1. The Mean Median Map Marc Chamberland
Department of Mathematics and Statistics
Grinnell College
Collaborator: Mario Martelli, Claremont McKenna College

2. What is the map? Start with the non-empty set Generate the sequence by
The Mean-Median Map
As usual,

3. Examples
We use bold to indicate that the sequence simply repeats that number thereafter.
{5} continues as {5}
{1,2,3} continues as {2}
{0,2,3} continues as { 3, 9/2,11/2, 3}
{0,4,5,6} continues as {15/2, 15/2, 17/2, 19/2, 51/4, 57/4, 15/2}
{0,4,5} continues as { 7, 13/2 15/2, 41/4, 47/4, 35/4, 37/4, 39/4, 41/4, 125/8, 135/8, 25/2, 13, 27/2, 14, 29/2, 15, 41/4}
Terminating Conjecture: Every initial point will become fixed in a finite number of steps.

4. Medians Theorem: The sequence of medians is montonic.
{5,5} continues as {5}
{1,2,3} continues as {2}
{0,2,3} continues as { 3, 9/2,11/2, 3}
medians: 2, 5/2, 3
{0,4,5,6} continues as {15/2, 15/2, 17/2, 19/2, 51/4, 57/4, 15/2}
medians: 9/2, 5, 11/2, 6, 27/4, 15/2
{0,4,5} continues as { 7, 13/2 15/2, 41/4, 47/4, 35/4, 37/4, 39/4, 41/4, 125/8, 135/8, 25/2, 13, 27/2, 14, 29/2, 15, 41/4}
medians: 4, 9/2, 5, 23/4, 13/2, 27/4, 7, 29/4, 15/2, 65/8, 35/4, 9, /4, 19/2, 39/4, 10, 41/4
Theorem: The sequence of medians is montonic.
Conjecture: The median occurs exactly twice before it settles permanently.

5. Properties with three initial points
Let M( a,b,c ) be the limiting median of the initial set {a,b,c} (if it exists). Then scaling and translation holds:
Let m(x) = M(0,x,1), so consider only 0<x<1.
m(1-x) = 1 – m(x), so consider only 1/2<x<1 (medians increase).
so consider only 1/2<x<2/3.

6. The map m on 1/2<x<2/3
Are you ready for something scary?

7. Steps to Convergence median
Steps needed until convergence. We believe this function is unbounded.

8. Maxima along the Way
median
Maximum in orbit

9. Near x=1/2 Theorem: for |e|<4/333.
Proof: The next iterates take the form 1/2 + re where r = 3, 6, 8, 13.5, 16.5, 15, …, , …, QED
All of these values of m are less than 1, and there are 70 iterates.
For larger values of e, the median reaches 1, so the limiting median must be at least 1. We found that
achieves a limiting median of 1 for 22 values of a.

10. General Form of m An experimental study suggests:
m is affine almost everywhere
all rational points are either local minima or in affine segments
(i.e. no local maxima or monotonic corners)
on either side of local minima, the # of steps to convergence is greater and the same on each side.
For example, experimental evidence near the local minimum at x=2/3 suggests (for sufficiently small positive e)
Steps:

11. Conclusion Questions: The function m is sensitive:
m(0.6) = steps = 32
m( ) = steps=12488
Using a CAS (like Maple) is imperative!
Questions:
Does m have an upper bound? Record: m(0.843) =
Is m continuous? Is it almost always affine?
Can one characterize the extrema? Can a rational point be a local maximum?
What if one starts with more than 3 points?