2026/07/21 by Maciej Janowicz, Arkadiusz Orłowski
#math.DS #math-ph #math.MP
We represent scalar logistic iterates as multiplication operators on L2([0,1]) and study their fixed matrix elements in the normalized shifted-Legendre basis. Three parameter regimes permit analytical control. At r=5/2, every fixed matrix element converges to 3δkl/5, where δkl is the Kronecker delta. At r=16/5, the attracting period-two orbit yields phase-resolved limits for the even and odd subsequences. At r=4, an exact Chebyshev-moment representation gives an O(4-n) approach of every fixed matrix element at iteration n to δkl/2. The case r=37/10 is treated only by a controlled finite numerical refinement study. Complementary finite diagnostics comprise matrix-element time dependence, a bifurcation-style plot, a time-averaged mean intensity, a normalized second-order intensity moment, and normalized scalar OTOC-type commutator correlation matrices. We also examine the separate finite-dimensional recursion Xk+1=R Xk(I-Xk)R^†, with fixed matrix R, using diagonal and tridiagonal amplitude profiles. These operator-valued calculations are exploratory finite-time numerics. The analytical statements concern fixed matrix elements for the specified basis indices; they are distinct from the finite-resolution observations and do not imply operator-norm convergence. A regularized phase-space lift is included as a controlled visualization.