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Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
ProofStructural stability of the inverse limit of endomorphisms
Pierre Bergerwork with A. Rovella and Kocsard
June 4, 2013
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
DefinitionA map is C r -structurally stable if it has the “same” dynamics as itsC r -perturbations.This usually means that ∀g ≈C r f there exists h ∈ Homeo(M) so that
g h = h f
Theorem (Palis-Smale, Robinson ⇐, Mane ⇒)
f ∈ Diff 1 is C 1-structurally stable ⇔ f satisfies Axiom A and the strongtransversallity condition (A.S.)
QuestionWhat if r = 1 + α?
Theorem (Sad-Mane-Sullivan, Lyubich)
The set of structurally stable rationnal functions of the sphere is dense.
Conjecture (Fatou ⇐ MLC)
A rational function which is structurally stable is A.S.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
DefinitionA map is C r -structurally stable if it has the “same” dynamics as itsC r -perturbations.This usually means that ∀g ≈C r f there exists h ∈ Homeo(M) so that
g h = h f
Theorem (Palis-Smale, Robinson ⇐, Mane ⇒)
f ∈ Diff 1 is C 1-structurally stable ⇔ f satisfies Axiom A and the strongtransversallity condition (A.S.)
QuestionWhat if r = 1 + α?
Theorem (Sad-Mane-Sullivan, Lyubich)
The set of structurally stable rationnal functions of the sphere is dense.
Conjecture (Fatou ⇐ MLC)
A rational function which is structurally stable is A.S.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
DefinitionA map is C r -structurally stable if it has the “same” dynamics as itsC r -perturbations.This usually means that ∀g ≈C r f there exists h ∈ Homeo(M) so that
g h = h f
Theorem (Palis-Smale, Robinson ⇐, Mane ⇒)
f ∈ Diff 1 is C 1-structurally stable ⇔ f satisfies Axiom A and the strongtransversallity condition (A.S.)
QuestionWhat if r = 1 + α?
Theorem (Sad-Mane-Sullivan, Lyubich)
The set of structurally stable rationnal functions of the sphere is dense.
Conjecture (Fatou ⇐ MLC)
A rational function which is structurally stable is A.S.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
DefinitionA map is C r -structurally stable if it has the “same” dynamics as itsC r -perturbations.This usually means that ∀g ≈C r f there exists h ∈ Homeo(M) so that
g h = h f
Theorem (Palis-Smale, Robinson ⇐, Mane ⇒)
f ∈ Diff 1 is C 1-structurally stable ⇔ f satisfies Axiom A and the strongtransversallity condition (A.S.)
QuestionWhat if r = 1 + α?
Theorem (Sad-Mane-Sullivan, Lyubich)
The set of structurally stable rationnal functions of the sphere is dense.
Conjecture (Fatou ⇐ MLC)
A rational function which is structurally stable is A.S.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
DefinitionThe inverse limit of f ∈ End r (M) := C r (M,M) is
←−M f := x = (xi )i ∈ MZ : xi+1 := f (xi )
The canonical action of f on←−M f by shift is denoted by
←−f .
d1(x , y) =∑
i 2−|i|d(xi , yi ).
Definitionf has C r -structurally stable inverse limit (C r -
←−−S .S .) if for every g ≈C r f
there exists h ∈ Homeo(←−M f ,←−Mg ) so that
h ←−f =←−g h
Theorem (Shub, Przytycki)
An Anosov endomorphism of the torus is structurally stable iff it is a
diffeomorphism or it is expanding. In general it is←−−S .S.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
DefinitionThe inverse limit of f ∈ End r (M) := C r (M,M) is
←−M f := x = (xi )i ∈ MZ : xi+1 := f (xi )
The canonical action of f on←−M f by shift is denoted by
←−f .
d1(x , y) =∑
i 2−|i|d(xi , yi ).
Definitionf has C r -structurally stable inverse limit (C r -
←−−S .S .) if for every g ≈C r f
there exists h ∈ Homeo(←−M f ,←−Mg ) so that
h ←−f =←−g h
Theorem (Shub, Przytycki)
An Anosov endomorphism of the torus is structurally stable iff it is a
diffeomorphism or it is expanding. In general it is←−−S .S.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
DefinitionThe inverse limit of f ∈ End r (M) := C r (M,M) is
←−M f := x = (xi )i ∈ MZ : xi+1 := f (xi )
The canonical action of f on←−M f by shift is denoted by
←−f .
d1(x , y) =∑
i 2−|i|d(xi , yi ).
Definitionf has C r -structurally stable inverse limit (C r -
←−−S .S .) if for every g ≈C r f
there exists h ∈ Homeo(←−M f ,←−Mg ) so that
h ←−f =←−g h
Theorem (Shub, Przytycki)
An Anosov endomorphism of the torus is structurally stable iff it is a
diffeomorphism or it is expanding. In general it is←−−S .S.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
DefinitionThe singular set of f ∈ C r (M,M) isSing(f ) := x ∈ M : Dx f not onto.
RemarkIf f ∈ C∞(M,M) is structurally stable, then Sing(f ) must be stable.
Theorem (Mather)
Smooth maps with stable singular set are C∞-generic.
RemarkNo satisfactory description of this generic set.
Theorem (Mane-Pugh, B.-Rovella)
There exit←−−S .S . endomorphisms such that robustly Sing(f ) ∩ Ωf 6= ∅.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
DefinitionThe singular set of f ∈ C r (M,M) isSing(f ) := x ∈ M : Dx f not onto.
RemarkIf f ∈ C∞(M,M) is structurally stable, then Sing(f ) must be stable.
Theorem (Mather)
Smooth maps with stable singular set are C∞-generic.
RemarkNo satisfactory description of this generic set.
Theorem (Mane-Pugh, B.-Rovella)
There exit←−−S .S . endomorphisms such that robustly Sing(f ) ∩ Ωf 6= ∅.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
DefinitionThe singular set of f ∈ C r (M,M) isSing(f ) := x ∈ M : Dx f not onto.
RemarkIf f ∈ C∞(M,M) is structurally stable, then Sing(f ) must be stable.
Theorem (Mather)
Smooth maps with stable singular set are C∞-generic.
RemarkNo satisfactory description of this generic set.
Theorem (Mane-Pugh, B.-Rovella)
There exit←−−S .S . endomorphisms such that robustly Sing(f ) ∩ Ωf 6= ∅.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Definitionf ∈ End1(M) is A.S. if
• Ωf = cl(Per(f )) is hyperbolic: ∃E s ⊂ TM|Ω(f ) s.t. ‖Df |E s‖ < 1and ‖(Df |TM/E s)−1‖ < 1.
• ∀x ∈ Ωf , ∀y ∈ Ω(←−f ), ∀z ∈W u
ε (y) ∩ f −n(W sε (x)) it holds:
Dz f n(TzW uε (y)) + Tf n(z)W
sε (x) = Tf n(z)M
Theorem (B.-Rovella, B.-Kocsard)
If f is A.S. then f is C 1-←−−S .S.
Example
• Pc(x) = x2 + c is AS and so C 1 −←−SS if it has an attracting
periodic orbit.
• Any Anosov endomorphism is AS and so C 1 −←−SS.
• Product of AS endomorphisms are AS and so C 1 −←−SS.
• (x1, . . . , xn) 7→ (x2k + c, x1, . . . xk−1, 0, . . . , 0) is AS and so C 1 −
←−SS.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Definitionf ∈ End1(M) is A.S. if
• Ωf = cl(Per(f )) is hyperbolic: ∃E s ⊂ TM|Ω(f ) s.t. ‖Df |E s‖ < 1and ‖(Df |TM/E s)−1‖ < 1.
• ∀x ∈ Ωf , ∀y ∈ Ω(←−f ), ∀z ∈W u
ε (y) ∩ f −n(W sε (x)) it holds:
Dz f n(TzW uε (y)) + Tf n(z)W
sε (x) = Tf n(z)M
Theorem (B.-Rovella, B.-Kocsard)
If f is A.S. then f is C 1-←−−S .S.
Example
• Pc(x) = x2 + c is AS and so C 1 −←−SS if it has an attracting
periodic orbit.
• Any Anosov endomorphism is AS and so C 1 −←−SS.
• Product of AS endomorphisms are AS and so C 1 −←−SS.
• (x1, . . . , xn) 7→ (x2k + c, x1, . . . xk−1, 0, . . . , 0) is AS and so C 1 −
←−SS.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Definitionf ∈ End1(M) is A.S. if
• Ωf = cl(Per(f )) is hyperbolic: ∃E s ⊂ TM|Ω(f ) s.t. ‖Df |E s‖ < 1and ‖(Df |TM/E s)−1‖ < 1.
• ∀x ∈ Ωf , ∀y ∈ Ω(←−f ), ∀z ∈W u
ε (y) ∩ f −n(W sε (x)) it holds:
Dz f n(TzW uε (y)) + Tf n(z)W
sε (x) = Tf n(z)M
Theorem (B.-Rovella, B.-Kocsard)
If f is A.S. then f is C 1-←−−S .S.
Example
• Pc(x) = x2 + c is AS and so C 1 −←−SS if it has an attracting
periodic orbit.
• Any Anosov endomorphism is AS and so C 1 −←−SS.
• Product of AS endomorphisms are AS and so C 1 −←−SS.
• (x1, . . . , xn) 7→ (x2k + c, x1, . . . xk−1, 0, . . . , 0) is AS and so C 1 −
←−SS.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Definitionf ∈ End1(M) is A.S. if
• Ωf = cl(Per(f )) is hyperbolic: ∃E s ⊂ TM|Ω(f ) s.t. ‖Df |E s‖ < 1and ‖(Df |TM/E s)−1‖ < 1.
• ∀x ∈ Ωf , ∀y ∈ Ω(←−f ), ∀z ∈W u
ε (y) ∩ f −n(W sε (x)) it holds:
Dz f n(TzW uε (y)) + Tf n(z)W
sε (x) = Tf n(z)M
Theorem (B.-Rovella, B.-Kocsard)
If f is A.S. then f is C 1-←−−S .S.
Example
• Pc(x) = x2 + c is AS and so C 1 −←−SS if it has an attracting
periodic orbit.
• Any Anosov endomorphism is AS and so C 1 −←−SS.
• Product of AS endomorphisms are AS and so C 1 −←−SS.
• (x1, . . . , xn) 7→ (x2k + c, x1, . . . xk−1, 0, . . . , 0) is AS and so C 1 −
←−SS.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Application to the renormalization of homoclinic tangencies inhigh dimension
Theorem (Mora)
There are open families (ft)t of homoclinic unfolding of a hyperbolicpoint P, such that there exist an open set U arbitrarily close to 0 and Varbitrarily close to P, and N >> 1 such that for t ∈ U:
Λ := ∩n≥0f nNt (V )
has a dynamics C 1-close to (x2k + c, x1, . . . xk−1, 0, . . . , 0)
Corollary
Λ is homeomorphic to the inverse limit of (x2k + c, x1, . . . xk−1).
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Description of←−−S .S .-coverings
Theorem (Aoki-Moriyasu-Sumi, B.-Rovella)
If a map is←−−S .S . without singularities, then it satisfies Axiom A and the
strong transversality conditions.
Corollary←−−S .S . covering are A.S .
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Adaption of Robbin-Robinson proof
Let f be an A.S . C 1-map. We want to find for every g C 1-close to f a
map h0 :←−M f 7→ M such that:
h0 ←−f = g h0.
indeed h := (h0 ←−f i )i semi conjugates
←−f and ←−g .
Proposition (B. Rovella)
If h is injective and C 0-close to (xi )i 7→ x0, then h is a homemorphism.
Proposition (Robbin)
If h0 is Lipschitz with small constant for the metricd∞(x , y) := supi d(xi , yi ) and uniformly close to (xi )i 7→ x0 small then his injective.
Hence it is sufficient to find for every g a map h0 :←−M f 7→ M
• which is uniformly close to (xi ) 7→ x0,
• which is d∞-Lipschitz with small constant,
• such that g h0 = h0 ←−f .
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Adaption of Robbin-Robinson proof
Let f be an A.S . C 1-map. We want to find for every g C 1-close to f a
map h0 :←−M f 7→ M such that:
h0 ←−f = g h0.
indeed h := (h0 ←−f i )i semi conjugates
←−f and ←−g .
Proposition (B. Rovella)
If h is injective and C 0-close to (xi )i 7→ x0, then h is a homemorphism.
Proposition (Robbin)
If h0 is Lipschitz with small constant for the metricd∞(x , y) := supi d(xi , yi ) and uniformly close to (xi )i 7→ x0 small then his injective.
Hence it is sufficient to find for every g a map h0 :←−M f 7→ M
• which is uniformly close to (xi ) 7→ x0,
• which is d∞-Lipschitz with small constant,
• such that g h0 = h0 ←−f .
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Adaption of Robbin-Robinson proof
Let f be an A.S . C 1-map. We want to find for every g C 1-close to f a
map h0 :←−M f 7→ M such that:
h0 ←−f = g h0.
indeed h := (h0 ←−f i )i semi conjugates
←−f and ←−g .
Proposition (B. Rovella)
If h is injective and C 0-close to (xi )i 7→ x0, then h is a homemorphism.
Proposition (Robbin)
If h0 is Lipschitz with small constant for the metricd∞(x , y) := supi d(xi , yi ) and uniformly close to (xi )i 7→ x0 small then his injective.
Hence it is sufficient to find for every g a map h0 :←−M f 7→ M
• which is uniformly close to (xi ) 7→ x0,
• which is d∞-Lipschitz with small constant,
• such that g h0 = h0 ←−f .
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Banachic formulation
Writing σ := exp−1 h0, it is sufficient to find for every g a map
σ0 :←−M f 7→ TM
• which is continuous and uniformly small,
• which is d∞-Lipschitz with small constant,
• such that φgf (σ0) = σ0, with φg
f := σ 7→ exp−1 g exp σ ←−f −1.
Put F? := σ 7→ Df σ ←−f −1. If J is a right inverse of F? − id :
(F? − id)J = id
then it is sufficient to find a fixed point σ of:
[(F? − id)− (φgf − id)] J = id − (φg
f − id) J,
such that σ0 := J σ is small.
ProblemF? does not preserve the d∞-Lipschitz section if f is not C 2, and is notinvertible if f has singularity.
Robinson issue for the C 1 case: replace Df by a smooth approximationof it in the expression of F?.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Banachic formulation
Writing σ := exp−1 h0, it is sufficient to find for every g a map
σ0 :←−M f 7→ TM
• which is continuous and uniformly small,
• which is d∞-Lipschitz with small constant,
• such that φgf (σ0) = σ0, with φg
f := σ 7→ exp−1 g exp σ ←−f −1.
Put F? := σ 7→ Df σ ←−f −1. If J is a right inverse of F? − id :
(F? − id)J = id
then it is sufficient to find a fixed point σ of:
[(F? − id)− (φgf − id)] J = id − (φg
f − id) J,
such that σ0 := J σ is small.
ProblemF? does not preserve the d∞-Lipschitz section if f is not C 2, and is notinvertible if f has singularity.
Robinson issue for the C 1 case: replace Df by a smooth approximationof it in the expression of F?.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Banachic formulation
Writing σ := exp−1 h0, it is sufficient to find for every g a map
σ0 :←−M f 7→ TM
• which is continuous and uniformly small,
• which is d∞-Lipschitz with small constant,
• such that φgf (σ0) = σ0, with φg
f := σ 7→ exp−1 g exp σ ←−f −1.
Put F? := σ 7→ Df σ ←−f −1. If J is a right inverse of F? − id :
(F? − id)J = id
then it is sufficient to find a fixed point σ of:
[(F? − id)− (φgf − id)] J = id − (φg
f − id) J,
such that σ0 := J σ is small.
ProblemF? does not preserve the d∞-Lipschitz section if f is not C 2, and is notinvertible if f has singularity.
Robinson issue for the C 1 case: replace Df by a smooth approximationof it in the expression of F?.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Banachic formulation
Writing σ := exp−1 h0, it is sufficient to find for every g a map
σ0 :←−M f 7→ TM
• which is continuous and uniformly small,
• which is d∞-Lipschitz with small constant,
• such that φgf (σ0) = σ0, with φg
f := σ 7→ exp−1 g exp σ ←−f −1.
Put F? := σ 7→ Df σ ←−f −1. If J is a right inverse of F? − id :
(F? − id)J = id
then it is sufficient to find a fixed point σ of:
[(F? − id)− (φgf − id)] J = id − (φg
f − id) J,
such that σ0 := J σ is small.
ProblemF? does not preserve the d∞-Lipschitz section if f is not C 2, and is notinvertible if f has singularity.
Robinson issue for the C 1 case: replace Df by a smooth approximationof it in the expression of F?.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
New ideas
For the endomorphism case we can have F? invertible if we extend thebundle TM. First trivialize TM to a certain M × RN . Then look at:
F : (x , u, v) ∈ M × RN × RN 7→ (Dx f px(u) + bv , bu)
where px : RN → TxM is a projection and b small and f a smoothapproximation of f . F is a homeomorphism.
Hence, with F# := σ 7→ F σ ←−f −1 and J the right inverse of
(id − F#), we want to find a fixed point σ of:
[(F# − id)− (φgf − id)] J = id − (φg
f − id) J,
such that σ0 := J σ is continuous and d∞-Lipschitz with small norms.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
New ideas
For the endomorphism case we can have F? invertible if we extend thebundle TM. First trivialize TM to a certain M × RN . Then look at:
F : (x , u, v) ∈ M × RN × RN 7→ (Dx f px(u) + bv , bu)
where px : RN → TxM is a projection and b small and f a smoothapproximation of f . F is a homeomorphism.
Hence, with F# := σ 7→ F σ ←−f −1 and J the right inverse of
(id − F#), we want to find a fixed point σ of:
[(F# − id)− (φgf − id)] J = id − (φg
f − id) J,
such that σ0 := J σ is continuous and d∞-Lipschitz with small norms.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
New ideas
The construction of the inverse J needs several analysis in the class ofd1-continuous and d∞-Lipschitz map.
Proposition (B. Kocsard)
For any covering (Ui )i of←−M f , there exists a partition of unity (ρi )i
subordinated to it and d∞-Lipschitz. For every manifold N, the
subspace of C 0(←−M f ,N) of d1-Lipschitz maps (and so d∞-Lip.) is dense.
Proof.Take ρ be a smooth bump function. Then for every r ∈ C 0(
←−M f ,Rn),
can be extended to a C 0-map in r ∈ C 0(MZ,Rn). We look then at:
r := x ∈←−M f ⊂ MZ 7→
∫y∈MZ ρ(
d1(x,y)
s)r(y)Leb⊗Z∫
y∈MZ ρ(d1(x,y)
s)Leb⊗Z
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
New ideas
The construction of the inverse J needs several analysis in the class ofd1-continuous and d∞-Lipschitz map.
Proposition (B. Kocsard)
For any covering (Ui )i of←−M f , there exists a partition of unity (ρi )i
subordinated to it and d∞-Lipschitz. For every manifold N, the
subspace of C 0(←−M f ,N) of d1-Lipschitz maps (and so d∞-Lip.) is dense.
Proof.Take ρ be a smooth bump function. Then for every r ∈ C 0(
←−M f ,Rn),
can be extended to a C 0-map in r ∈ C 0(MZ,Rn). We look then at:
r := x ∈←−M f ⊂ MZ 7→
∫y∈MZ ρ(
d1(x,y)
s)r(y)Leb⊗Z∫
y∈MZ ρ(d1(x,y)
s)Leb⊗Z
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
The expression of J is given by a convering (Ui )i of←−M f , over which
there are a (pseudo)-invariant splitting
R2N = E si ⊕ E u
i
and a partition of unity (γi )i subordinated to (Ui )i .
J := v 7→∑i
[−∞∑n=0
F n#(γiv
si ) +
−1∑n=−∞
F n#(γiv
ui )],
All these these object must be d1-continuous and d∞-Lipschitz. Also thesplitting must have an angle bounded from below when b approaches 0.
Structural stability ofthe inverse limit ofendomorphisms
Pierre Bergerwork with A. Rovella
and Kocsard
Structural stabilitytheorems
Main result
Proof
Unstable and Stable plane fields in the example
(x , y , z) 7→ (x2, y2, 0)