Asymptotics of one function on natural numbers

[dmitri....@gmail.com]

Below, N is the set of natural numbers.

Consider a function s: N->N such that s(x) = x + 1. Now, if f: N->N is
an arbitrary function, define a mapping from the set of all such
functions to N as given below:

p(f) = c(1)

where c is the composition of functions s and f.

Finally, define a new function g: N->N such that

g(x) = f(x) for all x != f^n(x)
g(x) = p(f) otherwise.

We will use a shorthand notation g = F(f) for the function g defined above.

Consider now the sequence of F repeatedly applied: F, F(F(f)),..., F^k(f),...

Question: how one will study the asymptotics of F^k at k->infinity for
various n and p?

Thanks,
Dmitri

[an...@euromake.com]

The construction is similar to pointer machines and specifically to Sch�nhage's storage modification machine (SMM) model. Although, it is indeed interesting to study if SMM is able to stay universal with only a single "set" instruction when the storage is an infinite directed graph.