We systematically eliminate access to configurations of an otherwise elementary thermal system model until only remains a subset of them of vanishing measure. The thermal system consists of 2^N distinguishable non-interacting degrees of freedom, each occupying energy levels of the form u_j,k = 2^j 2^2k, j,k=0,1,2,…, The system’s configurations are eliminated when their energies do not match the form 2^N-l, l=0,1,2,..., leading to a discrete scale invariant set of available configurations as Nà∞. In doing this we achieve the following results: 1) The constrained thermal system becomes an analogue of the dynamics towards the multifractal attractor at the period-doubling onset of chaos. 2) The statistical-mechanical properties of the thermal system depart from those of the ordinary Boltzmann-Gibbs (BG) form and acquire features from q-statistics. 3) Redefinition of energy levels as logarithms of the original ones recovers the BG scheme and the free energy Legendre transform property.We systematically eliminate access to configurations of an otherwise elementary thermal system model until only remains a subset of them of vanishing measure. The thermal system consists of 2^N distinguishable non-interacting degrees of freedom, each occupying energy levels of the form u_j,k = 2^j 2^2k, j,k=0,1,2,…, The system’s configurations are eliminated when their energies do not match the form 2^N-l, l=0,1,2,..., leading to a discrete scale invariant set of available configurations as Nà∞. In doing this we achieve the following results: 1) The constrained thermal system becomes an analogue of the dynamics towards the multifractal attractor at the period-doubling onset of chaos. 2) The statistical-mechanical properties of the thermal system depart from those of the ordinary Boltzmann-Gibbs (BG) form and acquire features from q-statistics. 3) Redefinition of energy levels as logarithms of the original ones recovers the BG scheme and the free energy Legendre transform property.