Heisenberg Uniqueness Pairs for a Hyperbola Branch: Supercritical Nonuniqueness for Shifted Lattice Crosses

Abstract

We study Heisenberg uniqueness for the positive hyperbola branch and shifted lattice crosses in the supercritical regime $q=\alpha\gamma>1$. 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 $v\in BV((1,q))$ 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 $L^1$ 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 $L^1((0,1))$, that $\Ran(I-Q)$ is not closed, and that, outside a countable set of algebraic values of $q>1$, the operator $\sum_{j=0}^{N-1}Q^jA:L^1((1,q))\to L^1((0,1))$ has norm $2N$ for every $N\ge1$.