Iteration process involving several functions

Question

I would like to carry out a following iteration process:
Apply function f1[a_,b_,c_,d_] to a starting list l1={a1,b1,c1,d1}, which will result in l2={a2,b2,c2,d2}. Then apply function f2[a_,b_,c_,d_] to list l2 to get l3={a3,b3,c3,d3}. Finally apply function f3[a_,b_,c_,d_] to list l3 to get l4={a4,b4,c4,d4}. As soon as this is over, I would like {a4,b4,c4,d4} to be the starting values of the process, in other words to start the procedure with values of l4 obtained in the previous step. This should repeat until l1 and l4 in the same step are identical. It would be also good if I could get lists l1,...,l4 from the last step returned at the end of the process. Any idea how can I do it in Mathematica?

I couldn't find a similar problem solved on the net. I read all about FixedPoint, NestWhile and Fold in the documentation, but I still don't know how to apply it to this problem, so I'd be grateful for all tips and advices.

Functions f1,...,f3 are a bit complicated and involve reading values from dll library. Let's say they look like this one:

f1[{a_,b_,c_,d_}]:={
  afrombc[b-0.2,c],
  b-0.2,
  c,
  dfromab[afrombc[b-0.2,c],b-0.2]
}

afrombc[] and dfromab[] are predefined functions connected with dll library.

Answer 1

This calls for FixedPoint. I add Composition and pure functions for enhanced value (in case you have many functions and want to chain them conveniently). The dummy functions here are ultimate simpletons, but the principle works as long as inputs and outputs for the functions are compatible:

{f1, f2, f3} = {#^(1/2) &, #^(1/3) &, #^(1/4) &};

FixedPoint[Composition[f3, f2, f1][#] &, .1]

1.

Equivalent function calls would be:

FixedPoint[f3@f2@f1[#] &, .1] 

or

FixedPoint[f3[f2[f1[#]]] &, .1]

Works also with lists in this case:

FixedPoint[Composition[f3, f2, f1][#] &, {0.1, .9, 1, 1.5}]

{1., 1., 1, 1.}

but here for exact numbers other than 1, you will be in trouble and you might want to use the third argument to FixedPoint, which defines the max. number of iterations:

FixedPoint[f3[f2[f1[#]]] &, 2, 20]

2^(1/4019988717840603673710821376)