  A new description of strange attractor systems through three geometrical and
dynamical invariants is provided. They are the correlation dimension
($\mathcal{D}$) and the correlation entropy ($\mathcal{K}$), both having
attracted attention over the past decades, and a new invariant called the
correlation concentration ($\mathcal{A}$) introduced in the present study. The
correlation concentration is defined as the normalised mean distance between
the reconstruction vectors, evaluated by the underlying probability measure on
the infinite-dimensional embedding space. These three invariants determine the
scaling behaviour of the system's R\'{e}nyi-type extended entropy, modelled by
Kummer's confluent hypergeometric function, with respect to the gauge parameter
($\rho$) coupled to the distance between the reconstruction vectors. The
entropy function reproduces the known scaling behaviours of $\mathcal{D}$ and
$\mathcal{K}$ in the 'microscopic' limit $\rho\to\infty$ while exhibiting a new
scaling behaviour of $\mathcal{A}$ in the other, 'macroscopic' limit $\rho\to
0$. The three invariants are estimated simultaneously via nonlinear regression
analysis without needing separate estimations for each invariant. The proposed
method is verified through simulations in both discrete and continuous systems.
