2 problems

Let $\mathcal{D}$ be the set of circle local homeomorphisms $f:\mathbb{S}^1\to\mathbb{S}^1$ that are topologically conjugate to the doubling map. For $f\in\mathcal{D}$, let $p$ be…

Let $n$ be a positive integer, and consider the $n\times n$ grid. Suppose it is covered by $m$ circles, with no two circles having equal radius. Distinct-radii circle-covering conj…