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I don't know if this is an elementary question or not. In what follows all maps are continuous

Suppose that $P:\mathbb{C}\rightarrow\mathbb{C}$ is a complex polynomial of degree $d>1$ and let $\mu$ be its invariant ergodic measure of maximal entropy. Now let $\Phi:\mathbb{C}\rightarrow S$ be a surjective map (also injective outside some real algebraic set) where $S\subset\mathbb{R}^n$ and take $Q:S\rightarrow S$. Let $\nu=\Phi_*(\mu)$ and let $\Psi:S\rightarrow\mathbb{C}$ be injective such that $\Phi\circ\Psi=Id_S$.

Suppose we have following relations $$\Phi\circ P=Q\circ\Phi,$$ and $$\Psi\circ Q=P\circ\Psi.$$

So my qestion is: Does it hold $h_{\mu}(P)=h_{\nu}(Q)$?

It is basic fact that $h_{\mu}(P)\geq h_{\nu}(Q)$, so proving other way around is trickier.

Improve this question
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  • $\begingroup$ for $z \mapsto z^2$, the support of the measure of maximal entropy is the circle, which is real algebraic. If you "crush" it by the map $\Phi$, you will get strictly lower entropy. same for $z^2-2$ where the support of the measure of maximal entropy is $[-2,2]$. of course these examples are exceptional and you may wish to exclude them $\endgroup$
    – Albert
    Commented Sep 10, 2014 at 17:16

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