Let me reply taking M to be the identity (indeed M is somewhat cosmetic to the discussion).

The identity $F\circ Ψ=DΨ\circ F$ is what one considers for example in the Grobman-Hartman theorem, passing from a dynamics to its linearization say at a fixed point. The possibilities for F are endless; $F$ would then be a topological conjugacy, perhaps locally, although maybe not very regular, more precisely at most Hölder continuous in general. Moreover, $F$ need not satisfy any invariance properties, which if I understand correctly is your main concern.

Invariance of dynamical system under a transformation
The graph below was derived from a raw seismogram recorded during an earthquake over a timespan from t=0 to t=1400 seconds (not shown on x-axis). 


The original seismogram $s(t)$ is not shown, but it is known that the original signal is the convolution between the source wave $w(t)$, the effects of the medium $g(t)$, and the effects from instrument itself $i(t)$; that is,  

$$s(t)=w(t)*g(t)*i(t)$$

The graph above is what remains after removing the instrument response as shown below:  



I think this instrument response is telling me that the amplitudes of the original seismogram $s(t)$ at frequencies higher than $0.02Hz$ were amplified at a constant gain of approximately $1E9$, while amplitudes lower than $0.01Hz$ are effectively dampened (by gains significantly smaller than at those frequencies higher than $0.02Hz$).

But what exactly can I effectively say about the shape of the original seismogram from this information?  If these frequencies are 'dampened' in the broadband representation above, what, if anything, can be said about the amplitudes and shapes observed in the original graph?  Can I simply say that longer waves should be present in the original signal but not say anything about when they occur?  Could I somehow convolve these graphs to generate the original signal (or at least closely approximate it)?
