I know that Galois groups let us determine when polynomials are solvable by radicals, and that Lie groups can be used to solve some differential equations. Does there exist an analogous group theory method that tells something about the solutions of recurrence relations? I am interested in both existence/Galois type results and symmetry methods of solution.
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If one considers a "recurrence relation" as an object that is ultimately equivalent to a dynamical system (in the sense of iterates of a selfmap of some set) (see e.g. How a group represents the passage of time?), then there are a few ideas that can be considered as analogues of Lie theory for recurrence relations analogous to Lie theory being analogous to Galois theory. In my humble opinion though how one handles such analogs depend highly on the nature of the "class" of dynamical system, and while there are general well-established streams the big picture is nowhere near complete.
Let $\alpha:G\curvearrowright X$ be an action of a group (or monoid) $G$ on a set $X$ (take $G$ to be nonnegative integers for iterates of a function). The object that is most straightforwardly analogous to a symmetry group for $\alpha$ would be its centralizer: this is the group of bijections $\Phi$ of $X$ such that $\forall t\in G: \Phi\circ \alpha_t=\alpha_t\circ \Phi$.
If the action is considered in the measurable category, then one arrives at fairly foundational ergodic theory/harmonic analysis (see e.g. Measure Preserving Self-Map of Compact Abelian Group Commuting with Ergodic Translation).
If $\alpha$ is a shift (in the sense of e.g. How does symbolic dynamics contribute to the study of dynamical systems?), and one considers its topological centralizer, then one is in the context of cellular automata theory (see e.g. works of T. Ceccherini-Silberstein and company or S. Schmieding and company), or from a more applied viewpoint theory of linear (or affine, or nonlinear, ...) time filters in signal processing (see e.g. Why do engineers use the Z-transform and mathematicians use generating functions? or https://youtu.be/qbx1RPGTkyU?feature=shared&t=5923).
If $\alpha$ is the iterates of a diffeomorphism, one arrives at the famous centralizer problem in smooth dynamics (see the references at Which matrices can be embedded in flows?).
The larger scope seems to be the problem of classification of dynamical systems; given $\alpha:G\curvearrowright X$ and $\beta:H\curvearrowright Y$, when is there a conjugacy $(\Phi,\Psi)$ so that $\Phi\circ \alpha_{g}(x)=\beta_{\Psi(g)}\circ \Phi(x)$? (see e.g. Examples of conjugate-like structures across mathematics)
As a final remark, there are symmetries that are attached to dynamical systems in more subtle ways, for instance certain spaces of cocycles/cohomologies (see e.g. Intuition of cocycles and their use in dynamical systems), spaces of joinings (see e.g. Characterization of a joining over a common subsystem.), groups arising from normal form theories (see e.g. works of B. Kalinin and V. Sadovskaya or The origin of the name homological equation), path groups or groupoids made up of concatenations of certain equivalence relations one can associate to a dynamical system (see e.g. works of K. Vinhage and company), groups that permute the cells of some equivalence relation preserved by the dynamical system (see e.g. works of D. Bohnet, or P. Carrasco),...
Alp Uzman
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