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I have an equation of the general form $$ X = S \cup T X $$ where $S \subset \mathbb R^n$ is a convex polytope (given by its bounding hyperplanes), $T\colon \mathbb R^n \to \mathbb R^n$ is a linear map, and the largest $X \subset \mathbb R^n$ must be found.

Is there an easy way to solve this and similar problems?

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  • $\begingroup$ Conditions on $T$ are in order; for instance, from the equation, the existence of $X$ (a bounded n-dimensional body) implies that $T$ maps a bounded nonempty open set into itself , which I think implies the spectral radius of $T$ is less than or equal to $1$. If $T$ has spectral radius less than $1$ then $X\mapsto S\cup TX$ is a contraction wrto the Hausdorff distance and has a unique fix point among closed bounded sets. $\endgroup$
    – Pietro Majer
    Commented Mar 17 at 17:20
  • $\begingroup$ Do you mean the smallest $X$? $\endgroup$
    – Geoffrey Irving
    Commented Mar 17 at 20:42
  • $\begingroup$ No, but I meant $\cap$ instead of $\cup$. Sorry. $\endgroup$
    – rimu
    Commented Mar 17 at 20:52
  • $\begingroup$ @rimu : One should not change the question so as to invalidate an existing answer. So, please roll back your edit, and then you can post a new question -- after re-reading it very carefully, so as not to waste other users' time. $\endgroup$
    – Iosif Pinelis
    Commented Mar 17 at 21:11
  • $\begingroup$ Done. (O rather undone...) $\endgroup$
    – rimu
    Commented Mar 18 at 19:22

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$\newcommand\R{\Bbb R}$If $X=S\cup T(X)$, then $$T(X)=T(S)\cup T^2(X),\quad X=S\cup T(S)\cup T^2(X), \ldots, \\ X=S\cup T(S)\cup\cdots\cup T^{n-1}(S)\cup T^n(X) \\ \subseteq Y:=S\cup T(S)\cup\cdots\cup T^{n-1}(S)\cup T^n(\R^n).$$ On the other hand, $Y=S\cup T(Y)$, since $T^{n+1}(\R^n)=T^n(\R^n)$.

So, the largest $X$ such that $X=S\cup T(X)$ is $Y$.

In particular, if $S$ is a singleton set $\{s\}$, then the largest $X$ such that $X=S\cup T(X)$ is $$\{s,Ts,\dots,T^{n-1}s\}\cup T^n(\R^n).$$ If e.g. $n=3$, $(e_1,e_2,e_3)$ is a basis of $\R^3$, $S=\{e_1\}$, $Te_1=e_2$, $Te_2=e_3$, and $Te_3=e_3$, then the largest $X$ such that $X=S\cup T(X)$ is $$\{e_1,e_2\}\cup\R e_3.$$

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