# Hyperoperators

## Complex Hyperoperators

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

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?