'Entropy Bagels' and Other Complex Structures Emerge From Simple Rules | Quanta Magazine

Abstraction: Complex fractal structures emerging from iterated simple polynomial rules

Key points:

• Mandelbrot set arises from iteration of f(x) = x² + c; "mind-blowing" complexity from simple rules (Georgia Tech's Matthew Baker)
• Lyubich (Stony Brook) proved in 1990s that vast majority of c values in [-2, 1/4] produce stable "hyperbolic" behavior (convergent cycles)
• Petsche and Noytaptim proved only three totally real values of c produce periodic cycles: 0, -1, and -2
• "Entropy bagel": fractal ring in complex plane discovered from Thurston's pre-2012 graph of entropy values for ~1 billion real c values plus their Galois conjugates
• Entropy measures how unpredictable the iterated sequence is; bagel reveals complex structure hidden in real-valued iteration
• Van Strien (Imperial College) working toward proof extending hyperbolicity to all real analytic functions, not just polynomials

Connections: William Thurston · Misha Lyubich · Dynamical Systems · Fractals · Chaos Theory · Complex Analysis

Source: https://www.quantamagazine.org/entropy-bagels-and-other-complex-structures-emerge-from-simple-rules-20240227/