  We investigate the Nth order amplitude squeezing in the fan-state |\xi
;2k,f>_F which is a linear superposition of the 2k-quantum nonlinear coherent
states. Unlike in usual states where an ellipse is the symbol of squeezing, a
4k-winged flower results in the fan state. We first derive the analytical
expression of squeezing for arbitrary k, N, f and then study in detail the case
of a laser-driven trapped ion characterized by a specific form of the nonlinear
function f. We show that the lowest order in which squeezing may appear and the
number of directions along which the amplitude may be squeezed depend only on k
whereas the precise directions of squeezing are determined also by the other
physical parameters involved. Finally, we present a scheme to produce such
fan-states.
