The dynamical model conjecture for motivic local systems on curves

Let $C/\mathbb F_q$ be a smooth compact geometrically connected curve of genus $g\ge 2$, and let $S$ be an oriented closed topological surface of genus $g$. Let $\Phi_C$ be an endomorphism of the tensor category of finite-dimensional complex local systems on $S$. For each rank $N\ge 1$, consider the moduli stack of irreducible rank-$N$ local systems and its natural symplectic form.

Dynamical model conjecture. There exists an endomorphism $\Phi_C$ such that:

• $\Phi_C$ is algebraic and defined over $\mathbb Q$, acting on each such moduli stack by a rational map defined over $\mathbb Q$;
• $\Phi_C$ multiplies the natural symplectic form on the moduli space of irreducible rank-$N$ local systems by $q$;
• for every $n,N\ge 1$, the set of isomorphism classes of irreducible motivic local systems of rank $N$ on $C\times_{\mathbb F_q}\mathbb F_{q^n}$ invariant under $\operatorname{Fr}^n$ is identified with the set of isomorphism classes of $\overline{\mathbb Q}$-local systems of rank $N$ on $S$ invariant under $\Phi_C^n$, compatibly with the relevant Galois symmetries and tensor constructions.

This strengthens the preceding dynamical proposal by requiring compatibility with the tensor structure, symplectic geometry, and all ranks. The source calls the formulation somewhat sloppy and provides no general construction.