  We study the non-perfect propagation of information to evolving
low-dimensional environment that includes self-evolution as well as noisy
initial states and analyze interrelations between the degree of objectivization
and environment parameters. In particular, we consider an analytical model of
three interacting qubits and derive its objectivity parameters. The numerical
analysis shows that the quality of the spectrum broadcast structure formed
during the interaction may exhibit non-monotonicity both in the speed of
self-dynamics of the environment as well as its mixedness. The former effect is
particularly strong, showing that -- considering part of the environment as a
measurement apparatus -- an increase of the external magnetic field acting on
the environment may turn the very vague measurement into close to ideal. The
above effects suggest that quantum objectivity may appear after increasing the
dynamics of the environment, although not with respect to the pointer basis,
but some other one which we call generalized pointer or indicator basis.
Furthermore, it seems also that when the objectivity is poor it may be
improved, at least by some amount, by increasing thermal noise. We provide
further evidence of that by analyzing the upper bounds on distance to the set
of states representing perfect objectivity in the case of a higher number of
qubits.
