The Directions That Only Stretch

Last timeRunning a Map Backwards

Some directions come out of a map pointing exactly where they went in, only longer or shorter, and finding them turns repeated application of a map into simple arithmetic.

Feed a map a direction and in general it comes out pointing somewhere else. That

is what a map does. But for most maps there are a few special directions that come

out pointing exactly where they went in, just longer or shorter or reversed. Along

those directions the map is not doing anything complicated. It is multiplying by a

number.

Finding those directions is worth real effort, because along them a map that is

otherwise an awkward grid of numbers becomes a single multiplication, and

multiplication by a number is something anyone can reason about over a hundred

repetitions.

FIG 1The whole definition

: a surviving direction, required to be non-zero or the statement says nothing
: the factor by which the map scales that direction
: where the map sends it, which is back along the same line

Read the equals sign as the demand being made. The left side is whatever the map does, which usually points somewhere new. The right side insists it points along the original line. Most directions fail this test; the ones that pass are the structure of the map.

The number is a rate

The factor attached to a surviving direction is not a length or a position. It is

a rate of change per application of the map, and that reading is the one that

makes it useful.

A factor larger than one means that direction grows a little more every time the

map runs. A factor between zero and one means it shrinks towards nothing. A

negative factor means it flips to the other side of the origin each time, so it

oscillates while growing or shrinking. A factor of exactly one means the direction

is completely untouched, which is how a steady state shows up: a configuration the

process leaves alone.

FIG 2Two directions under repetition

a direction with a factor above onea direction with a factor below one

Both components start at one and the map has no idea which is which. After a dozen applications the growing direction is roughly ten times its starting size and the shrinking one is almost gone. Nothing in the map changed between applications; the separation is entirely the compounding of two different rates.

Repetition sorts them out

Now the consequence that makes all of this matter. Applying a map a hundred times

is one product of a hundred copies, and along a surviving direction that product

is just the factor raised to the hundredth power.

Write any starting vector as a mixture of the surviving directions, assuming for

the moment that there are enough of them to do so. Each piece then evolves on its

own, multiplied by its own factor every time. After enough repetitions the piece

with the largest factor dwarfs the rest, no matter how small it was at the start,

because exponential growth at different rates separates without limit.

The lesson stops here

4 more paragraphs to go

You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.

The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.

See the plan The contents

This is the reading half

Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.

The contents

The Directions That Only Stretch

The number is a rate

Repetition sorts them out

4 more paragraphs to go

This is the reading half

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