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.

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.
