invariant Invariant 2002-02-22 11:40:24 2002-02-22 14:00:00 Definition \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand{\reals}{\mathbb{R}} \newcommand{\natnums}{\mathbb{N}} \newcommand{\cnums}{\mathbb{C}} \newcommand{\lp}{\left(} \newcommand{\rp}{\right)} \newcommand{\lb}{\left[} \newcommand{\rb}{\right]} \newcommand{\supth}{^{\text{th}}} \newtheorem{proposition}{Proposition} Let $A$ be a set, and $T:A\rightarrow A$ a transformation of that set. We say that $x\in A$ is {\em an invariant} of $T$ whenever $x$ is fixed by $T$: $$T(x)=x.$$ We say that a subset $B\subset A$ is {\em invariant with respect to $T$} whenever $$T(B)\subset B.$$ If this is so, the restriction of $T$ is a well-defined transformation of the invariant subset: $$T\Big|_B : B\rightarrow B.$$ The definition generalizes readily to a family of transformations with common domain $$T_i : A\rightarrow A,\quad i\in I$$ In this case we say that a subset is invariant, if it is invariant with respect to all elements of the family.