Min ZHOU

1 Introduction

It is well-known that the set of Morse functions is residual inCr(M,R)space,whereMis a closed manifold andr≥2.Let us extend this issue to a family of smooth functions.Letbe a family of smooth functions continuously depending on a parameterλ∈[0,1],it is natural to ask that whether there exists a residual setsuch that for eachV∈R and for eachλ∈[0,1],the functionFλ+Vis a Morse function.Unfortunately,it is not true even ifFλsmoothly depends on the parameter.Here is a counterexample.LetFλ:T→R be a family of functions such that for eachλ∈[0,1],Fλtakes its maximum attakes its minimum atandwhen the variables(λ,x)are restricted in a suitably small neighborhood of the originClearly,the pointis the non-degenerate critical point ofFλforλ>0.There exists no critical point ofFλin the neighborhood ofx=0 for.Forthe pointx=0 is a degenerate critical point.We note that the third derivative ofFλis bounded away from zero for allλwhenxremains close to the origin and the second derivative monotonously increases with respect toλ.Therefore,for anyC3-small perturbationV,certainλVexists such thatis small andFλV+Vhas a degenerate critical point close to the origin.

However,we are in different situation if we only consider the minimal as well as the maximal points of functions.Let[a]denote the integer part of the real numbera.The following theorem is the main result of this paper.

Theorem 1.1Let Fλ:T→Rbe a family of Cr-functions depending on the parameter λ∈[0,1].

(1)If r≥4and Fλis Lipschitz in the parameter λ,there exists an open-dense setO⊂Cr(T,R)such that for each V∈Oand each λ ∈[0,1],each global minimum as well as each global maximum of Fλ−V is non-degenerate.

(2)More generally,if Fλis α-H¨older continuous in λ(0<α≤1)and

then there exists an open-dense setsuch that for each V∈Oand each λ∈[0,1],certain weak non-degeneracy condition holds at each global minimum as well as each global maximum of Fλ−V:Some integer1≤?≤k exists such that

For the function of Lagrange action,the non-degeneracy of critical points corresponds to the hyperbolicity of periodic orbits of Lagrange flow(see[1]).It is closely related to the Ma˜n´e conjecture(see[2]),one can refer to[3]for some new progress in this problem.

We feel that the result can be extended to functions defined on closed smooth manifold with finite dimensions.

Conjecture 1.1Letbe a family of smooth functions,whereMis a closed smooth manifold andλ∈[0,1].IfFλis Lipschitz in the parameterλ,then some open-dense set O⊂C4(M,R)exists such that for eachV∈O and eachλ∈[0,1],each global minimum as well as each global maximum ofFλ−Vis non-degenerate.

2 Proof

We only need to prove the second part of the theorem,the first part is a special case of the second one.Obviously,the set O is the open set,as weak non-degeneracy of the critical point survives small perturbation.Therefore,we only need to show the density.Also,we only need to prove the non-degeneracy of the minimum,it is the same for the non-degeneracy of the maximum.Towards this goal,we introduce a set of small perturbations with 2k+2 parameters:

where I=[1,2].Let

We are going to show that,for any small numbers?,d>0,there exists∈I2k+2such that

holds for eachλ∈[0,1]whenever the pointx∗is a global minimizer ofFλ −V.It implies that there exists an even integer number 2≤j≤2k,such that thejth derivative ofFλ−Vatx∗is positive and theith derivative is equal to zero for eachi

By choosing sufficiently large integerN,the numbersandcan be set arbitrarily small,where the integerwill be determined later.Let

Given two points)andwe obtain two functionsVi(x)andin the form of Taylor series as shown in(2.2).Letforj=1,2,···,2k+1.We have ΔV(xi)=0 and

It follows that

LetWe construct a grid for the parametersby splitting the domain equally into a family of cuboids and setting the size by

These cuboids are denoted bycijwiththe cardinality of the set of the subscripts is up to the order

where the integer 0

if

with

The coefficients(a1,a2,···,a2k+1)depend on the positionxiand the parameters(A1,B1,···,Ak+1,Bk+1).The gird for(a1,a2,···,a2k+1)induces the partition for the parameters(A1,B1,···,Ak+1,Bk+1),determined by the equation

whereis a(2k+1)×(2k+2)matrix as following

in which each column has 2k+1 entries which take the form

where the integerjranges from 1 tok+1.

The coefficient matrixis non-singular for eachxi∈T.Indeed,and letM1be the(2k+1)×(2k+1)matrix constituted by first(2k+1)columns ofC,and letM2be the(2k+1)×(2k+1)matrix constituted by first 2kcolumns and the last column ofC.We find

where the constantM3is not equal to zero,and only depends on the integerk:

It induces a positive lower bound

Therefore,the grid forinduces a grid forwhich contains as many asdimensional strips(M4>0 is independent ofd),denoted bysijwithj∈Ji.Eachsijis mapped ontocijby(2.4).

Given certain parameterλ∈[0,1],if there exist Taylor coefficientswhich determine a perturbationVsuch that

then for any other Taylor coefficientssatisfying the condition

which determines another perturbationV?,one obtains from the formula(2.3)that

Under the map defined by(2.4),the inverse image of a cuboidcjwith the size

is a strip in the parameter space ofdenoted bysj(λ),with the Lebesgue measure as small asIf the cuboidcjis centered atthen for(2.5)holds.In other words,if,(2.5)holds.

Splitting the interval[0,1]equally into small sub-intervalsE?with the sizewe obtain as many assmall intervals.As the functionFλisα-H¨older continuous inλ,suitably large positive numberM5can be chosen so that

Therefore,forV∈V with,one has

Picking up one parameterλ?in each small intervalE?,we obtainstripsBy considering all small intervalsIiwithi=0,1,···,N−1,we find

where

if we choosep=2k+2 and set

Letting

we obtain the Lebesgue measure estimate

Obviously,for anyandi=0,1,2,···,N−1,(2.6)holds.It implies the density that all global minimal points ofFλ(·)satisfy the following property:There is

Lettingα=1,one immediately obtains the first part of the theorem.

[1]Cheng,C.Q.and Zhou,M.,Hyperbolicity of minimal periodic orbits,2013.arXiv:1302.4889

[2]Ma˜n´e,R.,Generic properties and problems of Lagrangian system,Nonlinearity,9,1996,323–334.

[3]Oliveira,E.R.,Generic properties of Lagrangians on surfaces:The Kupka-Smale theorem,Discrete Contin.Dyn.Syst.,21,2008,551–569.