Fractal dimension — flashcards

The Hausdorff dimension of a conformal repeller is the unique zero of the topological pressure function. An entire dynamical system is compressed into the root of one equation.

Effects mattering at every scale is not sufficient to make fractal machinery the right tool; what is required is invariance under a scaling with a power law. Multiscale relevance without self-similarity is the domain of renormalisation, not of dimension theory.

The fine structure constant α is a running coupling, not a scale-invariant quantity: verified numerically at two energy scales, α⁻¹(0)=137.036 against α⁻¹(M_Z)=128.957 (measured values), with a simple one-electron-loop estimate reproducing α⁻¹(M_Z)≈134.47 — capturing part but not all of the shift, the remainder named (muon, tau, hadronic vacuum polarization) rather than computed. This was deliberately set up as the inverted case relative to st/fractional-noether-and-dilation-current's 1/x² potential: both ask the identical diagnostic question — does this system have an exact dilation symmetry, evidenced by a genuine conserved current, or does the relevant coupling run with scale instead — with the 1/x² potential answering 'exact symmetry' and the fine structure constant answering 'runs' via the QED beta function (vacuum polarization screening).

Causal Dynamical Triangulations, Asymptotic Safety, and Loop Quantum Gravity converge on spacetime's spectral dimension — defined via a diffusion-process return probability — flowing from 4 at large scales to approximately 2 at the Planck scale, genuinely passing through fractional values along the way rather than jumping discretely. The definition itself was independently sanity-checked on ordinary integer-dimensional lattices before trusting it applied to the quantum-gravity claim: estimated dimension 0.97, 2.15, and 3.01 for true dimensions 1, 2, and 3 respectively.

st/scale-invariance establishes that fractal machinery needs a power law, not just multiscale relevance. What a discrete scaling ratio b (as opposed to continuous scale invariance under every ratio) adds: the zeta-type function Z(s) = 1/(1-λb^{-s}) has poles not at one real value but at a full periodic ladder s_k = ln(λ)/ln(b) + 2πik/ln(b), k ∈ Z — genuinely complex-valued, equally spaced in the imaginary direction. This ladder is not a formal curiosity: its real part matches the box-counting dimension of the Weierstrass function, D = 2 + ln(λ)/ln(b) (verified directly), and its imaginary spacing 2π/ln(b) matches, for the specific ratio given by the Feigenbaum constant δ, a numerically verified prediction of ≈4.077 for the period-doubling cascade's pole spacing — attributed to Lapidus's 'complex dimensions' program.

The pattern running through st/transforms-diagonalize-symmetry-groups — find the group being diagonalized, get a transform — has real limits, not just harder cases. The almost-Mathieu operator at golden-ratio flux is a concrete instance: verified numerically to have Cantor-set (singular continuous) spectrum with no spectral floor, a fact with no accessible 'which group?' answer of the kind the rest of the survey relies on — proved rigorously in general by the Ten Martini theorem (Avila–Jitomirskaya). A second, structurally different obstruction sits alongside it: non-type-I groups (e.g. free groups), whose von Neumann algebras are type II/III and admit no clean Peter–Weyl-style irreducible decomposition at all.

A binomial multiplicative cascade, Z_n(q)=(m₀^q+m₁^q)ⁿ, has scaling exponent τ(q)=−log₂(m₀^q+m₁^q); its q-derivative is the average Hölder exponent α(q)=τ'(q), and the Legendre transform f(α)=qα−τ(q) produces the full multifractal spectrum. Z_n(q) is then explicitly reframed as a statistical-mechanics partition function, E_I=−ln μ(I), β=q — a real algebraic identification (the same object under a relabeling that actually holds up, not a loose analogy), verified empirically at q=0,0.5,1,2,3,5 against the closed-form binomial expressions.

The log-periodic power law (LPPL) model used for critical-transition precursors is identified as exactly a spiral mode of st/spiral-transform-unifies-fourier-mellin in w=log(t_c−t): parameters are recovered accurately from noisy synthetic data, but usable lead-time collapses sharply and non-gracefully as the available data before t_c shrinks, rather than degrading gradually. A claim about the model's robustness to false positives, made from a single test on structureless (random-walk) data returning an unremarkable frequency parameter, was corrected after a proper multi-realization test: across many independent random walks, the fit typically returns large frequency values that would read as genuine log-periodicity — the model hallucinates apparent structure on pure noise most of the time, the opposite of the original claim.

Two constructions can give Cantor sets with the same Hausdorff dimension over R while behaving completely differently once the absolute value changes. A digit-restriction set (base-p digits confined to a subset of size d) has dimension log(d)/log(p) at every place where the construction makes sense — purely geometric, place-independent, verified matching a middle-thirds-style real Cantor set exactly. A Julia set's dimension is not: under good reduction (st/padic-reduction-and-chaos) it collapses to 0 in Q_p even where the real Julia set has dimension strictly between 0 and 1, and under bad reduction it takes its own value, log(2)/log(r) for expansion factor r=|c|_p^{1/2}, moving in the opposite direction from the real dimension as a parameter varies — with neither the sum nor the product of the two staying constant, so there is no product-formula analogue for dimension the way there is for absolute values themselves.

Rufus Bowen, 1979.

Rufus Bowen, 1979.

Bowen's formula computes the Hausdorff dimension of a Julia set as the zero of a pressure function. An entire dynamical system is summarised by a single number, and the pressure equation is the accounting for what that number is allowed to forget.

Hausdorff dimension takes values in a continuum and integer values are the exceptional slice. The familiar case is the degenerate one, which is the cleanest instance of this schema on the map.

Hausdorff and box dimension, taking values in a continuum. Integer dimension is the exceptional case rather than the default — which is the cleanest instance on this map of a familiar object turning out to be a degenerate slice of a general one.