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For every fixed $\epsilon$ $\in$ (0, 1), we construct an operator on theseparable Hilbert space which is $\delta$-hypercyclic for all $\delta$ $\in$($\epsilon$, 1) and which is not $\delta$-hypercyclic for all $\delta$ $\in$(0, $\epsilon$).

$\epsilon$-hypercyclic operators that are not $\delta$-hypercyclic for $\delta$ < $\epsilon$