Minimal distance between random orbits (Q6582356)

The authors study the minimum distance for orbit segments of quenched random dynamical systems. Under mild conditions on the phase space, they prove that the decay rate of the minimum distance is slower than in the deterministic and annealed cases. The decay rates of the minimum distance in the quenched case are determined by the maximum of two dimension-like quantities: one arising from the annealed dynamics and the other from the purely quenched setting. Below, we describe the results in more detail.\N\NLet \(\sigma : \Omega \to \Omega\) be an invertible map preserving a probability measure \(P\). Let \((X, d)\) be a metric space, and let\N\[\NS: (\omega, x) \mapsto (\sigma \omega, T_\omega(x))\N\]\Nbe a measurable map on \(\Omega \times X\), where \(T_\omega : X \to X\) is measurable for a.e. \(\omega \in \Omega\). The goal is to determine the limit of the quantity\N\[\N\min_{i,j < n} d(T^i_\omega(x), T^j_\omega(y))\N\]\Nfor typical \((x, y)\) and \(\omega\) as \(n \to \infty\), where\N\[\NT^n_\omega = T_{\sigma^{n-1} \omega} \circ T_{\sigma^{n-2} \omega} \circ \cdots \circ T_\omega\N\]\Nis the second component of \(S^n(\omega, x)\).\N\NRelevant probability measures in this setting are of the form \(\nu = P \otimes \mu_\omega\), where the family \(\{\mu_\omega\}\) is equivariant, i.e., \((T_\omega)_\ast \mu_\omega = \mu_{\sigma \omega}\). Let \(\mu = \int \mu_\omega \, dP(\omega)\) denote the integrated probability measure on \(X\). Define the following dimension-like quantities:\N\[\ND^{an} = \lim_{r \to 0} \frac{\log \int \mu(B(x, r)) \, d\mu(x)}{\log r},\N\]\Nand\N\[\ND^{qu} = \lim_{r \to 0} \frac{\log \int \mu_\omega(B(x, r)) \, d\mu_\omega(x) \, dP(\omega)}{\log r}.\N\]\N\NThe main result of the paper is as follows. Suppose \(X\) has bounded local complexity. If \(S\) is Lipschitz, fiberwise stretched exponential mixing, \(\omega \mapsto \mu_\omega\) is Lipschitz, and the base map is 4-mixing at stretched exponential speed, then\N\[\N\min_{i,j < n} d(T^i_\omega(x), T^j_\omega(y)) \sim n^{-\theta}, \quad \theta = \max\left\{\frac{2}{D^{an}}, \frac{1}{D^{qu}}\right\}.\N\]\N\NThe authors derive that in the deterministic or annealed cases, the result reduces to the corresponding known cases with \(\theta =\frac{2}{D^{an}}\).\N\NA metric space \((X, d)\) is said to have bounded local complexity if there exists a constant \(C > 0\) such that, for any sufficiently small \(r > 0\), there exists \(k(r) > 0\) and points \(x_i(r)\), \(i = 1, \ldots, k(r)\), such that the balls \(B(x_i, r)\) form a cover of \(X\), and each point \(x_i\) belongs to at most \(C\) balls.