δ ‐Almost Periodic Vectors for Bounded Linear Operators: Geometric and Dynamical Structure

We introduce δ ‐almost periodic vectors for bounded linear operators on Banach spaces, defined by uniform recurrence to the base point within a fixed tolerance δ ≥ 0. This yields a quantitative relaxation of the case δ = 0. We develop basic structural properties of the classes AP δ ( T ), including monotonicity in δ , positive homogeneity, forward invariance with scaled tolerance, orbit boundedness, closedness under power‐boundedness, and Borel regularity. For δ = 0, we show that under two‐sided power‐boundedness, AP 0 ( T ) coincides with the classical almost periodic set, while this identification fails in general. We also obtain geometric consequences in terms of finite thick coverings of the orbit. We further study the interaction with linear chaos. In infinite‐dimensional spaces, δ ‐almost periodic vectors are incompatible with hypercyclic vectors, and for hypercyclic operators, the sets AP δ ( T ) have empty interior. We also introduce the recurrence radius δ ∗ ( x ) and prove its lower semicontinuity under power‐boundedness. Finally, we prove that if T n x ⟶0, then x ∈ AP δ ( T )⇔ δ ≥ ‖ x ‖. In particular, strict contractions satisfy AP δ ( T ) = { x ∈ X : ‖ x ‖ ≤ δ }. We conclude with illustrative applications to unilateral weighted backward shifts on ℓ p . MSC2020 Classification: Primary 47A16; Secondary 47A35, 37B20, 43A60