Turings use of u, leading to K’ness
Since other parts of the argument have g = f and one sees the rayleigh quotient being applied to a space that is orthogonal to the constant functions we can see Turing focusing the rate of mixing (ie. using the spectral gap). But, he doesn’t reason in terms of matrices or the eigenspectrum. He reasons purely in terms of functional analysis, normed spaces, and metric – which of course Cambridge of the times was very specialized.
Normed spaces and continuity, and K:-
But we have to recall the context – that one is measuring the degree of dependency (between output and input bit flows, in power-walks).
Furthermore, one sees the pithily stated property of that which leads to protection against differential cryptanalysis:
When Turing discusses the wheel function in character terms (a. b-1) he is really discussing the difference between the two sides of the wheel (i.e. output to input)
The general theory of mixers (as Turing would have been thinking of such things) is at Rapidly Mixing Markov Chains- A Comparison of Techniques. One sees Goods coalescence thinking come into play, from Tunny era, AND one sees expander graphs and conductance (based on conditional probabilities of transitions between spaces):
One sees how expander graphs (which ensure there are no “capacity bottlenecks) play into this now – influencing the “rapidity” of the mixing (since maximum flow is arranged for).