Fixed points of maps on the space of rational functions

Abstract

Given integers $s,t$, define a function ${\varphi }_{s,t}$ on the space of all formal series expansions by ${\varphi }_{s,t}(\sum a_n{x}^n)=\sum a_{sn+t}{x}^n.$ For each function ${\varphi }_{s,t}$, we determine the collection of all rational functions whose Taylor expansions at zero are fixed by ${\varphi }_{s,t}$. This collection can be described as a subspace of rational functions whose basis elements correspond to certain $s$-cyclotomic cosets associated with the pair $(s,t)$.