I want to use this map as a sort of chaotic oscillator for audio, where lambda and w are widgets you can control from something like a touch surface in real time. The map is from C to C, and lambda, z, and w are all complex valued. I would output something like a projection of z, say the real part. The output projection could also be controlled by some 2D touch widget, something like y = Re [ P z ], where P is in C.
From what I have gathered, this map has a lot of structure in common with the tetration fractal, and it's also a generalization of the so called Circle Map. Assuming that the map always lands us inside some compact domain, such that we have some kind of radius of convergence. Big if true, I know! I have given some thought as to how to deal with divergence, but nothing super obvious stands out to me at this moment. My first thought was just return 1 / conjugate(z) whenever the magnitude is outside of the unit circle, but this hack makes the Taylor coefficients diverge hard when you try to find them numerically (using something akin to Runge-Kutta). For the same reason (convergence of the Taylor coefficients), I've given up on the idea to "tame" the map using a discrete time low pass filter in the loop.
The fractional iteration idea came up as a strategy to implement variable frequency. It's always nice to be able to control the pitch of your instrument. I could just sample the map and use standard bandlimited interpolation to peek between the iterations, but I know this will throw away a lot of the information and interesting behavior of the oscillator.
My first thought was to encode the map as a set of truncated Taylor series for the 0th, 1st, 2nd etc. partial derivative wrt z of the map. This can all be done "exactly" using rules for the differentiation of polynomials, together with a suitably truncated Taylor series for f(z) = exp(lambda (z - w)). But when the parameters lambda and w are varying, I need something like a numerical methods estimate for all those derivative polynomials, and it gets kind of scary.
For a fractional shift operator, I'm thinking of applying something like T^t = e^(t D), where D is the differential operator. So T^(t + 1/2) is the half iterate of my map at time t, etc. In general, we have the operator T^(t+alpha) is Identity + alpha D^1 / 1 + alpha^2 D^2 / 2 + alpha^3 D^3 / 6 + ... applied to my truncated Taylor polynomial on z[t], where the index t is a natural number.
Can this converge in any meaningful sense? I don't have the knowledge/tools needed to even begin to answer the question of viability.