Orbits, fixed points and periodic points

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

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 — The orbit comes from repeated iteration of an existing self-map.
• Sequence — An orbit is ordered by natural-number time.
• 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.