On the relation between Lyapunov exponents and exponential decay of correlations

Abstract

Abstract. Chaotic dynamics with sensitive dependence on initial conditions may result in exponential decay of correlation functions. We show that for one-dimensional interval maps the corresponding quantities, that is, Lyapunov exponents and exponential decay rates, are related. For piecewise linear expanding Markov maps observed via piecewise analytic functions we provide explicit bounds of the decay rate in terms of the Lyapunov exponent. In addition, we comment on similar relations for general piecewise smooth expanding maps.

On the mixing properties of piecewise expanding maps under composition with permutations, II: Maps of non-constant orientation

“…A nice property of such maps is that the corresponding transfer operator has a finite matrix representation. It is perhaps surprising that, even in this special situation, the precise determination of the mixing rate is already a non-trivial task [21,22]. On the other hand, using a Ulam-like construction [4,23], every expanding (non-linear) Markov map can be approximated by a sequence of piecewise linear maps.…”

“…On the other hand, using a Ulam-like construction [4,23], every expanding (non-linear) Markov map can be approximated by a sequence of piecewise linear maps. In many circumstances, we can use the mixing rate of high order piecewise linear maps to bound or approximate the mixing rate of the original non-linear maps [8,21,22]. (This is extended in [14,15] to the settings of multi-dimensional expanding maps and of Anosov maps.)…”

Anomalous transport and chaotic advection in homogeneous porous media

“…Although a formal link is yet to be established [28], chaotic dynamics and ergodicity are strongly associated with decaying correlations along Lagrangian trajectories [29]; it is this mechanism that suppresses longitudinal dispersion in boundary-dominated flows. Due to such ergodicity and the punctuated nature of stretching events at stagnation points, fluid transport and deformation in porous media flows may be described as a stochastic process (where the validity of this approximation is quantified by the infinite-time Lyapunov exponent λ ∞ ).…”

Chaos in wavy-stratified fluid-fluid flow

“…Figure 7 shows the ACF measurements from the experiments and numerical simulations. The dashed curve in Figure 7 represents this bound and it is seen that the relation 26,27 holds true for this case of wavy-stratified flow as well.…”