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Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
8
votes
Thurston-Cannon $S^2$-filling curves
There are multiple reasons that the Cannon-Thurston curves are related to the Peano curve. For instance, these are all continuous surjective paths onto 2-dimensional objects: for the original Peano cu …
4
votes
A question about a continuous curve in $\mathbb{R}^2$
Yes, this is true, and it is an application of compactness.
For the proof, first consider the continuous $\mathbb{R}^2$-valued function
$$F(t,t') = f(t')-f(t)
$$
defined on the compact set
$$D = \{( …
1
vote
A Fractional Linear Transformation Class Property
I'm going to answer the question under the assumption that the domain is the extended complex plane and the maps are holomorphic bijections. This simplifies the question considerably, because these ma …