We prove that entropy map is upper semi-continuous for C1 nonuniformly hyperbolic systems with domination, while it is not true for C1+alpha nonuniformly hyperbolic systems in general. This goes a little against a common intuition that conclusions are parallel between C1+domination systems and C1+alpha systems.

Topics: Mathematics, Dynamical Systems

Source: http://arxiv.org/abs/1407.8102

In this paper we mainly study the necessary conditions for the existence of functionally independent generalized rational first integrals of ordinary differential systems via the resonances. The main results extend some of the previous related ones, for instance the classical Poincar\'e's one \cite{Po}, the Furta's one, part of Chen's ones, and the Shi's one. The key point in the proof of our main results is that functionally independence of generalized rational functions implies the...

Topics: Mathematics, Classical Analysis and ODEs, Dynamical Systems

Source: http://arxiv.org/abs/1407.7948