Mathematical Notation of Sequence of Functions

Question

Let's say I have a finite set of functions $F=\{f_1,f_2,f_3,...,f_n\}$ and I want to show a recursive function that is constructed by an arbitrary sequence of applications of functions in $F$ to input $x$. For example, one such sequence could be $f_4(f_9(f_1(x)))$. Is there a nice mathematical way to show the set of all possible sequences of applying elements of $F$ to input $x$?

Answer 1

What you could write is just

$$\{(g_m \circ g_{m-1} \circ \ldots \circ g_1)(x) | m\leq k; g_i \in F\, \forall i=1,\ldots,m\}$$

This does cover repetitions and allows any sequence of $f_i$'s up to length $k$.

Answer 2

Just a suggestion..

Usually $\sigma(A)$ denotes the set of all possible permutations of elements of $A$

To simplify you could write $\sigma(n) = \sigma(\{1, \dots, n\}) $ and write then $f_{\sigma(n)}$ the set of all functions you describe

Or if you mean also subsequence (i.e. Permutations shorter than $n$) you could define $\sigma(n)= \cup_{i = 1}^n \sigma(\{1,\dots,i\})$

Answer 3

A possibly related concept is that of an Iterated Function System, and the Hutchinson operator: $$H(S) = \bigcup_{i=1}^n f_i(S)$$.

In this setting, the collection of all iterates of your collection of functions $\{f_i\}$ applied to $x$ could be described as $$\bigcup_{j=0}^\infty H^{j}(\{x\})$$ where the exponent represents repeated applications of the Hutchinson operator.

Answer 4

Here it is a recursive definition of $\Gamma(x)$, the orbit of $x$:

Let $\Gamma_1(x)=\{f_1(x),\ldots,f_n(x)\}$.

Let $\Gamma_{m+1}(x)=\{ f_1(y),\ldots,f_n(y) / y\in \Gamma_{m}(x)\}$.

Finally, $$\Gamma(x)=\bigcup_{m\geq 1}\Gamma_m(x).$$