Structural stability of the inverse limit of endomorphisms

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



and Kocsard


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.



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Main result

Proof


g h = h f










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Main result

Proof


g h = h f










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Main result

Proof


g h = h f










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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.



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Main result

Proof


←−M f := x = (xi )i ∈ MZ : xi+1 := f (xi )


←−f .

d1(x , y) =∑

i 2−|i|d(xi , yi ).




h ←−f =←−g h






and Kocsard


Main result

Proof


←−M f := x = (xi )i ∈ MZ : xi+1 := f (xi )


←−f .

d1(x , y) =∑

i 2−|i|d(xi , yi ).




h ←−f =←−g h






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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= ∅.



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Main result

Proof



Theorem (Mather)







and Kocsard


Main result

Proof



Theorem (Mather)







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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.



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Main result

Proof



• ∀x ∈ Ωf , ∀y ∈ Ω(←−f ), ∀z ∈W u



sε (x) = Tf n(z)M



Example


periodic orbit.




←−SS.



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Main result

Proof



• ∀x ∈ Ωf , ∀y ∈ Ω(←−f ), ∀z ∈W u



sε (x) = Tf n(z)M



Example


periodic orbit.




←−SS.



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Main result

Proof



• ∀x ∈ Ωf , ∀y ∈ Ω(←−f ), ∀z ∈W u



sε (x) = Tf n(z)M



Example


periodic orbit.




←−SS.



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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).



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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 .



and Kocsard


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 .



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Main result

Proof




h0 ←−f = g h0.


←−f and ←−g .











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Main result

Proof




h0 ←−f = g h0.


←−f and ←−g .











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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,


• 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?.



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Main result

Proof



σ0 :←−M f 7→ TM




f := σ 7→ exp−1 g exp σ ←−f −1.


(F? − id)J = id


[(F? − id)− (φgf − id)] J = id − (φg

f − id) J,






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Main result

Proof



σ0 :←−M f 7→ TM




f := σ 7→ exp−1 g exp σ ←−f −1.


(F? − id)J = id


[(F? − id)− (φgf − id)] J = id − (φg

f − id) J,






and Kocsard


Main result

Proof



σ0 :←−M f 7→ TM




f := σ 7→ exp−1 g exp σ ←−f −1.


(F? − id)J = id


[(F? − id)− (φgf − id)] J = id − (φg

f − id) J,






and Kocsard


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.



and Kocsard


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.



and Kocsard


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



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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



and Kocsard


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.



and Kocsard


Main result

Proof

Unstable and Stable plane fields in the example

(x , y , z) 7→ (x2, y2, 0)

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Structural stability of the inverse limit of endomorphisms