Devaney chaos, Li-Yorke chaos, and multi-dimensional Li-Yorke chaos for topological dynamics

Weakly Birkhoff recurrent switching signals, almost sure and partial stability of linear switched dynamical systems

In this paper, we mainly show the following two statements. (1) A discrete-time topological Markovian jump linear system is uniformly exponentially stable if and only if it is robustly periodically stable, by using a Gel׳fand–Berger–Wang formula proved here. (2) A random linear ODE driven by a semiflow with closing by periodic orbits property is uniformly exponentially stable if and only if it is robustly periodically stable, by using a perturbation technique of Shantao Liao and the semi-uniform ergodic theorems. Our proofs involve the ergodic theory in both of the above two cases. In addition, counterexamples are constructed to the robustness condition and to the spectral finiteness of linear cocycle.

Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations

Let be an ergodic measure-preserving Borel measurable transformation of a separable metric space X that is not necessarily compact, and suppose that is a T-subadditive sequence of -measurable upper-bounded functions. In this paper, we prove that, if the sets of n-discontinuities are of μ-measure 0 for all n ≥ 1 and if the growth rates then *(x) < 0 for all points x in the basin BT(μ) of (T, μ). We apply this to considering the Oseledets regular points.

Optimal state points of the subadditive ergodic theorem

Let be a Holder-continuous linear cocycle whose driving semiflow preserves a probability μ on a compact metric space X. Using the concept of two-sided quasi-Pesin orbits introduced in this paper, we show that the Lyapunov characteristic spectrum of μ can be approached arbitrarily by that of periodic points of f. So, if all periodic points of f have only positive Lyapunov exponents and such exponents are uniformly bounded away from zero, then μ also has only positive exponents. In our arguments, the exponential closing property of the driving semiflow is a basic condition, and it is easy to check that every C1-expanding map of a closed manifold obeys this closing property.Consequently, if f is a C1+Holder local diffeomorphism of a closed manifold and if it is Holder conjugated to some C1-expanding map, then f is itself expanding. This gives a positive answer to a question suggested by Anatole Katok.

On the approximation of Lyapunov exponents and a question suggested by Anatole Katok

In the paper, the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space ℝ d , not necessarily compact, by Liaowise spectral theorems that give integral expressions of Lyapunov exponents. In the context of smooth linear skew-product flows with Polish driving systems, the results are still valid. This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces.

Integral expressions of Lyapunov exponents for autonomous ordinary differential systems

We generalize an important theorem of Fred Galvin from the Stone-Cech compactification βT of any discrete semigroup T to any compact Hausdorff right-topological semigroup with a dense topological center; and then apply it to Ellis’ semigroups to prove that a point is distal if and only if it is IP*-recurrent, for any semiflow (T;X) with arbitrary compact Hausdorff phase space X not necessarily metrizable and with arbitrary phase semigroup T not necessarily cancelable.

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On Galvin’s theorem for compact Hausdorff right-topological semigroups with dense topological centers

In this paper, we simply prove, in the framework of Liao, the hyperbolicity of C
 1-star invariant sets of C
 1-class differential systems on a closed manifold of dimension ⩽ 4, without using the C
 1-connecting lemma and even the ergodic closing lemma.

XiongPing Dai

Papers

Integral expressions of Lyapunov exponents for autonomous ordinary differential systems

On Galvin’s theorem for compact Hausdorff right-topological semigroups with dense topological centers

Hyperbolicity of C 1-star invariant sets for C 1-class dynamical systems