Orbits & itineraries — flashcards

The initial claim — that Franklin's full-stabilizer fixed points and physics's reduced-stabilizer broken vacua are two opposite, exhaustive cases, with stabilizer size as the operative distinction (st/franklin-fixed-points-as-symmetry-breaking) — was tested directly and refuted. Mean-field Landau theory, F(m)=a(T−T_c)m²+bm⁴, shows m=0 keeps full ℤ₂ stabilizer at every temperature while flipping between being the realized (stable) state and an unrealized (unstable) one (F''(0)=2a(T−T_c), sign change confirmed symbolically at T=T_c) — stabilizer size alone doesn't determine selection. Feigenbaum's universal function g, the fixed point of the period-doubling operator T[g](x)=αg(g(x/α)), supplied a second refutation: g is unstable (its own instability eigenvalue is the Feigenbaum constant δ>1) yet still governs the entire period-doubling asymptotics — dynamical stability doesn't determine selection either. Feigenbaum's second constant α≈2.502907875 was independently verified via superstable-cycle-width ratios for f(x)=1−rx², giving successive ratios −3.219, −2.627, −2.528, −2.508, −2.504, converging toward the known value. Both refutations together forced a replacement: three independent selection mechanisms — algebraic cancellation (Franklin), energetic minimization (Landau, Higgs), and asymptotic dominance via a stable manifold (Feigenbaum) — rather than one axis (stabilizer size) or two opposite cases.

Any two complete decompositions of a polynomial into indecomposable factors under composition have the same length, and are related by explicit moves. Compositional structure is nearly unique, with Chebyshev polynomials and monomials the named exceptions.

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.

Fractional integration by parts, derived and verified using the Grünwald–Letnikov finite-difference definition of the fractional derivative (α=0.6: both sides of the identity matched to 10 significant digits, 0.22554001162413295 vs. 0.22554001162413304), establishes a genuine asymmetry absent at integer order: a left-sided fractional derivative under integration by parts produces a right-sided one. The resulting fractional Euler–Lagrange equation (Agrawal, 2002) mixes left and right fractional derivatives for exactly this reason — Riewe's original motivation (1996–97) was building genuine dissipation directly into a Lagrangian via this left/right mixing, rather than bolting on an external friction force.

Fractional Noether's theorem (Frederico–Torres, 2007) states that the conserved quantity satisfies a fractional differential condition rather than ordinary constancy — whether this still forces genuine constancy is convention-dependent: the Caputo derivative's kernel is exactly the constants, while the Riemann–Liouville derivative's kernel is larger, including non-constant power-law functions. The inverse-square potential V(x)=k/x² was identified and verified as the unique power-law potential whose full action (not just its equations of motion) has an exact dilation symmetry x→λx, t→λ²t; the resulting conserved quantity D=tH−½xp was verified numerically conserved to ~2×10⁻⁸ along an integrated trajectory, alongside H itself. The connection to conformal quantum mechanics (de Alfaro–Fubini–Furlan, 1976) and the full SO(2,1) symmetry (energy, dilation, and a third 'special conformal' generator) was named, with only D itself independently verified — the third generator and the full algebra were not constructed or checked here.

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.

Laskin's fractional Schrödinger equation — motivated by path integrals over Lévy flights rather than Brownian motion, replacing the ordinary Laplacian kinetic term with a fractional one |k|^α defined via Fourier multiplication — reuses the identical construction (replace an ordinary derivative's Fourier multiplier with |k|^α) already established for the fractional-derivative machinery in st/fractional-calculus-of-variations and st/fractional-hamiltonian-and-berry-keating, applied here to a new, quantum-mechanical setting rather than as a new mathematical tool. Time evolution under this equation was tested for linearity — the one structural property the no-cloning theorem actually depends on — via a split-step Fourier simulation: U(aψ₁+bψ₂) vs. aU(ψ₁)+bU(ψ₂) matched to 1.2×10⁻¹⁵ under α=1.4 evolution.

Non-Markovian open quantum systems and information backflow were addressed via the Breuer–Laine–Piilo measure (trace distance between two evolving states), distinguishing Markovian (one-way information loss) from non-Markovian (information returning to the system) dynamics on the exactly-solvable damped Jaynes–Cummings model in two regimes: strongly-damped (effectively Markovian — trace distance never increased across 499 simulated steps) and weakly-damped (non-Markovian — trace distance increased at 209 of 499 steps, with the excited-state population visibly reviving, e.g. dropping to 0.0028 then rising back to 0.13).

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.

z^s satisfies Laplace's equation ∇²u=0 for any complex s, not just integers — verified numerically to finite-difference precision — as an automatic consequence of being holomorphic; the general principle (any PDE invariant under both dilation and rotation has z^s as a separable eigenfunction) has named physical instances: spiral vortex sheets in 2D fluid dynamics (self-similar Kelvin–Helmholtz roll-up), and unstable photon orbits near a black hole's photon sphere, where the winding rate is tied to the orbit's own Lyapunov exponent. Applying the Joukowski map to a logarithmic spiral instead of the unit circle produces a 'damped Chebyshev' identity, T_n(cosh(a)cos(b)+i·sinh(a)sin(b)) = cosh(na)cos(nb)+i·sinh(na)sin(nb) — verified numerically to 15 digits — generalizing T_n(cosθ)=cos(nθ) by giving both frequency and growth rate simultaneously, and identified as the actual mechanism underneath the Chebyshev/Fourier correspondence, requiring no trigonometric identity manipulation.

For a composite signal built from several spiral modes, G'/G is dominated by the nearest pole (verified numerically) — the same logarithmic-derivative-locates-structure principle used elsewhere on this map, applied here to pole location rather than zero location. Locating that pole via ordinary Newton's method on G itself fails outright (the iteration diverges), for an identifiable reason: poles are repelling fixed points of Newton's method applied to G, since Newton's method finds zeros, not poles. Applying Newton's method to 1/G instead converges cleanly, to 17 digits in four steps. A further structural obstruction: a sum of m poles produces m−1 extraneous zeros of G itself, contaminating any single large contour meant to extract all modes at once — small, pole-isolating contours recover exact values instead, though no general fix for the multi-pole case (beyond this practical workaround) was found.

For a polynomial map over Q_p, whether iteration is chaotic is a question about reduction mod p, not about the field being non-archimedean per se. Under good reduction (e.g. f(x)=4x(1-x) at odd p), the multiplier stays a p-adic unit away from the critical point, so |f(x)-f(x')|_p = |x-x'|_p exactly at every step — valuations are preserved, not amplified, and the Julia set is empty (Rivera-Letelier; Benedetto). Under bad reduction (e.g. f(z)=z²+c with |c|_p>1), the identity f(z)-f(w)=(z-w)(z+w) instead gives genuine expansion by |c|_p^{1/2} per step on the Julia set: a repelling fixed point exists there, gaps between nearby points grow by an exact integer factor each iteration, giving a positive Lyapunov exponent and dynamics conjugate to the full 2-shift. So 'chaos is archimedean' is false as a general claim — the real dividing line is good vs. bad reduction, a condition on the map's coefficients relative to p, independent of which absolute value is in play.

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.

The Korteweg–de Vries equation is the motivating case for a 'nonlinear Fourier transform': inverse scattering (Gardner–Greene–Kruskal–Miura, 1967) linearizes KdV via an auxiliary linear scattering problem, in a role structurally analogous to the ordinary Fourier transform for linear PDEs but through a genuinely different mechanism — a Lax pair (L,A) with L̇=[A,L] (Lax, 1968). The general mechanism is elementary once stated: L̇=[A,L] implies L(t)=U(t)L(0)U⁻¹(t) for a unitary U(t) built from A, so eigenvalues are conserved by pure linear algebra — for any A(t) whatsoever, nothing specific to KdV. This is the same algebraic fact as the Heisenberg equation of motion, Ȯ=(i/ℏ)[H,O]. The single-soliton solution u=2κ²sech²(κ(x−4κ²t)) was verified to satisfy KdV exactly (symbolic residual reduces to 0), and its associated bound-state eigenfunction ψ=sech(κx) verified against −ψ''+V(x)ψ=−κ²ψ.

The free rigid body's (Euler top's) equations recast as Ṁ=[M,Ω], with M,Ω built from angular momentum and angular velocity, is a second, structurally independent instance of the same Lax-pair mechanism established for KdV (st/lax-pair-forces-isospectral-evolution) — a genuinely different setting (finite-dimensional classical mechanics rather than a nonlinear PDE) obeying the identical algebraic fact. Orientation is separately reconstructed via the quaternion commutator identity [p,q]=2(p×q), with q̇=½q⊗ω once ω(t) is known from the (nonlinear but now-solved) Euler equations. Verified numerically: |L| and total energy conserved to 5 digits over the integrated trajectory, and the quaternion commutator identity confirmed to reproduce the Lax equation directly.

Liouville–Arnold integrability requires n independent, Poisson-commuting conserved quantities on a 2n-dimensional phase space. Under this definition, exactly three heavy-top configurations are integrable: Euler's case (no gravity), Lagrange's case (I₁=I₂, center of mass on the symmetry axis), and Kovalevskaya's case (I₁=I₂=2I₃, center of mass in the equatorial plane — found via Painlevé analysis, S. Kovalevskaya, 1888); every other mass/inertia configuration is chaotic. Arnold's theorem (V. Arnold, 1966) reframes ideal 3D fluid flow as geodesic motion on the group of volume-preserving diffeomorphisms; energy and helicity are the known conserved quantities, far short of the infinitely many Liouville–Arnold would need, and after roughly 150 years of searching no Lax pair for Navier–Stokes is known.

The claim that a positive Lyapunov exponent is structurally incompatible with a Lax pair's existence is given a mechanism — SO(3)'s non-abelian structure (no shared eigenvectors between rotations about different axes; no nontrivial 1-dimensional representations) denies generic nonlinear systems the algebraic coincidence a Lax pair requires — and then tested directly: the Lorenz system (E. Lorenz, 1963), perturbed by 10⁻⁸, diverges exponentially (measured Lyapunov exponent ≈0.9, close to the commonly cited 0.9056), while the Euler top (st/euler-top-second-lax-pair-instance), perturbed identically, stays bounded at O(10⁻⁷) over the same integration window — the one system with a Lax pair, and the one without, behaving exactly as the claimed incompatibility predicts.

The Bak–Tang–Wiesenfeld sandpile model (1987), run on an L=60 lattice with threshold 4 over 40,000 grain drops, produces avalanches spanning four orders of magnitude (sizes 1 to 11,194) with no external tuning. Dhar's abelian sandpile group (D. Dhar, 1990) identifies recurrent sandpile configurations with spanning trees of the underlying graph via the Matrix-Tree theorem — verified directly on a 2×2 grid, where brute-force enumeration gives 192 recurrent configurations, exactly matching the determinant of the reduced graph Laplacian (192) computed independently via linear algebra. Separately, a fixed-energy (undriven, periodic-boundary) variant locates a genuine absorbing-to-active phase transition at ζ_c≈2.1–2.2, and the driven sandpile's self-organized density (ζ≈2.10) lands on that independently-located transition point.

Iterating n ↦ n/2 for even n and 3n+1 for odd n reaches 1 from every positive integer. Open.

Lothar Collatz, 1937.

Joseph Ritt, 1922.

Rufus Bowen, 1979.

A full orbit determines its seed; orbit statistics do not. Reversing the map is the question of how much of the seed survives being averaged.

Studying iteration in general and recovering the integer facts as corollaries, rather than building up from the integers and meeting iteration late. A claim about the right order to learn things in, not about what is true.

The alphabet size is the number of inverse branches of the map. Choosing base p and choosing to work p-adically are versions of the same choice, differing in how the refinement is oriented — disjoint balls rather than intervals sharing boundaries.

A continued fraction is the run-length encoding of a geodesic — the forward itinerary of the Gauss map. The digits stop being digits and become a record of where a point went.

The same reframe applied to a different object, and it inverts: a nested radical is a backward orbit. Running the two at once gives the natural extension — an interval crossed with a Cantor set.

A map applied over and over, and the symbolic record of where the point went. Continued fractions turn out to be run-length encodings of geodesics under exactly this construction.