$C^1$ weak Palis conjecture for nonsingular flows
Qianying Xiao Zuohuan Zheng
Discrete & Continuous Dynamical Systems - A 2018, 38(4): 1809-1832 doi: 10.3934/dcds.2018074

This paper focuses on generic properties of continuous dynamical systems. We prove $C^1$ weak Palis conjecture for nonsingular flows: Morse-Smale vector fields and vector fields admitting horseshoes are open and dense among $C^1$ nonsingular vector fields.

Our arguments contain three main ingredients: linear Poincaré flow, Liao's selecting lemma and the adapting of Crovisier's central model.

Firstly, by studying the linear Poincaré flow, we prove for a $C^1$ generic vector field away from horseshoes, any non-trivial nonsingular chain recurrent class contains a minimal set which is partially hyperbolic with 1-dimensional center with respect to the linear Poincaré flow.

Secondly, to understand the neutral behaviour of the 1-dimensional center, we adapt Crovisier's central model. The difficulties are that we can not build invariant plaque family of any time, the periodic point of a flow is not periodic for the discrete time map. Through delicate analysis of the center manifold of a periodic orbit near the partially hyperbolic set, we manage to yield nice periodic points such that their stable manifolds and unstable manifolds are well-placed for transverse intersection.

keywords: Nonsingular flow Morse-Smale system away from horseshoe linear Poincaré flow central model
Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications
Zuohuan Zheng Jing Xia Zhiming Zheng
Discrete & Continuous Dynamical Systems - A 2006, 14(3): 409-417 doi: 10.3934/dcds.2006.14.409
It has been established one-side uniform convergence in both the Birkhoff and sub-additive ergodic theorems under conditions on growth rates with respect to all the invariant measures. In this paper we show these conditions are both necessary and sufficient. These results are applied to study quasiperiodically forced systems. Some meaningful geometric properties of invariant sets of such systems are presented. We also show that any strange compact invariant set of a $\mathcal{C}^1$ quasiperiodically forced system must support an invariant measure with a non-negative normal Lyapunov exponent.
keywords: quasiperiodically forced systems. strange attractors Ergodic theorems

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