Gap-Word β-Shift

What It Is

Greedy β-expansion digits of a random x in [0,1) at base β ≈ 1.55, fixed by the aperiodic admissibility word 1 0 1 00 1 000 … (a 1 then a growing run of 0s). Because the β-expansion of 1 is aperiodic, β is a non-Parry number and the β-shift is NON-SOFIC: its Fischer cover (the number of distinct follower sets) grows without bound. Topological entropy log β ≈ 0.438, near-matched to the Golden-Mean β-Shift (sofic, β ≈ 1.618). The non-sofic pole of the matched-entropy contrast --- and, unlike the Sturmian words, non-sofic at POSITIVE entropy via the orbit-of-1 mechanism rather than an irrational rotation.

Interpretation

Standard analysis sees: few distinct values; anti-correlated (alternating); blue spectrum (high-frequency power); homoskedastic; low-dimensional. The atlas finds no named structure, but the source is distinctively extreme on 2-adic:distance_entropy (-2.7z) — beyond what the standard bank predicts for it. It sits beside Symbolic Henon in the atlas (standard-bank rank 40) — a neighbor conventional features miss.

Composition

Binary sequence — two distinct symbols.

Render Gallery

Native > phase_plotraw

Native > timeseriesraw

Native > triple_latticeraw

Boltzmann > lag_map

Boltzmann > spin_grid

Cayley > default

Chladni > free_plate_nodal

Chladni > plate_bottom

Chladni > plate_top

Heisenberg (Nil) (centered) > signed_log_z

Heisenberg (Nil) (centered) > xy_path

Hodge–Laplacian > default

Inflation (Substitution) > barcode

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

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

Ulam Spiral (Sacks) > default

Ulam Spiral (Square) > default

Visibility Graph > default

Wavelet Cascade > default

Atlas Position

Open in Atlas →

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