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We define and study the set ε(ρ) of end invariants of an SL(2,C) character ρ of the one-holed torus T. We show that the set ε(ρ) is the entire projective lamination space [inline-graphic 01] of T if and only if ρ corresponds to the dihedral representation or ρ is real and corresponds to an SU(2) representation; and that otherwise, ε(ρ) is closed and has empty interior in [inline-graphic 02]. For real characters ρ, we give a complete classification of ε(ρ), and show that ε(ρ) has either 0, 1 or infinitely many elements, and in the last case, ε(ρ) is either a Cantor subset of [inline-graphic 03] or is [inline-graphic 04] itself. We also give a similar> classification for "imaginary" characters where the trace of the commutator is less than 2. Finally, we show that for characters with discrete simple length spectrum (not corresponding to dihedral or SU(2) representations), ε(ρ) is a Cantor subset of [inline-graphic 04] if it contains at least three elements.

End invariants for SL(2,C) characters of the one-holed torus