Linear Relations of Finite Length Modules are Shift Equivalent to Maps

TL;DR

This work shows that linear relations on finite-length modules encode discrete dynamical information and, up to shift equivalence, are always reducible to bijective maps. By building and comparing the Leray functor and the Szymczak universal normal functor within the endomorphism category $\operatorname{End}\,\text{FRel}(R)$, the authors prove that each Szymczak class contains a canonical bijective representative $L(A,\alpha)$. They establish equivalence between the Leray and Szymczak constructions on finite-length modules and provide a concrete Z$_p$-case illustrating the classification of endomorphisms up to shift equivalence. The general finite-length result shows that, for such modules, the dynamical information carried by linear relations can be captured by bijections, enabling robust Conley-type analysis without requiring acyclicity. Together, these results yield a unified, computable framework linking linear relations, shift equivalence, and categorical invariants in discrete dynamics.

Abstract

Linear relations, defined as submodules of the direct sum of two modules, can be viewed as objects that carry dynamical information and reflect the inherent uncertainty of sampled dynamics. These objects also provide an algebraic structure that enables the definition of subtle invariants for dynamical systems. In this paper, we prove that linear relations defined on modules of finite length are shift equivalent to bijective mappings.

Figures (2)

• Figure 1: Linear relations over $\mathbb{Z}_3$ field with $\operatorname{id}_0$. Every box (color) shows one class of Szym-equivalent objects.
• Figure 2: Schematic depiction of the relation $\beta$ considered in the Appendix

Theorems & Definitions (43)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 3.1
• proof
• Definition 3.1
• Proposition 3.2
• proof
• Definition 4.1
• Theorem 4.1: Sz1995