  Some properties of the non--linear coherent states (NCS), recognized by Vogel
and de Matos Filho as dark states of a trapped ion, are extended to NCS on a
circle, for which the Wigner functions are presented. These states are obtained
by applying a suitable displacement operator $D_{h}(\alpha) $ to the vacuum
state. The unity resolutions in terms of the projectors $| \alpha, h> < \alpha,
h^{-1}| ,| \alpha, h^{-1}> < \alpha, h| $ are presented together with a measure
allowing a resolution in terms of $| \alpha, h> < \alpha, h| $. $D_{h}(\alpha)
$ is also used for introducing the probability distribution funtion
$\rho_{A,h}(z) $ while the existence of a measure is exploited for extending
the P-representation to these states. The weight of the n-th Fock state of the
NCS relative to a trapped ion with Lamb-Dicke parameter $\eta ,$ oscillates so
wildly as $n$ grows up to infinity that the normalized NCS fill the open circle
$\eta ^{-1}$ in the complex $\alpha $-plane. In addition this prevents the
existence of a measure including normalizable states only. This difficulty is
overcome by introducing a family of deformations which are rational functions
of n, each of them admitting a measure. By increasing the degree of these
rational approximations the deformation of a trapped ion can be approximated
with any degree of accuracy and the formalism of the P-representation can be
applied.
