# Iterations of a multivariable function

How do you define iterations of multivariable functions?

To be clear(example):

If $$f: \mathbb R^2 \to \mathbb R$$

How do you define

$$f \circ f$$, or $$f \circ \cdots \circ f$$?

I admit that this question sounds very odd, but I think I need to define or learn of this. (Why? I want to generalize this(Carleman matrix) to multivariable functions to solve this(Multivariable carleman matrix) or this(same but different sites) question!)

And I think this concept may be quite reasonable because there is a something like multiplication of matrices that have different dimensions.

My assumtion is that $$f \circ f \cdots \circ f : \text{also } \mathbb R^2 \to \mathbb R$$.

Any suggestions are appreciated.

• I suppose you could define $\tilde f:\mathbb R^2\to\mathbb R^2$ by $\tilde f(x,y) = (f(x,y),f(x,y))$, then investigate $f_n := f\circ \tilde f\circ \tilde f\circ\dots \circ \tilde f$. Sep 11 '18 at 15:29
• @MikeEarnest Worth trying! Sep 11 '18 at 15:36
• There are definitely many possible versions... $F(x,y) =(0,f(x,y)); (f(x,y),f(x,y));$ or even $(f(x,y),f(y,x))$ (which is sort of a more ‘symmetric’ version of $f$). Which one you should consider depends highly on the property you want them to satisfy Sep 11 '18 at 15:59

The composition is undefined as $$f \circ f=\mathbb{R}^2\xrightarrow{f}\mathbb{R}\xrightarrow{?}\underline{?}.$$ However if you have a function $\mathbb{R}^2\xrightarrow{F}\mathbb{R}^2,$ then we can easily form the composition.