Matpic
mathematics, in pictures
Concept

Markov partition

A division of a space into pieces that a map carries exactly across whole pieces, never part of the way into one. Orbits are then coded by walks on a finite graph, so counting periodic points and measuring entropy become computations with a matrix.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as topological entropy — the same set of essays touches all of them, so they are one junction rather than several.

A map drawn through the cycle 0 → 1/3 → 1, and the graph its pieces make. The graph of a map made of two straight pieces through a cycle of three points, with the cycle drawn as a staircase, beside a two-node graph showing which piece may follow which and the matrix of that graph.

A matrix that counts the returns

Draw a map straight through the cycle 0 → 1/3 → 1 and its two pieces carry each other in a fixed pattern: the left piece only across the right, the right across both. The orbits' words are then walks on a two-node graph, and the number of points that come back after n steps is the trace of that graph's matrix to the nth power — 1, 3, 4, 7, 11, 18 — each one checked by solving for the points exactly.

dynamics · Symbolic dynamics The logistic map applied 6 times at 3.5, 3.83, 4, and its folds. Side-by-side graphs of the logistic map composed with itself, one per parameter, each labelled with how many monotone pieces it has.

The folds that measure chaos

Apply the logistic map six times and its graph goes up and down 38 times at r = 3.5 and 64 times at r = 4. How fast that number of folds multiplies with each further step is the map's topological entropy: zero through the whole cascade of period doublings, log of the golden ratio in the window of three, log 2 at the top — and it never decreases as r rises.

dynamics · Symbolic dynamics

Named alongside it

The objects these essays reach for when they reach for this one.

Golden ratioPeriodic orbitTopological entropyChaosCharacteristic polynomialEigenvalueIntermediate value theoremLogistic mapLyapunov exponentMandelbrot setPeriod-doublingSymbolic dynamics

All concepts