Lecture 37 · Mathematics
A deck meets a quadratic map
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Meanwhile, read the outlineWhat you will learn
Define and distinguish real quadratic critical orbits, exact periodicity, superstability and boundedness.
Before this lecture
In this lecture
An outline of the lecture, scene by scene, with the statements that appear on screen. It follows the draft script, so details may change in production.
Introduce fixed-parameter iteration
- x₀ = 0; xₖ₊₁ = xₖ² + c
- c stays fixed
Read a cobweb correctly
- Vertical: evaluate f
- Horizontal: output becomes input
Define exact return and polynomial indexing
- Exact period = first positive return
- deg Pₖ = 2^(k−1)
Compute small critical cycles
- c*³ + 2c*² + c* + 1 = 0
Explain superstable
- (fⁿ)'(0) = product of orbit derivatives = 0
Prove the real bounded range's inclusion
- Real bounded parameters: −2 ≤ c ≤ 1/4
- β = (1 + √(1−4c))/2
Separate numerical and exact evidence
- Near return ≠ exact return
- Bounded so far ≠ proved bounded
Play and retrieve
- Bounded / critical periodic / eventually periodic
Companion interactive
Planned
Predict iterates and classify exact versus numerical behavior.
The interactive for this lecture is being built and will appear here.
Notes
Written notes for this lecture will appear here.
Sources
- Magical Mathematics, Chapter 5, pp. 72–78
How these sources were used
- Iteration and critical-cycle connection. Corrects the printed positive-parameter boundedness range and polynomial-degree indexing; no full dynamical correspondence proof claimed.