duaLab

Discrete dynamical system

Contents

1. Definition

A discrete dynamical system is a set with an endofunction, that is, an action of the additive monoid $\mathbb N$. These form the presheaf topos $\mathbf{Set}^{\mathbb N^{\mathrm{op}}}$. See Definition 2.1.

Definition 1.1 (Period and height)

For a connected system, the period is the length of its cycle, or zero if it has no cycle. Its core, when it exists, is the unique subobject isomorphic to $\mathbb Z/b\mathbb Z$ with successor, where $b$ is the period; for $b=0$ this means the bilateral successor system on $\mathbb Z$.

The height of a state is the number of iterations needed to reach the core. If there is no core, every state has infinite height. The height of the system is the supremum of its state heights. See Definitions 2.8, 5.4, 5.5 and 5.8.

2. Properties

Theorem 2.1 (Classification of quotients)★

The quotients and prequotients of this topos are both order-isomorphic to the ideals of $\mathbb N\times\mathbb N^{\mathrm{div}}$.

The first factor has the usual order. The second contains zero and is ordered by divisibility. An ideal is a nonempty downward-closed subset closed under finite joins. See Theorem 5.23.

Proposition 2.2 (Connected systems)★

Up to equivalence in the generative order, connected systems are classified by their height and period, ordered by the usual order on $\mathbb N\cup\{\infty\}$ and divisibility on $\mathbb N$, respectively.

See Propositions 5.18 and 5.19.

Essential quotients correspond to monoid epimorphisms from $\mathbb N$; these need not be surjective. See §6.2 and Appendix C.

4. References