2026/07/19 by Zhiqiang Wan
Mathematics · #math.CA #math.AP #math.DS
We study Heisenberg uniqueness for the positive hyperbola branch and shifted lattice crosses in the supercritical regime q=αγ>1. We resolve the infinite-dimensionality clause of the arbitrary-shift problem posed by Giri and Manna: for arbitrary shifts on both arms, the normalized pre-annihilator is infinite-dimensional. More precisely, every v∈ BV((1,q)) has a global BV pre-annihilating extension; the extension is unique unless both twisting phases are trivial, in which case its ambiguity is one-dimensional. The proof reduces the annihilation conditions to a graph equation for a twisted Perron--Frobenius operator and combines a phase-uniform Lasota--Yorke estimate with peripheral spectral rigidity. We also give an exact operator-theoretic normal form for the entire L1 pre-annihilator in terms of the maximal convergence domain of the associated Green series. Writing Q for the twisted product and A for the forcing operator, we show that Q has the closed unit disk as its spectrum on L1((0,1)), that \Ran(I-Q) is not closed, and that, outside a countable set of algebraic values of q>1, the operator ∑j=0N-1QjA:L1((1,q))→ L1((0,1)) has norm 2N for every N≥1.