RdL

Quasiperiodic Topology

See the Gallery for high-resolution versions of the published pictures and many other pictures and videos I could not include in the publications.
Keywords: Quasiperiodicity, Closed 1-forms, Foliations, Poisson Dynamics, Multivalued functions, Multivalued Morse Theory, Low-dimensional Topology

The study of quasiperiodic functions in more than one variable was started by S.P. Novikov in connection with his works on Solitons theory and on multivalued Morse theory.

Definition. Let $\pi_n:\mathbb R^n\to\mathbb T^n$ be the canonical projection of the Euclidean $n$-space into the $n$-Torus. We say that a function $$f:\mathbb R^k\to\mathbb R$$ is quasiperiodic with $n$ quasiperiods if there are ia a map $F:\mathbb T^n\to\mathbb R$ and an affine embedding $\psi:\mathbb R^k\to\mathbb R^n$ such that $$ f = F\circ\pi_n\circ\psi $$ and there is no smaller $n$ for which this can be done.

Roughly speaking, a quasiperiodic map in $k$ variables with $n$ quasiperiods is obtained by setting $n-k$ affine relations among the $n$ variables of a $n$-periodic function.

Example 1 The function $f(x) = \cos(2\pi x)+\cos(\sqrt{2}\,2\pi x)$ is a quasiperiodic function in 1 variable with 2 quasiperiods. In this case $F(x,y)=\cos(2\pi x)+\cos(2\pi y)$ and $\psi(x)=(x,\sqrt{2}\,x)$.

Example 2 The function $f(x,y) = \cos(2\pi x)+\cos(2\pi y)+\cos(2\pi(\sqrt{2}\, x+\sqrt{3}\, x-1))$ is a quasiperiodic function in 2 variables with 3 quasiperiodis. In this case $F(x,y,z)=\cos(2\pi x)+\cos(2\pi y)+\cos(2\pi z)$ and $\psi(x,y)=(x,y,\sqrt{2}\, x+\sqrt{3}\, x-1)$.

One of the main problems of this field is: study the topology of the level sets of a quasiperiodic function.

So far, I have been studying the simplest non-trivial case: level sets of quasiperiodic functions in 2 variables with 3 quasiperiods. This is equivalent to studying the topology of plnar sections of 3ply periodic surfaces. See the Gallery for a list of most cases studied numerically so far.

For more results and examples, see

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