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The Shub conjecture on topological entropy $h(f)$ of self map f on manifold M says that the topological entropy is greater (or equal) than (to) the log of maximum absolute values of the eigenvalues of the linear map $f_*$ induced on Homologies.

Are there some Polynomial entropy version of this conjecture on compact topological manifolds or even topological space?

The polynomial entropy is described here

https://link.springer.com/article/10.1134/S156035472304007X

I asked the question in Physicsoverflow and in a comment form in RG too.

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    $\begingroup$ You would replace the exponential growth rate for the action on the homology by the new growth rate: look at the limits of $\|f_*^n\|/\log n$. But it is only of interest if the spectrum of $f_*$ is contained in the imaginary axis. $\endgroup$
    – John B
    Commented Jul 9 at 20:40
  • $\begingroup$ @JohnB Thank you! are there some dynamical interpretations for this spectrum condition? $\endgroup$
    – Ali Taghavi
    Commented Jul 9 at 20:49
  • $\begingroup$ @JohnB Please expand your comment $\endgroup$
    – Ali Taghavi
    Commented Jul 9 at 21:15
  • $\begingroup$ @JohnB are there examples of non vanishing of the induced homology map $f_*$ but the topological entropy $h(f)=0$? $\endgroup$
    – Ali Taghavi
    Commented Jul 9 at 21:17
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    $\begingroup$ There is nothing to expand in my comment, sorry, it is only my answer to your question "Are there ..." I have never seen it written anywhere and yes, of course, $\log n$ can be replaced by any other rate. Is this interesting? Maybe, probably not (but if so it could perhaps help understanding the dynamics of zero entropy maps). It is doubtful for example that Yomdin's proof for $C^\infty$ maps extends to other growth rates (sorry, too technical to explain here why). $\endgroup$
    – John B
    Commented Jul 10 at 1:42

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