A symbolic description of the processes of complex systems (Q1903464)

In a first part, the standard complexity of a binary word is introduced and characterized. From a finite binary word, on can construct in a natural way a function taking values in \(\{+ 1,- 1\}\), constant on subintervals of equal length corresponding to a subdivision of an interval \([a, b]\subset \mathbb{R}\). From the value at \(b\) of the iterated integral (up to order \(r\)) of this function, one can associate in a unique way a system of equations \[ \sum^{i= p}_{i= 1} x^m_i- \sum^{j= q}_{j= 1} y^m_j= N_m, \] where \(0\leq m\leq r- 1\) and the solutions are integers. The initial function is completely determined by the \(x_i\) and the \(y_j\). The structural complexity of the binary word is the minimum \(r\) such that there is a unique integer solution to this system of equations. This system determines a compact Riemann surface whose genus is a bound of the structural complexity of the underlying binary word. (This is shown by referring to the zeta function of this surface and to Weil's proved hypothesis.) To a symbolic coding of the iterated dynamics of \(z_{k+ 1}= h_\lambda(z_k)\), (\(z_0= 1\), \(h_\lambda\) is a unimodal mapping from \([0, 1]\) to \([0, 1]\)), which undergoes a cascade of period doubling as the parameter \(\lambda\) varies, corresponds a certain sequence of infinite binary words with periodic structure; from one word to the next in the sequence, a period doubling occurs and it is shown that the structural complexity increases by one.