The Wolfram Language has flexible capabilities for handling iterated maps, as well as highly optimized algorithms for common objects of investigation such as Julia sets and the Mandelbrot set.

Iterating Arbitrary Functions

Nest — iterate a function

NestList  ▪  NestGraph  ▪  NestWhile  ▪  NestWhileList  ▪  FixedPoint  ▪  FixedPointList

ReplaceRepeated — do repeated substitutions

Groupings — generate all possible nestings from a list

Substitution Systems

SubstitutionSystem — string, list, or array substitution system

RulePlot — display rules and evolutions for substitution systems

ArrayFilter  ▪  ArrayFlatten

Geometric Iteration

AnglePath — compute a "turtle graphics" path from turns and moves

AnglePath3D — compute a path from successive rotations in 3D

Space-Filling Curves

HilbertCurve — generate a Hilbert curve in any dimension

PeanoCurve  ▪  SierpinskiCurve  ▪  KochCurve

Complex Iterated Maps

JuliaSetPlot — plot Julia sets of arbitrary rational functions

JuliaSetPoints  ▪  JuliaSetIterationCount

MandelbrotSetPlot — plot the Mandelbrot set at any resolution

MandelbrotSetMemberQ  ▪  MandelbrotSetDistance  ▪  MandelbrotSetIterationCount

JuliaSetBoettcher  ▪  MandelbrotSetBoettcher

Fractal Regions

CantorMesh  ▪  MengerMesh  ▪  SierpinskiMesh

Fractal Functions

CantorStaircase  ▪  MinkowskiQuestionMark

Discrete Recurrence Relations

RecurrenceTable — create a table of values from recurrence relations

Substitution Sequences

ThueMorse  ▪  RudinShapiro

Iterated Boolean Functions

CellularAutomaton — arbitrary cellular automaton rules in any number of dimensions

Iterated String Substitutions

StringReplace

StringReplaceList — generate a multiway system with string replacements