TDA homeworks

question 1: Peano squares:

• Construct an onto (surjective) mapping from $I^2$ to $I^4$ (here $I=[0,1]$ is the unit interval).
• Can it be one-to-one?
• Is it Hölder? what is its exponent?
• Is it the best possible?
• What is the best possible (the higher the better) Hölder exponent you can achieve for an onto mapping $I^2\to I^3$?

question 2: $G(4,2)$

Consider the space $X$ of straight lines in $\Real^3$.

• Is it a manifold? if so, present charts that cover it.
• Is it a compact space?
• If $L$ is a smooth loop in $\Real^3$ (i.e., the image of the embedding $\lambda:S^1\to \Real^3$), what is the dimension of the set of lines intersecting it?
• What maximal number of generically embedded loops in $\Real^3$ that can be intersected by a straight line?

question 3: Takens’ embeddings

Let $X\cong S^1$ with coordinate $x\in \Real/2\pi\Int$, and $\phi:x\mapsto x+\alpha+\beta\sin(x)$. Take some generic $h(x)$ (e.g., a trigonometric polynomial $h(x)=\sum_k a_k\cos(kx++b_k\sin(kx))$ and construct Takens’ map $X\to\Real^3$. Write (use any tools!) code generating this embedding, and produce pictures of it for different $\alpha,\beta$.
Comment on what transpires.