Thank you very much for your answer

@WillSawin the rate of convergence has a standard definition in numerical analysis: Let $a_n$ positive goes to zero then the rate is unique q for which the limit $\frac{a_{n+1}}{a_n^q}$ is finite and non zero. In case of non convergence of the later limit you may replace by limsup or lim inf appropriately. now if $a_n$ goes to $\ell$ you consider $|a_n-\ell$ instead of $a_n$. An interesting example is that rate of convergence of secant algorithm is the golden number

Your Lie group consideration is very interesting I will read it and think about its detail. Thanks again for your answer

the locally constant condition for k remind me the following situation: In the book Differential forms in algebraic topology" By Bott, &tu one find the definition of "Space of differentiial forms with vertical compact support ". then it is claimed that this space is closed under differentiation. according to the definitioj of the first version of the book, it is not realy true that the space is d-invariant. But in the second version they correct the definition. So that is my motivation to say "Polynomial growth with locally constant rate"

Thank you very much for your very interesting answer. I think you upgrade and modifed my question. To be honnest I did not meant the first version. I meant the space of all functions fiberwise dominated by $|V|^k$ for some $k$. As you said under Poisson bracket we may lose this property unless (I guess) we assume that k is locally constant. I beleive that the interesting points you mentioned (in 2 interpretation of the question) can be expanded.

@AnthonyQuas is injectivity of f the point you are indicating to?or you meqn consideration of a precise example of $f$?

@AnthonyQuas I think topologicql entropy is defined for non injective maps too. Here $f$ is a map on the interval which is composition of the projection $\pi_2$ with a space filling curve. so we have a dynamic on the interval

@AnthonyQuas Thank you for your comment. In fact by this post I am curious about the diversity of space filling curve from the entropy point of view. What quantities can be realized as the entropy of some potential filling curve? Can one say that the entropy is always non zero?

@JohnB As you pointed out, my motivation for this question was just zero entropy maps

@JohnB BTW in the question I did not say that we formally replace topological entropy with polynomial one(without any other possible change). I would appreciate if you read my question again

@JohnB are there examples of non vanishing of the induced homology map $f_*$ but the topological entropy $h(f)=0$?