# Orbits, fixed points and periodic points

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Repeated iteration of a self-map forms an orbit. Fixed points have period 1; the least positive period of a periodic point must be distinguished from an arbitrary return time.

mko-orbit-fixed-periodic · concept · v0.10.0



## Mathematical statement

For a self-map T:X→X, the forward orbit of x is x,T(x),T²(x),…. A point is fixed when T(x)=x. If there is a least positive integer p such that T^p(x)=x, it is periodic with least period p. Eventually entering a cycle does not make the starting point itself periodic.





## Representative formula (exact TeX)

T(x)=x





## Assumptions


- Exercises use complete finite residue-class states and affine self-maps, allowing enumeration until the first repeat.
- Classification applies only to the specified finite model. An arbitrary finite prefix proves no nonperiodicity claim for a general infinite system.





## Symbol scope


- T — state-update self-map
- x — specified initial state





## Intuition, definition, and misconceptions

A fixed point is the special case with least period 1. The equality T^p(x)=x establishes only a return time unless smaller positive return times have been excluded.

Every orbit of a finite self-map eventually repeats. If the first cycle starts strictly after the initial state, that starting point is preperiodic rather than periodic.

A complete finite table permits exact least-period classification of its points, but does not establish the periodic structure of continuous systems or parameter families.

Do not call every return time the least period or label a transient initial state periodic merely because it eventually enters a cycle.





## Adopted derivation guide

[Orbits, fixed points and periodic points: classifying all six states of T(x)=(2x+1) mod 6](https://ocme.evemisslab.com/en/derivations/derivation-mko-orbit-fixed-periodic-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Discrete-time dynamical system](https://ocme.evemisslab.com/en/mko/mko-discrete-dynamical-system/) — The orbit comes from repeated iteration of an existing self-map.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — An orbit is ordered by natural-number time.
- [Equation](https://ocme.evemisslab.com/en/mko/mko-equation/) — Fixed-point and periodicity conditions are iterate equations.





## Verification boundary

A complete finite model can classify its specified points; a finite prefix alone proves no general nonperiodicity result.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add preperiodicity, attracting cycles, stability and parameter bifurcations.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/orbit-fixed-periodic.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-orbit-fixed-periodic/index.md)