From Iterated Functions

Complex Hyperoperators[edit]

Theorem: Complex Hyperoperator Theorem The hyperoperator \(a \rightarrow b \rightarrow n \) with \(a,b \in \mathbb{C}\) and \(n \in \mathbb{N^+}\) can be constructed given \(\lambda \neq 0\) and \(\lambda^k \neq 1\) where \(k \in \mathbb{N^+}\).

Proof: Assume that the function is entire with \(z_0 \in \mathbb{C}\).

If the function is not \(f(z)=z\) it has a fixed point not at infinity, \(z_0\), such that \(f(z_0)=z_0\). The Taylor series of \(f^t(z)\) can be constructed for the complex plane if \(a \rightarrow b \rightarrow k\) can be constructed for every \(1 \leq k < n\).

Prove by induction.

Basis Steps:

Case \(n=1\). Exponentiation is entire and can be constructed.

Case \(n=k-1\). Assume \(f(z)=a \rightarrow z \rightarrow (k-1)\) with convergence closed under the composition of entire functions. (Induction Hypothesis)

Induction Step:

Case \(n=k\). The equation \(a \rightarrow b \rightarrow k = z_0+f^t(b)\) can be constructed for all \(k\). \(\blacksquare\)

Future Research[edit]

Further work on generalizing the domain of hyperoperators to Banach space.

The hyperoperators become simple at \((1+u) \rightarrow v \rightarrow k\) for small values of \(u\) as \(1 \rightarrow v \rightarrow k = 1\).

A general answer to the question "Do all maps have flows?" Research in complex dynamics indicates the importance of symmetry in the simplification of the discrete time version of \(f^t(x)\). In which symmetries does $t$ simplify from a natural number to a real or complex number?