Correlational properties of Chebyshev chaotic sequences (Q2703254)

A chaotic sequence \(X_1,X_2,\dots\) is considered, where \(X_{i+1}=\tau(X_i), \tau\colon[-1,1]\to[0,1]\) is a Chebyshev polynomial of degree \(k\). Using the Perron-Frobenius operator and the equidistributivity property, the authors evaluate \(E(X_n^{N_1}X_{n+l}^{N_2})\) and the characteristic function of \((X_n,\dots,X_{n+l})\). E.g. it is shown that for any NEWLINE\[NEWLINEk>\max(N_1,\dots,N_{m-1}),\;E(X_n^{N_1}X_{n+l_1}^{N_2}\cdots X_{n+l_{m-1}}^{N_2}) =E(X^{N_1})E(X^{N_2})\dots E(X^{N_m})NEWLINE\]NEWLINE for any \(l_1,\dots,l_{m-1}\) and any \(N_m\).