A dynamical property for planar homeomorphisms and an application to the problem of canonical position around an isolated fixed point

The first theorem in this paper gives a criterion for detecting a fixed point of a planar homeomorphism into a disc. We then use this result to re-prove the path-connectedness of the space of all planar homeomorphisms fixing only one point, with a given Lefschetz index.     

某四阶微分算子的一个正则性定理

本文给出了某带参数λ的四阶微分算子B_λ的一个正则性定理,由此我们分别获得了关于该算子在“非线性情形”的二个同胚类及在“线性情形”的三个线性同构类。这对描述某飞行器在其运行过程中某些类双向稳定性质是有用与方便的。     

New results of some existence theorems on nonlinear boundary value problems

IntroductionSeveralpapershavebeenwrittenaboutequationwhereLisalineardifferentialexpression,NacontinuouslyFrechetdifferentialoperatorsuchthatL N'(u)satisfiescertainconditions,andFanoperator,usuallywithboundedrange.AconcreteexampleiswhereLisaseconddifferentialoperator,gsatisfiessomeconditionsrelatingtoL,andfisbounded.AmainlyusedtechniqueisSchauderfixedpointtheorem.Byestablishingapriorbound,itisnaturaltoobtaintheexistCnceofthesolutions.Butthismethodcannotgettheuniquenessofthesolution,noristh…     

On the generic behavior of the metric entropy, and related quantities,of uniformly continuous maps over Polish metric spaces

In this work, we show that if f is a uniformly continuous map defined over a Polish metric space, then the set of f-invariant measures with zero metric entropy is a $G_{\delta }$ set (in the weak topology). In particular, this set is generic if the set of f-periodic measures is dense in the set of f-invariant measures. This settles a conjecture posed by Sigmund (Trans. Amer. Math. Soc. 190 (1974), 285–299), which states that the metric entropy of an invariant measure of a topological dynamical system that satisfies the periodic specification property is typically zero. We also show that if X is compact and if f is an expansive or a Lipschitz map with a dense set of periodic measures, typically the lower correlation entropy for $q\in (0,1)$ is equal to zero. Moreover, we show that if X is a compact metric space and if f is an expanding map with a dense set of periodic measures, then the set of invariant measures with packing dimension, upper rate of recurrence and upper quantitative waiting time indicator equal to zero is residual.