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

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

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.

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, 37/4, 19/2, 39/4, 10, 41/4 Theorem: The sequence of medians is montonic. Conjecture: The median occurs exactly twice before it settles permanently.

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.

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

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

Maxima along the Way median Maximum in orbit

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, …, 117.5625, …, 41.625. 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.

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: 4 60 60

Conclusion Questions: The function m is sensitive: m(0.6) = 2.4 steps = 32 m(0.600001) = 2.6353166429240703582763671875 steps=12488 Using a CAS (like Maple) is imperative! Questions: Does m have an upper bound? Record: m(0.843) = 4.64546875 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?