# How can the topological entropy and $L^2$ mixing rate be related?

For a product of otherwise identical systems evolving at different rates, the toplogical entropy and a quantity very closely related to (indeed, identifiable with a nondegenerate variant of) the $L^2$ mixing rate behave quite similarly.

Specifically, the topological entropy of such a flow satisfies

$h(\prod_m \phi_{c_m t}) = \sum_m c_m \cdot h(\phi_t)$,

where we assume $c_m > 0$. Meanwhile, if $f$ is a function extremizing the Rayleigh quotient $\mathcal{E}(f)/Var(f)$ associated to a Markov generator $Q$ (here $\mathcal{E}$ is the Dirichlet form and the implied measure is obtained from $Q$) and we consider the tensor sum $Q^\otimes := \sum_m I^{\otimes(m-1)} \otimes c_m Q \otimes I^{\otimes(N-m)}$, the corresponding quotient

$\mathcal{E}^\otimes(f^{\otimes N})/Var(f^{\otimes N}) = \sum_m c_m \cdot \mathcal{E}(f)/Var(f)$

is also an extremum (but not a global one, as that corresponds to ignoring all the factors save the one with the least $c_m$). Ignoring degenerate extrema leads to the variant of the $L^2$ mixing rate alluded to above (and recovers the usual mixing rate for non-product systems).

Now in some sense the above is a glorified scaling argument, and not much more. But on the other hand the two concepts provide probably the most natural rates in their respective domains of applicability--except when they overlap, in which case it would be nice to know if their similar scaling behavior is more than a coincidence.

So: is there a concrete sense in which topological entropy and $L^2$ mixing rates have or can be related?

I am aware of work by Fernandes et al involving conductances, but could only scrounge up one reference which doesn't seem particularly helpful (though I plan to look at it more closely, it is confusing at first glance). Similarly, I am also aware of the recent paper by Richeson and Wiseman which touches on chain recurrence rates. But I am hoping for something more specifically geared to $L^2$ versus some kind of symbolic or topological mixing rate, or on the other hand a more universal point of view encompassing most or all notions of mixing rate.

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