I would like to know if the following concept has a name. I call it a "conservation law", but if it's defined in the literature it's probably called something else. I'm interested in any results based on it, and in its applications to dynamical systems theory.
A "conservation law" is a (in general non-bijective) function $c:X \to Y$. We say that a function $f:X \to X$ obeys the conservation law if $c\circ f = c$. (The interpretation is that $X$ is the state space of a discrete-time dynamical system, and $c$ is something that assigns a different value to every point in that space. If $f$ is the state transition function of the dynamical system then the value of $c(x)$ will be constant over time if and only if $f$ obeys the conservation law $c$.) Note that all functions obey the trivial conservation law ($c(x)=0$ for all $x$), and if $c$ is a bijection then only the identity function obeys it.
We may consider the set of bijections from $X$ to $X$ that obey a given conservation law. It is straightforward to show that these form a group under function composition. This association between groups and conservation laws is intriguing, because it has something of the flavour of Noether's theorem in Lagrangian mechanics.
I'm interested in whether this has been noted before, and whether there's anything useful you can do with it. In particular, I'd like to know if there's some sense in which one can go the other way, and associate a conservation law to a permutation group, rather than the other way around.
A bit of abstract nonsense shows that one can always associate what you call a conservation law $c$ to a given transformation $u:X\to X$.
Namely, consider the equivalence relation $R^u$ on $X$ such that $xR^uy$ if and only if $u^n(x)=y$ or $u^n(y)=x$ for some $n$. Let $X^u=X/R^u$, then the canonical equivalence function $c^u:X\to X^u$ is a conservation law for $u$. Recall that $c^u$ is defined uniquely by the condition that, for each $x$ in $X$, $x\in c^u(x)$.
Note that $c^u$ is the most detailed conservation law for $u$ available in the sense that, for any other conservation law $c:X\to Y$, if $c^u(x)=c^u(y)$ then $c(x)=c(y)$ (in other words, there exists a factorization $c=g\circ c^u$ of $c$ for some function $g:X^u\to Y$).