessible route into the topic.
Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2022
For any simple polygon P we compute the optimal upper and lower angle bounds for triangulating P ... more For any simple polygon P we compute the optimal upper and lower angle bounds for triangulating P with Steiner points, and show that these bounds can be attained (except in one special case). The sharp angle bounds for an N-gon are computable in time O(N), even though the number of triangles needed to attain these bounds has no bound in terms of N alone. In general, the sharp upper and lower bounds cannot both be attained by a single triangulation, although this does happen in some cases. For example, we show that any polygon with minimal interior angle θ has a triangulation with all angles in the interval I = [θ, 90 • − min(36 • , θ)/2], and for θ ≤ 36 • both bounds are best possible. Surprisingly, we prove the optimal angle bounds for polygonal triangulations are the same as for triangular dissections. The proof of this verifies, in a stronger form, a 1984 conjecture of Gerver.
Proceedings of the American Mathematical Society
We show that the self-similar set known as the "antenna set" has the property that inf f dim(f (X... more We show that the self-similar set known as the "antenna set" has the property that inf f dim(f (X)) = 1 (where the infimum is over all quasiconformal mappings of the plane), but that this infimum is not attained by any quasiconformal map; indeed, is not attained for any quasisymmetric map into any metric space.
Abstract. We show that for any K-quasiconformal map of the upper half plane to itself and any &qu... more Abstract. We show that for any K-quasiconformal map of the upper half plane to itself and any "> 0, there is a (K + ")-quasiconformal map of the half plane with the same boundary values which is also biLipschitz with respect to the hyperbolic metric. 1.
Transactions of the American Mathematical Society, 1996
We show that for any analytic set A A in R d \mathbf {R}^d , its packing dimension dim P ( A ) ... more We show that for any analytic set A A in R d \mathbf {R}^d , its packing dimension dim P ( A ) \dim _{\mathrm {P}}(A) can be represented as sup B { dim H ( A × B ) − dim H ( B ) } , \; \sup _B \{ \dim _{\mathrm {H}} (A \times B) -\dim _{\mathrm {H}}(B) \} \, , \, where the supremum is over all compact sets B B in R d \mathbf {R}^d , and dim H \dim _{\mathrm {H}} denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if dim P ( A ) > d \dim _{\mathrm {P}} (A) > d . In contrast, we show that the dual quantity inf B { dim P ( A × B ) − dim P ( B ) } , \; \inf _B \{ \dim _{\mathrm {P}}(A \times B) -\dim _{\mathrm {P}}(B) \} \, , \, is at least the “lower packing dimension” of A A , but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)
Proceedings of the American Mathematical Society, Series B, 2022
We show that if γ is a curve in the unit disk, then arclength on γ is a Carleson measure iff the ... more We show that if γ is a curve in the unit disk, then arclength on γ is a Carleson measure iff the image of γ has finite length under every conformal map of the disk onto a bounded domain with a rectifiable boundary.
Proceedings of the American Mathematical Society, 1996
We show that a function f f on the unit disk extends continuously to M \mathcal M , the maximal i... more We show that a function f f on the unit disk extends continuously to M \mathcal M , the maximal ideal space of H ∞ ( D ) H^\infty (\mathbb D) iff it is uniformly continuous (in the hyperbolic metric) and close to constant on the complementary components of some Carleson contour.
Proceedings of the American Mathematical Society, 1999
There is a quasiconformal mapping f f of R 3 {\Bbb R}^3 to itself such that the image of R 2 × { ... more There is a quasiconformal mapping f f of R 3 {\Bbb R}^3 to itself such that the image of R 2 × { 0 } {\Bbb R}^2 \times \{0\} contains no rectifiable curves.
Discrete & Computational Geometry, 2016
We show that any planar PSLG with n vertices has a conforming triangulation by O(n 2.5) nonobtuse... more We show that any planar PSLG with n vertices has a conforming triangulation by O(n 2.5) nonobtuse triangles, answering the question of whether a polynomial bound exists. The triangles may be chosen to be all acute or all right. A nonobtuse triangulation is Delaunay, so this result improves a previous O(n 3) bound of Eldesbrunner and Tan for conforming Delaunay triangulations. In the special case that the PSLG is the triangulation of a simple polygon, we will show that only O(n 2) elements are needed, improving an O(n 4) bound of Bern and Eppstein. We also show that for any ǫ > 0, every PSLG has a conforming triangulation with O(n 2 /ǫ 2) elements and with all angles bounded above by 90 • + ǫ. This improves a result of S. Mitchell when ǫ = 3 8 π = 67.5 • and Tan when ǫ = 7 30 π = 42 • .
Discrete & Computational Geometry, 2016
We prove that every planar straight line graph with n vertices has a conforming quadrilateral mes... more We prove that every planar straight line graph with n vertices has a conforming quadrilateral mesh with O(n 2) elements, all angles ≤ 120 • and all new angles ≥ 60 •. Both the complexity and the angle bounds are sharp.
Pacific Journal of Mathematics, 1996
We refine a theorem of Beardon and Maskit by showing that a Kleinian group is geometrically finit... more We refine a theorem of Beardon and Maskit by showing that a Kleinian group is geometrically finite if and only if its limit set consists entirely of conical limit points and parabolic fixed points.
Revista Matemática Iberoamericana, 2015
We apply a recent result of the first author to prove the following result: any continuum in the ... more We apply a recent result of the first author to prove the following result: any continuum in the plane can be approximated arbitrarily closely in the Hausdorff topology by the Julia set of a postcritically finite polynomial with two finite postcritical points.
Pacific Journal of Mathematics, 1989
Let Ωi and Ω 2 be two disjoint, simply connected domains in the plane, and let ω { and ω 2 be har... more Let Ωi and Ω 2 be two disjoint, simply connected domains in the plane, and let ω { and ω 2 be harmonic measures associated to Ωi and Ω 2. We present necessary and sufficient conditions for a>\ and ωi to be mutually singular.
Pacific Journal of Mathematics, 2005
We construct examples of H ∞ functions f on the unit disk such that the push-forward of Lebesgue ... more We construct examples of H ∞ functions f on the unit disk such that the push-forward of Lebesgue measure on the circle is a radially symmetric measure µ f in the plane, and we characterize which symmetric measures can occur in this way. Such functions have the property that { f n } is orthogonal in H 2 , and provide counterexamples to a conjecture of W. Rudin, independently disproved by Carl Sundberg. Among the consequences is that there is an f in the unit ball of H ∞ such that the corresponding composition operator maps the Bergman space isometrically into a closed subspace of the Hardy space.
Journal of Functional Analysis, 1997
Consider a planar Brownian motion run for finite time. The frontier or "outer boundary" of the pa... more Consider a planar Brownian motion run for finite time. The frontier or "outer boundary" of the path is the boundary of the unbounded component of the complement. Burdzy (1989) showed that the frontier has infinite length. We improve this by showing that the Hausdorff dimension of the frontier is strictly greater than 1. (It has been conjectured that the Brownian frontier has dimension 4/3, but this is still open.) The proof uses Jones's Traveling Salesman Theorem and a self-similar tiling of the plane by fractal tiles known as Gosper Islands.
International Journal of Mathematics and Mathematical Sciences, 2003
We study the conformality problems associated with quasiregular mappings in space. Our approach i... more We study the conformality problems associated with quasiregular mappings in space. Our approach is based on the concept of the infinitesimal space and some new Grötzsch-Teichmüller type modulus estimates that are expressed in terms of the mean value of the dilatation coefficients.
Bulletin of the American Mathematical Society, 1991
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Papers by Christopher Bishop