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# Endomorphisms of degree d oftheona sphere with infinite fibers on a dense subset

Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$ disjoint n-dimensional open discs in $S^n$. Then collapsing $K$ to a point we find that $S^n/K\simeq \bigvee _{i=1}^d S^n$. Identifying (choosing a homemomorphism) each of the sphere in the disjoint union with $S^n$ (with the approriate orientation), the composition of the two maps $$S^n\rightarrow \bigvee _{i=1}^d S^n \rightarrow S^n,$$ gives us a map $\phi_d:S^{n}\rightarrow S^n$ of degree $d$.

In general, if $f:S^n\rightarrow S^n$ is a map of degree $d$ and $x\in S^{n}$ is such that the fiber $f^{-1}(x)$ is finite, then one has from excision theorem that $$\sum_{y\in f^{-1}(x)} deg_f(y)=d.$$

Q1: How would you construct a (continuous) map $f:S^n\rightarrow S^n$ of degree $d$ such that on a dense subset of points $X\subseteq S^{n}$ one has that for every $x\in X$ that the fiber $f^{-1}(x)$ is infinite?

Q2: Having a map $f$ as in Q1 and a point $x\in X$, is it possible to take some kind natural average sum over the local degrees of the elements of $f^{-1}(x)$ in such a way that the sum converges to $d$ (you may assume that $S^n$ is endowed with a metric if you think it helps) ?

added: Note that if one constructs a map $f:S^n\rightarrow S^n$ of degree one with infinite fibers (on a dense set) then by post-composing with a map of degree $d$ we obtain a map of degree $d$ which satisfies all the conditions.

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# Endomorphisms of degree d of the sphere with infinite fibers onadensesubset

Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$ disjoint n-dimensional open discs in $S^n$. Then collapsing $K$ to a point we find that $S^n/K\simeq \bigvee _{i=1}^d S^n$. Identifying (choosing a homemomorphism) each of the sphere in the disjoint union with $S^n$ (with the approriate orientation), the composition of the two maps $$S^n\rightarrow \bigvee _{i=1}^d S^n \rightarrow S^n,$$ gives us a map $\phi_d:S^{n}\rightarrow S^n$ of degree $d$.

In general, if $f:S^n\rightarrow S^n$ is a map of degree $d$ and $x\in S^{n}$ is such that the fiber $f^{-1}(x)$ is finite, then one has from excision theorem that $$\sum_{y\in f^{-1}(x)} deg_f(y)=d.$$

Q1: How would you construct a (continuous) map $f:S^n\rightarrow S^n$ of degree $d$ with such that on a point dense subset of points $x\in X\subseteq S^{n}$ such one has that for every $x\in X$ that the fiber $f^{-1}(x)$ is infinite?

Q2: When the fiber Having a map $f^{-1}(x)$ is infinitef$as in Q1 and a point$x\in X$, is it possible to take some kind natural average sum over the local degrees of the elements of$f^{-1}(x)$in such a way that the sum converges to$d$(you may assume that$S^n\$ is endowed with a metric if you think it helps) ?

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