  Two new notions of equivalence for representations of a Toeplitz algebra
$\mathcal{E}_n$, $n<\infty$, on a common Hilbert space are defined. Our main
results apply to $C^*$-dynamics and the conjugacy of certain $*$-endomorphisms.
One particular case of the relations is shown to coincide with the multiplicity
of a representation. Previously known results due to Laca and Enomoto-Watatani
are recovered as special cases.
