Noether's theorems roughly state that if the dynamics of a physical system are invariant w.r.t. a certain transformation, then there is a corresponding invariant.
But as far as I know, these theorems rely on continuity assumptions.
Is there an analogous version of Noether's theorems for discrete time systems?
Here's a simple version of Noether's theorem for quantum discrete-time dynamical systems, by which I just mean a pair consisting of a Hilbert space $H$ and a unitary operator $U : H \to H$; time evolution at discrete time $n$ is given by the $n^{th}$ power $U^n$.
Suppose such a system has a $G$-symmetry, meaning that a group (discrete or otherwise) $G$ acts by unitary operators on $H$ and that this action commutes with the action of $U$. Then:
Noether's theorem: $U$ preserves every isotypic component of $H$.
What this means in more physical language is that if some $v \in H$ "transforms under" some irreducible representation of $G$ then so will $U v$, hence so will $U^n v$ for all $n$.
This is a very straightforward calculation, and gives us conserved quantities of the form "which isotypic component is a given vector in?" which are discrete if $G$ is compact (e.g. spin) and can be continuous if not (e.g. momentum).
In the classical case, I only know how to write down a version of Noether's theorem for, say, continuous symmetries on a symplectic manifold, but time evolution doesn't need to be continuous.
