The X-ray transform on Anosov manifolds A survey of recent results - PowerPoint PPT Presentation

X-ray transform on Anosov manifolds Geodesic X-ray transform The marked length spectrum The X-ray transform on Anosov manifolds A survey of recent results Thibault Lefeuvre Joint works with Sébastien Gouëzel, Colin Guillarmou, Gerhard Knieper Institut de Mathématique d’Orsay April 15th 2019 Thibault Lefeuvre The X-ray transform on Anosov manifolds

X-ray transform on Anosov manifolds Setting of the problem Geodesic X-ray transform Livsic theorems The marked length spectrum ( M , g ) smooth closed connected n -dimensional Riemannian manifold, X ∈ C ∞ ( M , T M ) smooth vector field generating a transitive Anosov flow ( ϕ t ) t ∈ R , i.e. such that there exists a continuous flow-invariant splitting T M = E s ⊕ E u ⊕ R X , with: � d ϕ t ( v ) � ≤ Ce − λ t � v � , ∀ v ∈ E s , ∀ t ≥ 0 , � d ϕ t ( v ) � ≤ Ce − λ | t | � v � , ∀ v ∈ E u , ∀ t ≤ 0 , where the constants C , λ > 0 are uniform, � · � = g ( · , · ) 1 / 2 , Thibault Lefeuvre The X-ray transform on Anosov manifolds

X-ray transform on Anosov manifolds Setting of the problem Geodesic X-ray transform Livsic theorems The marked length spectrum G set of periodic orbits. Definition (X-ray transform) � ℓ ( γ ) 1 I : C 0 ( M ) → ℓ ∞ ( G ) , If : G ∋ γ �→ � δ γ , f � := f ( ϕ t z ) d t , ℓ ( γ ) 0 where z ∈ γ , ℓ ( γ ) is the period of γ . Definition can be restricted to other regularities: C α (Hölder), H s (Sobolev) for s > n 2 , ... Question: can we describe the kernel of I on functions with prescribed regularity? I ( Xu ) = 0, for any u ∈ C ∞ ( M ) ; Xu is called a coboundary. Thibault Lefeuvre The X-ray transform on Anosov manifolds

X-ray transform on Anosov manifolds Setting of the problem Geodesic X-ray transform Livsic theorems The marked length spectrum Theorem ( Livsic ’72 ) Let α ∈ ( 0 , 1 ) . Given f ∈ C α ( M ) such that If = 0 , there exists u ∈ C α ( M ) such that f = Xu . Moreover, u is unique up to an additive constant. Classical Livsic theorem was also proved: in smooth regularity i.e. f , u ∈ C ∞ ( M ) ( de la Llave-Marco-Moriyon ’86 ), in Sobolev regularity i.e. f , u ∈ H s ( M ) ( Guillarmou ’17 ). Other natural questions: What if If ≥ 0 instead of If = 0? (Positive version of Livsic theorem) What if If ≃ ε (i.e. � If � ℓ ∞ := sup γ ∈G | If ( γ ) | ≤ ε )? (Approximate Livsic theorem) What if If ( γ ) = 0 for all periodic orbits γ of length ℓ ( γ ) ≤ L ? (Finite Livsic theorem) Thibault Lefeuvre The X-ray transform on Anosov manifolds

X-ray transform on Anosov manifolds Setting of the problem Geodesic X-ray transform Livsic theorems The marked length spectrum Theorem ( Lopes-Thieullen ’04 , Positive Livsic theorem) Let α ∈ ( 0 , 1 ) . There exists β ∈ ( 0 , α ) , C > 0 such that the following holds. Let f ∈ C α ( M ) such that If ≥ 0 . Then, there exists u , h ∈ C β ( M ) such that Xu ∈ C β ( M ) , h ≥ 0 and f = Xu + h . (In particular, f ≥ Xu .) Moreover, � h � C β + � Xu � C β ≤ C � f � C α . Thibault Lefeuvre The X-ray transform on Anosov manifolds

X-ray transform on Anosov manifolds Setting of the problem Geodesic X-ray transform Livsic theorems The marked length spectrum Theorem ( Gouëzel-L. ’19 , Approximate Livsic theorem) Let α ∈ ( 0 , 1 ) . There exists β ∈ ( 0 , α ) , ν > 0 such that the following holds. For any ε > 0 small enough, given f ∈ C α ( M ) such that � f � C α ≤ 1 and � If � ℓ ∞ ≤ ε , there exists u , h ∈ C β ( M ) such that Xu ∈ C β ( M ) , � h � C β ≤ ε ν and f = Xu + h . Theorem ( Gouëzel-L. ’19 , Finite Livsic theorem) Let α ∈ ( 0 , 1 ) . There exists β ∈ ( 0 , α ) , µ > 0 such that the following holds. For any L > 0 large enough, given f ∈ C α ( M ) such that � f � C α ≤ 1 and If ( γ ) = 0 for all γ ∈ G such that ℓ ( γ ) ≤ L , there exists u , h ∈ C β ( M ) such that Xu ∈ C β ( M ) , � h � C β ≤ L − µ and f = Xu + h . This implies that � If � ℓ ∞ ≤ L − µ . (Second theorem is actually a corollary of the proof of the first one.) Thibault Lefeuvre The X-ray transform on Anosov manifolds

X-ray transform on Anosov manifolds Setting of the problem Geodesic X-ray transform Livsic theorems The marked length spectrum Idea for the first theorem: find a periodic orbit of length ε − β 1 ( β 1 < 1) that is ε β 2 -dense in M and yet ε β 3 -separated. Then, mimick the proof of the classical Livsic theorem. ∼ " β 3 Σ (transverse section) Thibault Lefeuvre The X-ray transform on Anosov manifolds

X-ray transform on Anosov manifolds Stability estimates Geodesic X-ray transform Idea of proof The marked length spectrum ( M , g ) smooth connected closed Riemannian manifold with Anosov geodesic flow ( ϕ t ) t ∈ R on its unit tangent bundle M := SM . We call ( M , g ) an Anosov Riemannian manifold. C is the set of free homotopy classes; there exists a unique closed geodesic in each free homotopy class c ∈ C ( Klingenberg ’74 ). We identify G and C . C ∞ ( M , ⊗ m S T ∗ M ) is the vector-space of smooth symmetric m -tensors ( m ∈ N ). Thibault Lefeuvre The X-ray transform on Anosov manifolds

X-ray transform on Anosov manifolds Stability estimates Geodesic X-ray transform Idea of proof The marked length spectrum Symmetric tensors on M can be seen as functions on the unit tangent bundle SM , polynomial in the spheric variable. Given m ∈ N , f ∈ C 0 ( M , ⊗ m S T ∗ M ) , we define π ∗ m f ∈ C 0 ( SM ) by π ∗ m f : ( x , v ) �→ f x ( v , ..., v ) . Definition (Geodesic X-ray transform) S T ∗ M ) → ℓ ∞ ( C ) , I m : C 0 ( M , ⊗ m � ℓ ( γ c ) 1 I m f = I π ∗ m f : C ∋ c �→ f γ c ( t ) (˙ γ c ( t ) , ..., ˙ γ c ( t )) d t , ℓ ( γ c ) 0 with γ c unique closed geodesic in c . Question: Kernel of the X-ray transform? Thibault Lefeuvre The X-ray transform on Anosov manifolds

X-ray transform on Anosov manifolds Stability estimates Geodesic X-ray transform Idea of proof The marked length spectrum Tensor decomposition: f = Dp + h , with D := σ ◦ ∇ ( ∇ Levi-Civita connexion, σ symmetrization operator of tensors), D ∗ h = 0 where D ∗ is the formal adjoint of D . We call Dp the potential part and h the solenoidal part of f . I m ( Dp ) = 0, that is { potential tensors } ⊂ ker I m . I m is said to be s(olenoidal)-injective when this is an equality. Conjecture I m is s-injective whenever ( M , g ) is an Anosov Riemannian manifold. Known results when ( M , g ) Anosov; I m is s-injective for: any m ∈ N on surfaces ( Paternain-Salo-Uhlmann ’14 , Guillarmou ’17 ), any m ∈ N in any dimension, in nonpositive curvature ( Croke-Sharafutdinov ’98 ), m = 0 , 1 in any dimension ( Dairbekov-Sharafutdinov-11 ). Thibault Lefeuvre The X-ray transform on Anosov manifolds

X-ray transform on Anosov manifolds Stability estimates Geodesic X-ray transform Idea of proof The marked length spectrum Question: Once we have s-injectivity, can we obtain a stability estimate of the form � f � H 1 ≤ C � I m f � H 2 , ∀ f solenoidal , for some well-chosen spaces H 1 , 2 ? Theorem ( Guillarmou-L. ’18 , Gouëzel-L. ’19 ) For all exponents n / 2 < s < r , there exists C , ν > 0 such that the following holds. For all solenoidal tensors f such that � f � H r ≤ 1 , one has: � f � H s ≤ C � I m f � ν ℓ ∞ Now, recall the Finite Livsic theorem: Theorem ( Gouëzel-L. ’19 , Finite Livsic theorem) For any L > 0 large enough, given f ∈ C α ( M ) such that � f � C α ≤ 1 and If ( γ ) = 0 for all γ ∈ G such that ℓ ( γ ) ≤ L , there exists u , h ∈ C β ( M ) such that Xu ∈ C β ( M ) , � h � C β ≤ L − µ and f = Xu + h . This implies that � If � ℓ ∞ ≤ L − µ . Thibault Lefeuvre The X-ray transform on Anosov manifolds

X-ray transform on Anosov manifolds Stability estimates Geodesic X-ray transform Idea of proof The marked length spectrum Combining the two previous theorems, we obtain the Corollary ( Gouëzel-L. ’19 ) For all exponents n / 2 < s < r , there exists µ > 0 such that the following holds. For any L > 0 large enough, given any solenoidal tensor f such that � f � H r ≤ 1 and I m f ( c ) = 0 for all c ∈ C such that ℓ ( γ c ) ≤ L , one has: � f � H s ≤ L − µ . (In particular, L = + ∞ is the Classical Livsic theorem.) Thibault Lefeuvre The X-ray transform on Anosov manifolds

X-ray transform on Anosov manifolds Stability estimates Geodesic X-ray transform Idea of proof The marked length spectrum The X-ray transform I has bad analytic properties (in particular, it maps to functions on a discrete set). Idea (Guillarmou ’17): Mimick the case of a simple manifold with boundary (smwb). On a ( x; v ) smwb, we can write the normal operator ` − ( x; v ) ` + ( x; v ) � + ∞ M I ∗ I = e tX d t −∞ � ℓ + ( x , v ) (i.e. I ∗ If ( x , v ) = ℓ − ( x , v ) f ( ϕ t ( x , v )) d t ). Then I ∗ m I m = π m ∗ I ∗ I π ∗ m is a Ψ DO of order -1, elliptic on solenoidal tensors. If R ± ( λ ) := ( X ± λ ) − 1 denotes the resolvent of the generator of the geodesic flow, then I ∗ I = R + ( 0 ) − R − ( 0 ) . Thibault Lefeuvre The X-ray transform on Anosov manifolds