EDIT: From here (https://www.mathworks.com/help/signal/ref/butter.html) it appears that an Butterworth filter has a non-linear phase lag at the normalized frequency of 0.5 onwards. Does that mean that if the signal has multiple frequencies beyond 0.5, its output after passing through this filter will be distorted by these frequencies?
NOTE: To help clarify things, this question had been modified from the original one below.
A butterworth low pass filter would have a large phase lag for signals passing through at higher frequencies. [EDIT: The assumption is, for some filters to work, you'd have to do some kind of transform on the input (into the s domain), multiply that by the filter's s-domain transfer function (such as ones from here ece.uic.edu/~jmorisak/blpf.html), then to see the time-domain filtered output, the resulting output has to undergo an inverse transform.] I envision that this means that the output signal coming out of the filter, when converted back into time domain, would come out distorted due to this lag. [EDIT: The assumption is, phase lag is the same as a time delay between the input and the output, and if the input has more than one frequency component, the different components would be delayed differently, and end up being combined with another signal output at a later point in time.] Would there be similar distortions (i.e.: as if there were a phase lag) for an IIR time-domain digital filter?