Attractlet attractlet model · § 7.4

The Bernoulli Shift

The textbook chaotic map: double a number, drop the whole part, repeat. Nothing settles, nothing comes back, and on a real machine it collapses to zero — because the digits it runs on are handed in from outside.

The map is as simple as a chaotic system gets: take a number between 0 and 1, double it, and throw away the whole part, so 0.63 becomes 0.26 becomes 0.52, and so on. Two starts a millionth apart are pushed twice as far apart every step until they share nothing. There is no attractor to fall onto and no point a nudge returns to. What looks like endless motion is really the map spending one binary digit of the starting number per step. A real computer holds only about fifty-two of them, so a typical start runs out of digits and collapses to exactly zero and stays there. The motion was never the map’s own; it was reading out a supply of digits, and when the supply is gone the motion stops.

Double, Drop, Repeat

the unit interval · 0 on the left, 1 on the right

points still moving—

collapsed to zero—

twin separation—

digits left in a typical start—

step count—

maprunning

iterating

The motion is a supply of digits being spent, one per step. The step count is the map’s clock, handed in from outside; the map keeps no time of its own. Stop the map and every point freezes. Lower the precision slider and points run out of digits sooner and collapse to zero faster; raise it and the collapse is only postponed, never removed. The histogram along the bottom is the spread of the whole cloud: whatever shape you start with, it flattens toward uniform.

Why This Is an Attractlet

the kernel’s own classification

x_n+1 = 2·x_n mod 1
x₀ in [0, 1) — equivalently, shift the binary point one place right and drop the leading digit

The kernel’s Attractlet Model (Formal or Symbolic) class explicitly includes “simulations and numerical solvers” (ATTRACTLET, § 7.4). The Bernoulli shift is a numerical iteration and nothing more. Whatever it does, it does inside the arithmetic, and the arithmetic runs only while a machine runs it.

It is supplied twice over. The iterator supplies every step. And the starting number supplies the answer in advance: the map does no work but reveal, one binary digit at a time, digits that were already written into x₀. Reading the map forward is reading those digits off; there is no dynamics adding anything the start did not already contain.

That is why a real computer collapses it. A double holds about fifty-two binary digits. After that many steps the map has shifted every real digit away and only the padding zeros remain, so the point lands on exactly 0 and stays — not a fixed point it was drawn to, but the machine running out of supply. Turn the precision down and it happens sooner; turn it up and it is merely delayed.

The Shape of the Basin

none for a point; uniform for a cloud

For a single point there is no basin at all. A basin is the set of starts that return to the same place after a nudge, and here a nudge is doubled every step: two starts a millionth apart are a thousandth apart in ten steps and share nothing in twenty. Press Release twins and watch them fly apart. Nothing returns to anything.

For a spread of points there is a different kind of return, and it is the honest one to report. Press Scatter new starts with any lopsided cloud: the histogram along the bottom flattens toward uniform and stays flat. The uniform distribution is invariant under the map — but that is a statement about ensembles, not about any point holding itself together. No individual is doing anything but being read out and, on a real machine, running dry.

None of this moves the classification. § 7.4: “No attractlet may be promoted to attractor status by complexity, longevity, apparent stability, mathematical elegance, or external coupling strength.” The Bernoulli shift is the cleanest chaos in the textbook, and none of that counts. It is order handed in as digits; remove the supply and it is gone.

The doubling map (dyadic transformation) is standard in ergodic theory; see e.g. Devaney, R. L. (1989). An Introduction to Chaotic Dynamical Systems (2nd ed.). Addison-Wesley. Author, title and edition stated with confidence; pagination not independently re-verified here.

← Back to Attractlets