Bernoulli Shift
chaos · 36 views
chaos
What It Is
Simplest exactly solvable chaotic map: x(n+1) = 2x mod 1. Maximal entropy h=log2, uniform invariant measure, Bernoulli process on binary digits
Interpretation
Standard analysis sees: rich, high-entropy values; high-dimensional / space-filling. The atlas finds no named structure, but the source is distinctively extreme on Zariski:residual_slope (-2.7z) — beyond what the standard bank predicts for it. It sits beside Tent Map in the atlas (standard-bank rank 69) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.31
asymmetry0.51
occupancy0.88
short-range corr0.52
long-range memory0.50
spectral colour0.48
periodicity0.25
complexity0.56
time-irreversibility0.32
volatility clustering0.21
multifractality0.27
dimensionality0.96
nonstationarity0.33
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Zariski:residual_slope | -2.8z | bank-miss 3.6σ |
G2 Root System:short_long_ratio | -2.0z | bank-miss 1.0σ |
Composition
dtypefloat64
range[6.107e-05, 0.9999]
unique values13867 / 16384
mean ± std0.499 ± 0.287
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Bernoulli_Shift.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Bernoulli_Shift.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Bernoulli_Shift.png)
Inflation (Substitution) > barcode
/d_curve/Bernoulli_Shift.png)
Inflation (Substitution) > d_curve
Information Theory > default
Julia Set > escape_histogram
Klein Bottle > default
Laplacian > default
Logarithmic Spiral > default
Möbius-S³ > mobius_cylinder
Möbius-S³ > s2_base_map
Ordinal Partition > default
/phi_spectrum/Bernoulli_Shift.png)
Penrose (Quasicrystal) > phi_spectrum
Persistent Homology > barcode
Persistent Homology > h1_diagram
Predictability > default
Recurrence Quantification > default
Spectral Analysis > default
Spectral Graph > default
Spirograph > mode_spectrum
Spirograph > phasor_trajectory
Symplectic > flux_grid
Symplectic > phase_space
/default/Bernoulli_Shift.png)
Ulam Spiral (Sacks) > default
/default/Bernoulli_Shift.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Baker Map | 3.67 | |
| LFSR (16-bit) | 4.12 | cross-domain |
| Tent Map | 4.23 |
Which Geometries Light Up
G2 Root System › G2 Root System:short_long_ratio | rank 294/298 | 0.2996 |
Modular Residue › Modular Residue:drift_anomaly | rank 295/298 | 0.0027 |
Symplectic › Symplectic:phase_reflection_symmetry | rank 294/298 | -0.0361 |
Zariski › Zariski:residual_slope | rank 296/298 | -5.5469 |
← / → within domain · ⇧← / ⇧→ alphabetical · ⇧← / ⇧→ inside an open render = same view across sources