Transfer equation

Transfer equation is relation between some holomorphic function $$h$$, called transfer function and another function $$F$$, called Superfunction, expressed with

$$F(z\!+\!1)=h(F(z))$$

for all $$z\in C \subseteq \mathbb C$$.

Physical sense of the transfer equation

The transfer equation describes the evolution of some isolated system in the discrete time, when the state in the next moment o time is completely determined by its state in the previous moment of time; this determinism is expressed with the transfer function $$h$$.

In principle $$F(z)$$ may belong some Hilbert space; then the transfer equation may correspond to the evolution of the state of a system in the discrete time $$z$$; then $$h$$ is analogy of the Hamiltonian. The extension to the complex, and, in particular, to the real values of time $$z$$ refers to the conventional Quantum mechanics, where time appears as real parameter.

It happened, that even in the case of simple, "zero-dimensional" Hilbert space, when $$h$$ and $$F$$ have values from the set of complex numbers, the problem of finding of superfunction $$F$$ for given transfer function $$h$$ is not trivial. This determined the interest to the superfunctions from the set of holomorphic functions of complex variable. The extension to the complex values of $$z$$ happens to be essential for the establishment of the unique superfunctions for some simple transfer dunctions auch as exponential or the logistic operator.

Abel function and Abel equation

The Inverse of the Superfunction, $$G=F^{-1}$$ is called Abel function. Within some domain $$D\in \mathbb C$$, the Abel function satisfies the Abel equation

$$G(h(z))=G(z)+1$$

The Abel function, together with the corresponding superfunction $$F$$, allows to express the iteration of the transfer function $$h$$ in the following form:

$$h^c(z)=F(c+G(z))$$

for all z from some domain in the set of complex numbers, where the number $$c$$ of iterations have no need to be integer; and the relation

$$h^c(h^d(z))=h^{c+d}(z)$$

for all complex $$c,d,z$$ from some domain which may cover significant part of the complex plane.

In particular, $$h^{-1}$$ is the inverse function, $$h^0$$ is the identity function and $$h^1=h$$.

Examples

Several examples of pairs (superfunction, Abel function) are suggested in the Moscow University Physics Bylletin [1]. In particular, the tetration satisfies the transfer equation with the transfer function exp.

The superfunction and its inverse for the subsequence of the Collatz sequence are considered in the article [2].

The superfunction for the holomorphic extension of the second iteration of the Optimized Collatz operator are considered in the article Holomorphic_extension_of_the_Collatz_subsequence.

References

1. http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
2. http://www.springerlink.com/content/u712vtp4122544x4 D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. (Russian version: p.24-31)