Semi-uniform ergodic theorems and applications to forced systems

Abstract

In nonlinear dynamics an important distinction exists between uniform bounds on growth rates, as in the definition of hyperbolic sets, and non-uniform bounds as in the theory of Liapunov exponents. In rare cases, for instance in uniquely ergodic systems, it is possible to derive uniform estimates from non-uniform hypotheses. This allowed one of us to show in a previous paper that a strange non-chaotic attractor for a quasiperiodically forced system could not be the graph of a continuous function. This had been a conjecture for some time. In this paper we generalize the uniform convergence of time averages for uniquely ergodic systems to a broader range of systems. In particular, we show how conditions on growth rates with respect to all the invariant measures of a system can be used to derive one-sided uniform convergence in both the Birkhoff and the sub-additive ergodic theorems. We apply the latter to show that any strange compact invariant set for a quasiperiodically forced system must support an invariant measure with a non-negative maximal normal Liapunov exponent; in other words, it must contain some 'non-attracting' orbits. This was already known for the few examples of strange non-chaotic attractors that have rigorously been proved to exist. Finally, we generalize our semi-uniform ergodic theorems to arbitrary skew product systems and discuss the application of such extensions to the existence of attracting invariant graphs.

Datko-Pazy conditions for nonuniform exponential stability

“…We shall now show that Theorem 1 when combined with the result obtained by Morris [18] (building on the earlier work of Schreiber [30] and Sturman and Stark [32]) gives also new conditions for uniform exponential stability of continuous cocycles, i.e. of cocycles with the property that A : M → B(X) is a continuous map.…”

Global attractors of pinched skew products

“…The study of pinched skew products is interesting in that some rigorous results are possible, the systems are simple to construct, and they give an indication of the possible behaviour of more general quasiperiodically forced systems. It is worth noting that the existence of stable continuous invariant curves, which has not been a focus of this note, can be explored using the techniques of Stark and Sturman (Stark 1997;Sturman and Stark 2000). This exploits the Liapunov exponent analysis and can be used to show that the conditions of Lemma 7 hold.…”

“…See (Sturman and Stark 2000) for results involving Liapunov exponents of the system. Of course, if the pinched skew product is invariant under a symmetry which maps φ to ψ and vice versa, then either φ and ψ are both continuous or neither is continuous.…”

Hyperbolicity and integral expression of the Lyapunov exponents for linear cocycles

“…Recently, this topic becomes more and more interesting; see [44,23,5,12,41,46,13,14,20,19] amongst others. In this paper, we consider a linear skew-product system…”