My dynamics course has talked a bit about the Pomeau-Mannville map, defined as $f(x)=x + x^{1+p}\mod 1,p > 0, 0\leq x\leq 1$. I was wondering whether there is any known trick or a special function that allows us to write explicitly the inverse branches of $f$. Plotting the function shows two inverse branches over the unit interval. But I don't have any clue on how to find the inverse, nor do I know whether it is possible
2026-04-04 07:55:19.1775289319
Is it possible to derive an explicit expression for the inverse branches of $f(x)=x + x^{1+p}\mod 1, p > 0, 0\leq x\leq 1$?
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If you can already plot the function, there's very simple steps to find the inverse functions by mirroring with y=x line: $$(x,f(x)) \\\rightarrow (f(y),y) $$
Usually it continues like this: $$ \rightarrow (f^{-1}(f(t)),f^{-1}(t)) \\\rightarrow ((f^{-1} \circ f)(t),f^{-1}(t)) \\\rightarrow (Id_t(t),f^{-1}(t)) \\\rightarrow (x,f^{-1}(x))$$, but the 2nd one in the chain already describes the inverse function graph.