KAM Tori Are No More Than Sticky Rome, 58 February 2019 David - PowerPoint PPT Presentation

Gevrey regularity: Given α ě 1, L ą 0 and K a product of closed Euclidean balls and tori, we define uniformly Gevrey- p α, L q fcns: G α, L p R M ˆ K q : “ t f P C 8 p R M ˆ K q | � f � α, L ă 8u , ℓ P N N L | ℓ | α ℓ ! α � B ℓ f � C 0 p R M ˆ K q � f � α, L : “ ř (with ℓ ! “ ℓ 1 ! . . . ℓ N !). Nice properties: Banach algebra, Cauchy-Gevrey inequalities, the flow of a Gevrey vector field is a Gevrey map, etc. Gevrey functions with compact support: if α ą 1, z P T ˆ R and ν ą 0, G α, L p T ˆ R q contains a function 0 ď η z ,ν ď 1 such that η z ,ν ” 1 on B p z , ν { 2 q , η z ,ν ” 0 on B p z , ν q c 3/13

Gevrey regularity: Given α ě 1, L ą 0 and K a product of closed Euclidean balls and tori, we define uniformly Gevrey- p α, L q fcns: G α, L p R M ˆ K q : “ t f P C 8 p R M ˆ K q | � f � α, L ă 8u , ℓ P N N L | ℓ | α ℓ ! α � B ℓ f � C 0 p R M ˆ K q � f � α, L : “ ř (with ℓ ! “ ℓ 1 ! . . . ℓ N !). Nice properties: Banach algebra, Cauchy-Gevrey inequalities, the flow of a Gevrey vector field is a Gevrey map, etc. Gevrey functions with compact support: if α ą 1, z P T ˆ R and ν ą 0, G α, L p T ˆ R q contains a function 0 ď η z ,ν ď 1 such that η z ,ν ” 1 on B p z , ν { 2 q , η z ,ν ” 0 on B p z , ν q c , 1 � η z ,ν � α, L ď exp p c ν ´ α ´ 1 q . 3/13

Gevrey regularity: Given α ě 1, L ą 0 and K a product of closed Euclidean balls and tori, we define uniformly Gevrey- p α, L q fcns: G α, L p R M ˆ K q : “ t f P C 8 p R M ˆ K q | � f � α, L ă 8u , ℓ P N N L | ℓ | α ℓ ! α � B ℓ f � C 0 p R M ˆ K q � f � α, L : “ ř (with ℓ ! “ ℓ 1 ! . . . ℓ N !). Nice properties: Banach algebra, Cauchy-Gevrey inequalities, the flow of a Gevrey vector field is a Gevrey map, etc. Gevrey functions with compact support: if α ą 1, z P T ˆ R and ν ą 0, G α, L p T ˆ R q contains a function 0 ď η z ,ν ď 1 such that η z ,ν ” 1 on B p z , ν { 2 q , η z ,ν ” 0 on B p z , ν q c , 1 � η z ,ν � α, L ď exp p c ν ´ α ´ 1 q . echet space G α, L p R M ˆ K q : cover the factor R M by an Fr´ increasing sequence of closed balls B R j , choose L j “ 2 ´ j L , get a complete metric space with translation-invariant distance d α, L . 3/13

Theorem ... D Gevrey perturbations of H 0 so that invariant tori are no more than doubly exponentially stable... 4/13

Theorem ... D Gevrey perturbations of H 0 so that invariant tori are no more than doubly exponentially stable... This is about the existence of “diffusive” invariant tori. 4/13

Theorem ... D Gevrey perturbations of H 0 so that invariant tori are no more than doubly exponentially stable... This is about the existence of “diffusive” invariant tori. Definition Given a transformation T (or a flow) on a metric space p M , d q and ν ą 0, we say that: 4/13

Theorem ... D Gevrey perturbations of H 0 so that invariant tori are no more than doubly exponentially stable... This is about the existence of “diffusive” invariant tori. Definition Given a transformation T (or a flow) on a metric space p M , d q and ν ą 0, we say that: A point z of M is ν -diffusive if there exist an initial condition ˆ z P M and a positive integer (or real) t such that d p ˆ z , z q ď ν , t ď E p ν q and d p T t ˆ z , z q E p ν q “ e e C ν ´ γ (with C , γ ą 0 to be chosen later) 4/13

Theorem ... D Gevrey perturbations of H 0 so that invariant tori are no more than doubly exponentially stable... This is about the existence of “diffusive” invariant tori. Definition Given a transformation T (or a flow) on a metric space p M , d q and ν ą 0, we say that: A point z of M is ν -diffusive if there exist an initial condition ˆ z P M and a positive integer (or real) t such that d p ˆ z , z q ď ν , t ď E p ν q and d p T t ˆ z , z q E p ν q “ e e C ν ´ γ (with C , γ ą 0 to be chosen later) 4/13

Theorem ... D Gevrey perturbations of H 0 so that invariant tori are no more than doubly exponentially stable... This is about the existence of “diffusive” invariant tori. Definition Given a transformation T (or a flow) on a metric space p M , d q and ν ą 0, we say that: A point z of M is ν -diffusive if there exist an initial condition ˆ z P M and a positive integer (or real) t such that d p ˆ z , z q ď ν , t ď E p ν q and d p T t ˆ z , z q ě E p 2 ν q . E p ν q “ e e C ν ´ γ (with C , γ ą 0 to be chosen later) 4/13

Theorem ... D Gevrey perturbations of H 0 so that invariant tori are no more than doubly exponentially stable... This is about the existence of “diffusive” invariant tori. Definition Given a transformation T (or a flow) on a metric space p M , d q and ν ą 0, we say that: A point z of M is ν -diffusive if there exist an initial condition ˆ z P M and a positive integer (or real) t such that d p ˆ z , z q ď ν , t ď E p ν q and d p T t ˆ z , z q ě E p 2 ν q . A subset X of M is ν -diffusive if all points in X are ν -diffusive. E p ν q “ e e C ν ´ γ (with C , γ ą 0 to be chosen later) 4/13

Theorem ... D Gevrey perturbations of H 0 so that invariant tori are no more than doubly exponentially stable... This is about the existence of “diffusive” invariant tori. Definition Given a transformation T (or a flow) on a metric space p M , d q and ν ą 0, we say that: A point z of M is ν -diffusive if there exist an initial condition z P M and a positive integer (or real) t such that d p ˆ ˆ z , z q ď ν , t ď E p ν q and d p T t ˆ z , z q ě E p 2 ν q . A subset X of M is ν -diffusive if all points in X are ν -diffusive. A subset X of M is diffusive if there exists a sequence ν n Ñ 0 such that X is ν n -diffusive for each n . E p ν q “ e e C ν ´ γ (with C , γ ą 0 to be chosen later) 4/13

Coordinates p θ 1 , . . . , θ n , r 1 , . . . , r n q in T n ˆ R n 5/13

Coordinates p θ 1 , . . . , θ n , r 1 , . . . , r n q in T n ˆ R n or p θ 1 , . . . , θ n , τ, r 1 , . . . , r n , s q in T n ` 1 ˆ R n ` 1 , where n : “ N ´ 1 ě 2. 5/13

Coordinates p θ 1 , . . . , θ n , r 1 , . . . , r n q in T n ˆ R n or p θ 1 , . . . , θ n , τ, r 1 , . . . , r n , s q in T n ` 1 ˆ R n ` 1 , where n : “ N ´ 1 ě 2. It is equivalent to consider a non-autonomous Hamiltonian h p θ, r , t q on T n ˆ R n which depends 1-periodically on the time t or 5/13

Coordinates p θ 1 , . . . , θ n , r 1 , . . . , r n q in T n ˆ R n or τ “ B H B s “ 1 ] [ 9 p θ 1 , . . . , θ n , τ, r 1 , . . . , r n , s q in T n ` 1 ˆ R n ` 1 , where n : “ N ´ 1 ě 2. It is equivalent to consider a non-autonomous Hamiltonian h p θ, r , t q on T n ˆ R n which depends 1-periodically on the time t or autonomous Hamiltonian H p θ, τ, r , s q “ s ` h p θ, r , τ q . 5/13

Coordinates p θ 1 , . . . , θ n , r 1 , . . . , r n q in T n ˆ R n or τ “ B H B s “ 1 ] [ 9 p θ 1 , . . . , θ n , τ, r 1 , . . . , r n , s q in T n ` 1 ˆ R n ` 1 , where n : “ N ´ 1 ě 2. It is equivalent to consider a non-autonomous Hamiltonian h p θ, r , t q on T n ˆ R n which depends 1-periodically on the time t or autonomous Hamiltonian H p θ, τ, r , s q “ s ` h p θ, r , τ q . For arbitrary ω P R n , non-autonomous 1-periodic perturbations of h 0 p r q : “ p ω, r q ` 1 2 p r , r q are equivalent to 5/13

Coordinates p θ 1 , . . . , θ n , r 1 , . . . , r n q in T n ˆ R n or τ “ B H B s “ 1 ] [ 9 p θ 1 , . . . , θ n , τ, r 1 , . . . , r n , s q in T n ` 1 ˆ R n ` 1 , where n : “ N ´ 1 ě 2. It is equivalent to consider a non-autonomous Hamiltonian h p θ, r , t q on T n ˆ R n which depends 1-periodically on the time t or autonomous Hamiltonian H p θ, τ, r , s q “ s ` h p θ, r , τ q . For arbitrary ω P R n , non-autonomous 1-periodic perturbations of h 0 p r q : “ p ω, r q ` 1 2 p r , r q are equivalent to certain autonomous perturbations of the integrable Hamiltonian H 0 p r , s q : “ s ` h 0 p r q 5/13

Coordinates p θ 1 , . . . , θ n , r 1 , . . . , r n q in T n ˆ R n or τ “ B H B s “ 1 ] [ 9 p θ 1 , . . . , θ n , τ, r 1 , . . . , r n , s q in T n ` 1 ˆ R n ` 1 , where n : “ N ´ 1 ě 2. It is equivalent to consider a non-autonomous Hamiltonian h p θ, r , t q on T n ˆ R n which depends 1-periodically on the time t or autonomous Hamiltonian H p θ, τ, r , s q “ s ` h p θ, r , τ q . For arbitrary ω P R n , non-autonomous 1-periodic perturbations of h 0 p r q : “ p ω, r q ` 1 2 p r , r q are equivalent to certain autonomous perturbations of the integrable Hamiltonian H 0 p r , s q : “ s ` h 0 p r q , for which T p r , s q : “ T n ` 1 ˆ tp r , s qu is an invariant quasi-periodic torus with frequencies 9 θ “ ω ` r , 9 τ “ 1 (for arbitrary r and s ). 5/13

Coordinates p θ 1 , . . . , θ n , r 1 , . . . , r n q in T n ˆ R n or τ “ B H B s “ 1 ] [ 9 p θ 1 , . . . , θ n , τ, r 1 , . . . , r n , s q in T n ` 1 ˆ R n ` 1 , where n : “ N ´ 1 ě 2. It is equivalent to consider a non-autonomous Hamiltonian h p θ, r , t q on T n ˆ R n which depends 1-periodically on the time t or autonomous Hamiltonian H p θ, τ, r , s q “ s ` h p θ, r , τ q . For arbitrary ω P R n , non-autonomous 1-periodic perturbations of h 0 p r q : “ p ω, r q ` 1 2 p r , r q are equivalent to certain autonomous perturbations of the integrable Hamiltonian H 0 p r , s q : “ s ` h 0 p r q , for which T p r , s q : “ T n ` 1 ˆ tp r , s qu is an invariant quasi-periodic torus with frequencies 9 θ “ ω ` r , 9 τ “ 1 (for arbitrary r and s ). THEOREM 1 5/13

Coordinates p θ 1 , . . . , θ n , r 1 , . . . , r n q in T n ˆ R n or τ “ B H B s “ 1 ] [ 9 p θ 1 , . . . , θ n , τ, r 1 , . . . , r n , s q in T n ` 1 ˆ R n ` 1 , where n : “ N ´ 1 ě 2. It is equivalent to consider a non-autonomous Hamiltonian h p θ, r , t q on T n ˆ R n which depends 1-periodically on the time t or autonomous Hamiltonian H p θ, τ, r , s q “ s ` h p θ, r , τ q . For arbitrary ω P R n , non-autonomous 1-periodic perturbations of h 0 p r q : “ p ω, r q ` 1 2 p r , r q are equivalent to certain autonomous perturbations of the integrable Hamiltonian H 0 p r , s q : “ s ` h 0 p r q , for which T p r , s q : “ T n ` 1 ˆ tp r , s qu is an invariant quasi-periodic torus with frequencies 9 θ “ ω ` r , 9 τ “ 1 (for arbitrary r and s ). THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that 5/13

Coordinates p θ 1 , . . . , θ n , r 1 , . . . , r n q in T n ˆ R n or τ “ B H B s “ 1 ] [ 9 p θ 1 , . . . , θ n , τ, r 1 , . . . , r n , s q in T n ` 1 ˆ R n ` 1 , where n : “ N ´ 1 ě 2. It is equivalent to consider a non-autonomous Hamiltonian h p θ, r , t q on T n ˆ R n which depends 1-periodically on the time t or autonomous Hamiltonian H p θ, τ, r , s q “ s ` h p θ, r , τ q . For arbitrary ω P R n , non-autonomous 1-periodic perturbations of h 0 p r q : “ p ω, r q ` 1 2 p r , r q are equivalent to certain autonomous perturbations of the integrable Hamiltonian H 0 p r , s q : “ s ` h 0 p r q , for which T p r , s q : “ T n ` 1 ˆ tp r , s qu is an invariant quasi-periodic torus with frequencies 9 θ “ ω ` r , 9 τ “ 1 (for arbitrary r and s ). THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , 5/13

Coordinates p θ 1 , . . . , θ n , r 1 , . . . , r n q in T n ˆ R n or τ “ B H B s “ 1 ] [ 9 p θ 1 , . . . , θ n , τ, r 1 , . . . , r n , s q in T n ` 1 ˆ R n ` 1 , where n : “ N ´ 1 ě 2. It is equivalent to consider a non-autonomous Hamiltonian h p θ, r , t q on T n ˆ R n which depends 1-periodically on the time t or autonomous Hamiltonian H p θ, τ, r , s q “ s ` h p θ, r , τ q . For arbitrary ω P R n , non-autonomous 1-periodic perturbations of h 0 p r q : “ p ω, r q ` 1 2 p r , r q are equivalent to certain autonomous perturbations of the integrable Hamiltonian H 0 p r , s q : “ s ` h 0 p r q , for which T p r , s q : “ T n ` 1 ˆ tp r , s qu is an invariant quasi-periodic torus with frequencies 9 θ “ ω ` r , 9 τ “ 1 (for arbitrary r and s ). THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and 5/13

Coordinates p θ 1 , . . . , θ n , r 1 , . . . , r n q in T n ˆ R n or τ “ B H B s “ 1 ] [ 9 p θ 1 , . . . , θ n , τ, r 1 , . . . , r n , s q in T n ` 1 ˆ R n ` 1 , where n : “ N ´ 1 ě 2. It is equivalent to consider a non-autonomous Hamiltonian h p θ, r , t q on T n ˆ R n which depends 1-periodically on the time t or autonomous Hamiltonian H p θ, τ, r , s q “ s ` h p θ, r , τ q . For arbitrary ω P R n , non-autonomous 1-periodic perturbations of h 0 p r q : “ p ω, r q ` 1 2 p r , r q are equivalent to certain autonomous perturbations of the integrable Hamiltonian H 0 p r , s q : “ s ` h 0 p r q , for which T p r , s q : “ T n ` 1 ˆ tp r , s qu is an invariant quasi-periodic torus with frequencies 9 θ “ ω ` r , 9 τ “ 1 (for arbitrary r and s ). THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . 5/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω Diophantine ñ T p 0 , s q is doubly exponentially stable: 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω Diophantine ñ T p 0 , s q is doubly exponentially stable: for any ν -close initial condition, the orbit stays within distance 2 ν 1 c ν ´ ` ` α p τ ` 1 q ˘˘ from T p 0 , s q during time exp exp . 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω τ -Diophantine ñ T p 0 , s q is doubly exponentially stable: for any ν -close initial condition, the orbit stays within distance 2 ν 1 c ν ´ ` ` α p τ ` 1 q ˘˘ from T p 0 , s q during time exp exp . 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω τ -Diophantine ñ T p 0 , s q is doubly exponentially stable: for any ν -close initial condition, the orbit stays within distance 2 ν 1 c ν ´ ` ` α p τ ` 1 q ˘˘ from T p 0 , s q during time exp exp . Theorem 1 shows that we cannot expect in general a stability better than doubly exponential. 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω τ -Diophantine ñ T p 0 , s q is doubly exponentially stable: for any ν -close initial condition, the orbit stays within distance 2 ν 1 c ν ´ ` ` α p τ ` 1 q ˘˘ from T p 0 , s q during time exp exp . Theorem 1 shows that we cannot expect in general a stability better than doubly exponential. Our “diffusiveness exponent” in E p ν q “ exp p exp p C ν ´ γ qq is γ “ 1 α ´ 1 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω τ -Diophantine ñ T p 0 , s q is doubly exponentially stable: for any ν -close initial condition, the orbit stays within distance 2 ν 1 c ν ´ ` ` α p τ ` 1 q ˘˘ from T p 0 , s q during time exp exp . Theorem 1 shows that we cannot expect in general a stability better than doubly exponential. Our “diffusiveness exponent” in E p ν q “ exp p exp p C ν ´ γ qq is γ “ 1 α ´ 1 , exponents do not match 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω τ -Diophantine ñ T p 0 , s q is doubly exponentially stable: for any ν -close initial condition, the orbit stays within distance 2 ν 1 c ν ´ ` ` α p τ ` 1 q ˘˘ from T p 0 , s q during time exp exp . Theorem 1 shows that we cannot expect in general a stability better than doubly exponential. Our “diffusiveness exponent” in E p ν q “ exp p exp p C ν ´ γ qq is γ “ 1 α ´ 1 , exponents do not match yet. 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω τ -Diophantine ñ T p 0 , s q is doubly exponentially stable: for any ν -close initial condition, the orbit stays within distance 2 ν 1 c ν ´ ` ` α p τ ` 1 q ˘˘ from T p 0 , s q during time exp exp . Theorem 1 shows that we cannot expect in general a stability better than doubly exponential. Our “diffusiveness exponent” in E p ν q “ exp p exp p C ν ´ γ qq is γ “ 1 α ´ 1 , exponents do not match yet. THEOREM 2 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω τ -Diophantine ñ T p 0 , s q is doubly exponentially stable: for any ν -close initial condition, the orbit stays within distance 2 ν 1 c ν ´ ` ` α p τ ` 1 q ˘˘ from T p 0 , s q during time exp exp . Theorem 1 shows that we cannot expect in general a stability better than doubly exponential. Our “diffusiveness exponent” in E p ν q “ exp p exp p C ν ´ γ qq is γ “ 1 α ´ 1 , exponents do not match yet. THEOREM 2 Given α ą 1, L ą 0, ε ą 0, there are h P G α, L p T n ˆ R n ˆ T q 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω τ -Diophantine ñ T p 0 , s q is doubly exponentially stable: for any ν -close initial condition, the orbit stays within distance 2 ν 1 c ν ´ ` ` α p τ ` 1 q ˘˘ from T p 0 , s q during time exp exp . Theorem 1 shows that we cannot expect in general a stability better than doubly exponential. Our “diffusiveness exponent” in E p ν q “ exp p exp p C ν ´ γ qq is γ “ 1 α ´ 1 , exponents do not match yet. THEOREM 2 Given α ą 1, L ą 0, ε ą 0, there are h P G α, L p T n ˆ R n ˆ T q and X ε Ă r 0 , 1 s with Leb p X ε q ě 1 ´ ε 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω τ -Diophantine ñ T p 0 , s q is doubly exponentially stable: for any ν -close initial condition, the orbit stays within distance 2 ν 1 c ν ´ ` ` α p τ ` 1 q ˘˘ from T p 0 , s q during time exp exp . Theorem 1 shows that we cannot expect in general a stability better than doubly exponential. Our “diffusiveness exponent” in E p ν q “ exp p exp p C ν ´ γ qq is γ “ 1 α ´ 1 , exponents do not match yet. THEOREM 2 Given α ą 1, L ą 0, ε ą 0, there are h P G α, L p T n ˆ R n ˆ T q and X ε Ă r 0 , 1 s with Leb p X ε q ě 1 ´ ε such that d α, L p h 0 , h q ă ε , 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω τ -Diophantine ñ T p 0 , s q is doubly exponentially stable: for any ν -close initial condition, the orbit stays within distance 2 ν 1 c ν ´ ` ` α p τ ` 1 q ˘˘ from T p 0 , s q during time exp exp . Theorem 1 shows that we cannot expect in general a stability better than doubly exponential. Our “diffusiveness exponent” in E p ν q “ exp p exp p C ν ´ γ qq is γ “ 1 α ´ 1 , exponents do not match yet. THEOREM 2 Given α ą 1, L ą 0, ε ą 0, there are h P G α, L p T n ˆ R n ˆ T q and X ε Ă r 0 , 1 s with Leb p X ε q ě 1 ´ ε such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and 6/13

THEOREM 1 Given α ą 1, L ą 0, ε ą 0, there is h P G α, L p T n ˆ R n ˆ T q such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and the tori T p 0 , s q Ă T n ` 1 ˆ R n ` 1 are invariant and diffusive for H . Note: ω τ -Diophantine ñ T p 0 , s q is doubly exponentially stable: for any ν -close initial condition, the orbit stays within distance 2 ν 1 c ν ´ ` ` α p τ ` 1 q ˘˘ from T p 0 , s q during time exp exp . Theorem 1 shows that we cannot expect in general a stability better than doubly exponential. Our “diffusiveness exponent” in E p ν q “ exp p exp p C ν ´ γ qq is γ “ 1 α ´ 1 , exponents do not match yet. THEOREM 2 Given α ą 1, L ą 0, ε ą 0, there are h P G α, L p T n ˆ R n ˆ T q and X ε Ă r 0 , 1 s with Leb p X ε q ě 1 ´ ε such that d α, L p h 0 , h q ă ε , the Hamiltonian vector field generated by H : “ s ` h p θ, r , τ q is complete and, for each r P p X ε ` Z q ˆ R n ´ 1 and s P R , the torus T p r , s q “ T n ` 1 ˆ tp r , s qu Ă T n ` 1 ˆ R n ` 1 is invariant and diffusive for H . 6/13

Method: Obtain first a discrete version of the results 7/13

Method: Obtain first a discrete version of the results by “Herman’s synchronized diffusion” mechanism... 7/13

Method: Obtain first a discrete version of the results by “Herman’s synchronized diffusion” mechanism... Discrete version in the case n “ 2: 7/13

Method: Obtain first a discrete version of the results by “Herman’s synchronized diffusion” mechanism... Discrete version in the case n “ 2: Phase space M 1 ˆ M 2 » T 2 ˆ R 2 with M 1 : “ T ˆ R , M 2 : “ T ˆ R . T 0 : “ F 0 ˆ G 0 : M 1 ˆ M 2 ý Unperturbed integrable system: with F 0 : M 1 ý and G 0 : M 2 ý defined by F 0 p θ 1 , r 1 q : “ p θ 1 ` ω 1 ` r 1 , r 1 q , G 0 p θ 2 , r 2 q : “ p θ 2 ` ω 2 ` r 2 , r 2 q . 7/13

Method: Obtain first a discrete version of the results by “Herman’s synchronized diffusion” mechanism... Discrete version in the case n “ 2: Phase space M 1 ˆ M 2 » T 2 ˆ R 2 with M 1 : “ T ˆ R , M 2 : “ T ˆ R . T 0 : “ F 0 ˆ G 0 : M 1 ˆ M 2 ý Unperturbed integrable system: with F 0 : M 1 ý and G 0 : M 2 ý defined by F 0 p θ 1 , r 1 q : “ p θ 1 ` ω 1 ` r 1 , r 1 q , G 0 p θ 2 , r 2 q : “ p θ 2 ` ω 2 ` r 2 , r 2 q . T 0 : “ T 2 ˆ tp 0 , 0 qu invariant torus with frequency ω “ p ω 1 , ω 2 q . 7/13

Method: Obtain first a discrete version of the results by “Herman’s synchronized diffusion” mechanism... Discrete version in the case n “ 2: Phase space M 1 ˆ M 2 » T 2 ˆ R 2 with M 1 : “ T ˆ R , M 2 : “ T ˆ R . T 0 : “ F 0 ˆ G 0 : M 1 ˆ M 2 ý Unperturbed integrable system: with F 0 : M 1 ý and G 0 : M 2 ý defined by F 0 p θ 1 , r 1 q : “ p θ 1 ` ω 1 ` r 1 , r 1 q , G 0 p θ 2 , r 2 q : “ p θ 2 ` ω 2 ` r 2 , r 2 q . T 0 : “ T 2 ˆ tp 0 , 0 qu invariant torus with frequency ω “ p ω 1 , ω 2 q . ⊲ Φ H = time-1 map of the Hamiltonian flow Notation: H „ „ „ e.g. T 0 “ Φ ω 1 r 1 ` ω 2 r 2 ` 1 2 p r 2 1 ` r 2 2 q . 7/13

Method: Obtain first a discrete version of the results by “Herman’s synchronized diffusion” mechanism... Discrete version in the case n “ 2: Phase space M 1 ˆ M 2 » T 2 ˆ R 2 with M 1 : “ T ˆ R , M 2 : “ T ˆ R . T 0 : “ F 0 ˆ G 0 : M 1 ˆ M 2 ý Unperturbed integrable system: with F 0 : M 1 ý and G 0 : M 2 ý defined by F 0 p θ 1 , r 1 q : “ p θ 1 ` ω 1 ` r 1 , r 1 q , G 0 p θ 2 , r 2 q : “ p θ 2 ` ω 2 ` r 2 , r 2 q . T 0 : “ T 2 ˆ tp 0 , 0 qu invariant torus with frequency ω “ p ω 1 , ω 2 q . ⊲ Φ H = time-1 map of the Hamiltonian flow Notation: H „ „ „ e.g. T 0 “ Φ ω 1 r 1 ` ω 2 r 2 ` 1 2 p r 2 1 ` r 2 2 q . THEOREM 1 follows easily from THEOREM 1’ D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that 7/13

Method: Obtain first a discrete version of the results by “Herman’s synchronized diffusion” mechanism... Discrete version in the case n “ 2: Phase space M 1 ˆ M 2 » T 2 ˆ R 2 with M 1 : “ T ˆ R , M 2 : “ T ˆ R . T 0 : “ F 0 ˆ G 0 : M 1 ˆ M 2 ý Unperturbed integrable system: with F 0 : M 1 ý and G 0 : M 2 ý defined by F 0 p θ 1 , r 1 q : “ p θ 1 ` ω 1 ` r 1 , r 1 q , G 0 p θ 2 , r 2 q : “ p θ 2 ` ω 2 ` r 2 , r 2 q . T 0 : “ T 2 ˆ tp 0 , 0 qu invariant torus with frequency ω “ p ω 1 , ω 2 q . ⊲ Φ H = time-1 map of the Hamiltonian flow Notation: H „ „ „ e.g. T 0 “ Φ ω 1 r 1 ` ω 2 r 2 ` 1 2 p r 2 1 ` r 2 2 q . THEOREM 1 follows easily from THEOREM 1’ D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that (1) u and v are flat for r 1 “ 0, � u � α, L ` � v � α, L ă ε , 7/13

Method: Obtain first a discrete version of the results by “Herman’s synchronized diffusion” mechanism... Discrete version in the case n “ 2: Phase space M 1 ˆ M 2 » T 2 ˆ R 2 with M 1 : “ T ˆ R , M 2 : “ T ˆ R . T 0 : “ F 0 ˆ G 0 : M 1 ˆ M 2 ý Unperturbed integrable system: with F 0 : M 1 ý and G 0 : M 2 ý defined by F 0 p θ 1 , r 1 q : “ p θ 1 ` ω 1 ` r 1 , r 1 q , G 0 p θ 2 , r 2 q : “ p θ 2 ` ω 2 ` r 2 , r 2 q . T 0 : “ T 2 ˆ tp 0 , 0 qu invariant torus with frequency ω “ p ω 1 , ω 2 q . ⊲ Φ H = time-1 map of the Hamiltonian flow Notation: H „ „ „ e.g. T 0 “ Φ ω 1 r 1 ` ω 2 r 2 ` 1 2 p r 2 1 ` r 2 2 q . THEOREM 1 follows easily from THEOREM 1’ D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that (1) u and v are flat for r 1 “ 0, � u � α, L ` � v � α, L ă ε , (2) T 0 is invariant and diffusive for T : “ Φ v ˝ p Φ u ˝ F 0 q ˆ G 0 ` ˘ . 7/13

THEOREM 1’ D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that (1) u and v are flat for r 1 “ 0, � u � α, L ` � v � α, L ă ε , (2) T 0 is invariant and diffusive for T : “ Φ v ˝ p Φ u ˝ F 0 q ˆ G 0 ` ˘ . 8/13

THEOREM 1’ D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that (1) u and v are flat for r 1 “ 0, � u � α, L ` � v � α, L ă ε , (2) T 0 is invariant and diffusive for T : “ Φ v ˝ p Φ u ˝ F 0 q ˆ G 0 ` ˘ . There is also a THEOREM 2’ which implies THEOREM 2... 8/13

THEOREM 1’ D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that (1) u and v are flat for r 1 “ 0, � u � α, L ` � v � α, L ă ε , (2) T 0 is invariant and diffusive for T : “ Φ v ˝ p Φ u ˝ F 0 q ˆ G 0 ` ˘ . There is also a THEOREM 2’ which implies THEOREM 2... Key proposition: localized diffusive orbits: 8/13

THEOREM 1’ D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that (1) u and v are flat for r 1 “ 0, � u � α, L ` � v � α, L ă ε , (2) T 0 is invariant and diffusive for T : “ Φ v ˝ p Φ u ˝ F 0 q ˆ G 0 ` ˘ . There is also a THEOREM 2’ which implies THEOREM 2... Key proposition: localized diffusive orbits: 1 PROPOSITION Let γ “ α ´ 1 . For any ν ą 0 small enough and r P R , there exist D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that ¯ 8/13

THEOREM 1’ D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that (1) u and v are flat for r 1 “ 0, � u � α, L ` � v � α, L ă ε , (2) T 0 is invariant and diffusive for T : “ Φ v ˝ p Φ u ˝ F 0 q ˆ G 0 ` ˘ . There is also a THEOREM 2’ which implies THEOREM 2... Key proposition: localized diffusive orbits: 1 PROPOSITION Let γ “ α ´ 1 . For any ν ą 0 small enough and r P R , there exist D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that ¯ r ` ν q , � u � α, L ` � v � α, L ď e ´ c ν ´ γ , (1) u ” 0 , v ” 0 for r 1 R p ¯ r ´ ν, ¯ 8/13

THEOREM 1’ D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that (1) u and v are flat for r 1 “ 0, � u � α, L ` � v � α, L ă ε , (2) T 0 is invariant and diffusive for T : “ Φ v ˝ p Φ u ˝ F 0 q ˆ G 0 ` ˘ . There is also a THEOREM 2’ which implies THEOREM 2... Key proposition: localized diffusive orbits: 1 PROPOSITION Let γ “ α ´ 1 . For any ν ą 0 small enough and r P R , there exist D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that ¯ r ` ν q , � u � α, L ` � v � α, L ď e ´ c ν ´ γ , (1) u ” 0 , v ” 0 for r 1 R p ¯ r ´ ν, ¯ (2) the set T ˆ p ¯ r ´ ν, ¯ r ` ν q ˆ M 2 is invariant and ν -diffusive for T : “ Φ v ˝ p Φ u ˝ F 0 q ˆ G 0 ` ˘ . 8/13

THEOREM 1’ D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that (1) u and v are flat for r 1 “ 0, � u � α, L ` � v � α, L ă ε , (2) T 0 is invariant and diffusive for T : “ Φ v ˝ p Φ u ˝ F 0 q ˆ G 0 ` ˘ . There is also a THEOREM 2’ which implies THEOREM 2... Key proposition: localized diffusive orbits: 1 PROPOSITION Let γ “ α ´ 1 . For any ν ą 0 small enough and r P R , there exist D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that ¯ r ` ν q , � u � α, L ` � v � α, L ď e ´ c ν ´ γ , (1) u ” 0 , v ” 0 for r 1 R p ¯ r ´ ν, ¯ (2) the set T ˆ p ¯ r ´ ν, ¯ r ` ν q ˆ M 2 is invariant and ν -diffusive for T : “ Φ v ˝ p Φ u ˝ F 0 q ˆ G 0 ` ˘ . PROP ñ THEOREM 1’: take ν “ ν n “ 10 ´ n ε , ¯ r “ ¯ r n “ 2 ν n and add up the corresponding u n ’s and v n ’s... (Disjoint supports!) 8/13

THEOREM 1’ D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that (1) u and v are flat for r 1 “ 0, � u � α, L ` � v � α, L ă ε , (2) T 0 is invariant and diffusive for T : “ Φ v ˝ p Φ u ˝ F 0 q ˆ G 0 ` ˘ . There is also a THEOREM 2’ which implies THEOREM 2... Key proposition: localized diffusive orbits: 1 PROPOSITION Let γ “ α ´ 1 . For any ν ą 0 small enough and r P R , there exist D u P G α, L p M 1 q , v P G α, L p M 1 ˆ M 2 q such that ¯ r ` ν q , � u � α, L ` � v � α, L ď e ´ c ν ´ γ , (1) u ” 0 , v ” 0 for r 1 R p ¯ r ´ ν, ¯ (2) the set T ˆ p ¯ r ´ ν, ¯ r ` ν q ˆ M 2 is invariant and ν -diffusive for T : “ Φ v ˝ p Φ u ˝ F 0 q ˆ G 0 ` ˘ . PROP ñ THEOREM 1’: take ν “ ν n “ 10 ´ n ε , ¯ r “ ¯ r n “ 2 ν n and add up the corresponding u n ’s and v n ’s... (Disjoint supports!) PROP ñ THEOREM 2’: more elaborate... 8/13

Herman’s synchronized diffusion mechanism relies on a “coupling lemma” which was used in the construction of examples 9/13

Herman’s synchronized diffusion mechanism relies on a “coupling lemma” which was used in the construction of examples with drifting orbits, biasymptotic to infinity, with diffusion speed bounded from below (J.-P.Marco-D.S. 2003) 9/13

Herman’s synchronized diffusion mechanism relies on a “coupling lemma” which was used in the construction of examples with drifting orbits, biasymptotic to infinity, with diffusion speed bounded from below (J.-P.Marco-D.S. 2003) with wandering polydiscs (J.-P.Marco-D.S. 2004) 9/13

Herman’s synchronized diffusion mechanism relies on a “coupling lemma” which was used in the construction of examples with drifting orbits, biasymptotic to infinity, with diffusion speed bounded from below (J.-P.Marco-D.S. 2003) with wandering polydiscs (J.-P.Marco-D.S. 2004), with estimates for their size in L.Lazzarini-J.-P.Marco-D.S. 2018 9/13

Herman’s synchronized diffusion mechanism relies on a “coupling lemma” which was used in the construction of examples with drifting orbits, biasymptotic to infinity, with diffusion speed bounded from below (J.-P.Marco-D.S. 2003) with wandering polydiscs (J.-P.Marco-D.S. 2004), with estimates for their size in L.Lazzarini-J.-P.Marco-D.S. 2018 with q th iterate containing a subsystem isomorphic to a q Z ˆ t ω 1 , ω 2 u Z giving rise to a skew-product defined on 1 random walk of step 1 q for r 1 (J.-P.Marco-D.S. 2004) 9/13

Herman’s synchronized diffusion mechanism relies on a “coupling lemma” which was used in the construction of examples with drifting orbits, biasymptotic to infinity, with diffusion speed bounded from below (J.-P.Marco-D.S. 2003) with wandering polydiscs (J.-P.Marco-D.S. 2004), with estimates for their size in L.Lazzarini-J.-P.Marco-D.S. 2018 with q th iterate containing a subsystem isomorphic to a q Z ˆ t ω 1 , ω 2 u Z giving rise to a skew-product defined on 1 random walk of step 1 q for r 1 (J.-P.Marco-D.S. 2004) with a subsystem isomorphic to a transitive system on p T ˆ R q n ´ 1 ˆ t ω 1 , . . . , ω r u Z , 9/13

Herman’s synchronized diffusion mechanism relies on a “coupling lemma” which was used in the construction of examples with drifting orbits, biasymptotic to infinity, with diffusion speed bounded from below (J.-P.Marco-D.S. 2003) with wandering polydiscs (J.-P.Marco-D.S. 2004), with estimates for their size in L.Lazzarini-J.-P.Marco-D.S. 2018 with q th iterate containing a subsystem isomorphic to a q Z ˆ t ω 1 , ω 2 u Z giving rise to a skew-product defined on 1 random walk of step 1 q for r 1 (J.-P.Marco-D.S. 2004) with a subsystem isomorphic to a transitive system on p T ˆ R q n ´ 1 ˆ t ω 1 , . . . , ω r u Z , with convergence in law to a Brownian motion of the n ´ 1 first action variables after rescaling, 9/13

Herman’s synchronized diffusion mechanism relies on a “coupling lemma” which was used in the construction of examples with drifting orbits, biasymptotic to infinity, with diffusion speed bounded from below (J.-P.Marco-D.S. 2003) with wandering polydiscs (J.-P.Marco-D.S. 2004), with estimates for their size in L.Lazzarini-J.-P.Marco-D.S. 2018 with q th iterate containing a subsystem isomorphic to a q Z ˆ t ω 1 , ω 2 u Z giving rise to a skew-product defined on 1 random walk of step 1 q for r 1 (J.-P.Marco-D.S. 2004) with a subsystem isomorphic to a transitive system on p T ˆ R q n ´ 1 ˆ t ω 1 , . . . , ω r u Z , with convergence in law to a Brownian motion of the n ´ 1 first action variables after rescaling, ergodic if n “ 2 or 3 (D.S. 2006, unpublished) 9/13

Herman’s synchronized diffusion mechanism relies on a “coupling lemma” which was used in the construction of examples with drifting orbits, biasymptotic to infinity, with diffusion speed bounded from below (J.-P.Marco-D.S. 2003) with wandering polydiscs (J.-P.Marco-D.S. 2004), with estimates for their size in L.Lazzarini-J.-P.Marco-D.S. 2018 with q th iterate containing a subsystem isomorphic to a q Z ˆ t ω 1 , ω 2 u Z giving rise to a skew-product defined on 1 random walk of step 1 q for r 1 (J.-P.Marco-D.S. 2004) with a subsystem isomorphic to a transitive system on p T ˆ R q n ´ 1 ˆ t ω 1 , . . . , ω r u Z , with convergence in law to a Brownian motion of the n ´ 1 first action variables after rescaling, ergodic if n “ 2 or 3 (D.S. 2006, unpublished) with a non-resonant elliptic fixed point attracting an orbit (B.Fayad-J.-P.Marco-D.S. 2018). 9/13

Herman’s mechanism: 10/13

Herman’s mechanism: Fine-tuned coupling of two twist maps: 10/13

Herman’s mechanism: Fine-tuned coupling of two twist maps: At exactly one point z ˚ of a well chosen periodic orbit of period q of the first twist map F “ Φ u ˆ F 0 : M 1 “ T ˆ R ý 10/13

Herman’s mechanism: Fine-tuned coupling of two twist maps: At exactly one point z ˚ of a well chosen periodic orbit of period q of the first twist map F “ Φ u ˆ F 0 : M 1 “ T ˆ R ý , the coupling will push the orbits in the second annulus M 2 “ T ˆ R upward, along a fixed vertical ∆ 10/13

Herman’s mechanism: Fine-tuned coupling of two twist maps: At exactly one point z ˚ of a well chosen periodic orbit of period q of the first twist map F “ Φ u ˆ F 0 : M 1 “ T ˆ R ý , the coupling will push the orbits in the second annulus M 2 “ T ˆ R upward, along a fixed vertical ∆, by an amount 1 { q that sends an invariant curve whose rotation number is a multiple of 1 { q exactly to another one having the same property. 10/13

Herman’s mechanism: Fine-tuned coupling of two twist maps: At exactly one point z ˚ of a well chosen periodic orbit of period q of the first twist map F “ Φ u ˆ F 0 : M 1 “ T ˆ R ý , the coupling will push the orbits in the second annulus M 2 “ T ˆ R upward, along a fixed vertical ∆, by an amount 1 { q that sends an invariant curve whose rotation number is a multiple of 1 { q exactly to another one having the same property. The dynamics of the q th iterate of the coupled map on the line t z ˚ u ˆ ∆ Ă M 1 ˆ M 2 will thus drift at a linear speed 10/13

Herman’s mechanism: Fine-tuned coupling of two twist maps: At exactly one point z ˚ of a well chosen periodic orbit of period q of the first twist map F “ Φ u ˆ F 0 : M 1 “ T ˆ R ý , the coupling will push the orbits in the second annulus M 2 “ T ˆ R upward, along a fixed vertical ∆, by an amount 1 { q that sends an invariant curve whose rotation number is a multiple of 1 { q exactly to another one having the same property. The dynamics of the q th iterate of the coupled map on the line t z ˚ u ˆ ∆ Ă M 1 ˆ M 2 will thus drift at a linear speed: after q 2 iterates the point will have moved by 1 in the second action coordinate r 2 10/13

Herman’s mechanism: Fine-tuned coupling of two twist maps: At exactly one point z ˚ of a well chosen periodic orbit of period q of the first twist map F “ Φ u ˆ F 0 : M 1 “ T ˆ R ý , the coupling will push the orbits in the second annulus M 2 “ T ˆ R upward, along a fixed vertical ∆, by an amount 1 { q that sends an invariant curve whose rotation number is a multiple of 1 { q exactly to another one having the same property. The dynamics of the q th iterate of the coupled map on the line t z ˚ u ˆ ∆ Ă M 1 ˆ M 2 will thus drift at a linear speed: after q 2 iterates the point will have moved by 1 in the second action coordinate r 2 , and after q 3 it will have moved by q . 10/13

Herman’s mechanism: Fine-tuned coupling of two twist maps: At exactly one point z ˚ of a well chosen periodic orbit of period q of the first twist map F “ Φ u ˆ F 0 : M 1 “ T ˆ R ý , the coupling will push the orbits in the second annulus M 2 “ T ˆ R upward, along a fixed vertical ∆, by an amount 1 { q that sends an invariant curve whose rotation number is a multiple of 1 { q exactly to another one having the same property. The dynamics of the q th iterate of the coupled map on the line t z ˚ u ˆ ∆ Ă M 1 ˆ M 2 will thus drift at a linear speed: after q 2 iterates the point will have moved by 1 in the second action coordinate r 2 , and after q 3 it will have moved by q . The diffusing orbits obtained this way are bi-asymptotic to infinity: their r 2 -coordinates travel from ´8 to `8 at average speed 1 { q 2 . 10/13

Coupling lemma 11/13

Coupling lemma F : M 1 ý and G 0 : M 2 ý diffeomorphisms z ˚ P M 1 a q -periodic for F f : M 1 Ñ R and g : M 2 Ñ R (Hamiltonian) functions. 11/13

Coupling lemma F : M 1 ý and G 0 : M 2 ý diffeomorphisms z ˚ P M 1 a q -periodic for F f : M 1 Ñ R and g : M 2 Ñ R (Hamiltonian) functions. Then T : “ Φ f b g ˝ p F ˆ G 0 q : M 1 ˆ M 2 ý satisfies z ˚ , Φ g ˝ G q T q p z ˚ , z 2 q “ ` ˘ 0 p z 2 q for all z 2 P M 2 . We have denoted by f b g the function p z 1 , z 2 q ÞÑ f p z 1 q g p z 2 q . 11/13

Coupling lemma F : M 1 ý and G 0 : M 2 ý diffeomorphisms z ˚ P M 1 a q -periodic for F f : M 1 Ñ R and g : M 2 Ñ R (Hamiltonian) functions. Synchronization Assumption f p F s p z ˚ qq “ 0 , d f p F s p z ˚ qq “ 0 f p z ˚ q “ 1 , d f p z ˚ q “ 0 , for 1 ď s ď q ´ 1. Then T : “ Φ f b g ˝ p F ˆ G 0 q : M 1 ˆ M 2 ý satisfies z ˚ , Φ g ˝ G q T q p z ˚ , z 2 q “ ` ˘ 0 p z 2 q for all z 2 P M 2 . We have denoted by f b g the function p z 1 , z 2 q ÞÑ f p z 1 q g p z 2 q . 11/13

Coupling lemma F : M 1 ý and G 0 : M 2 ý diffeomorphisms z ˚ P M 1 a q -periodic for F f : M 1 Ñ R and g : M 2 Ñ R (Hamiltonian) functions. Synchronization Assumption f p F s p z ˚ qq “ 0 , d f p F s p z ˚ qq “ 0 f p z ˚ q “ 1 , d f p z ˚ q “ 0 , for 1 ď s ď q ´ 1. Then T : “ Φ f b g ˝ p F ˆ G 0 q : M 1 ˆ M 2 ý satisfies z ˚ , Φ g ˝ G q T q p z ˚ , z 2 q “ ` ˘ 0 p z 2 q for all z 2 P M 2 . We have denoted by f b g the function p z 1 , z 2 q ÞÑ f p z 1 q g p z 2 q . The point is that Φ g p z 2 q f p z 1 q , Φ f p z 1 q g p z 2 q Φ f b g p z 1 , z 2 q “ ` ˘ for all p z 1 , z 2 q . 11/13

T : “ Φ f b g ˝ p F ˆ G 0 q : M 1 ˆ M 2 ý satisfies z ˚ , Φ g ˝ G q T q p z ˚ , z 2 q “ ` ˘ 0 p z 2 q for all z 2 P M 2 . 12/13

T : “ Φ f b g ˝ p F ˆ G 0 q : M 1 ˆ M 2 ý satisfies z ˚ , Φ g ˝ G q T q p z ˚ , z 2 q “ ` ˘ 0 p z 2 q for all z 2 P M 2 . ψ : “ Φ g ˝ G q 0 : M 2 ý appears as a subsystem of T q : M 1 ˆ M 2 ý 12/13

T : “ Φ f b g ˝ p F ˆ G 0 q : M 1 ˆ M 2 ý satisfies z ˚ , Φ g ˝ G q T q p z ˚ , z 2 q “ ` ˘ 0 p z 2 q for all z 2 P M 2 . ψ : “ Φ g ˝ G q 0 : M 2 ý appears as a subsystem of T q : M 1 ˆ M 2 ý To prove PROP: sin p 2 πθ 2 q Use g p r 2 , θ 2 q “ ´ 1 , so ψ = rescaled standard map 2 π q ψ p θ 2 , r 2 q “ p θ 2 ` q p ω 2 ` r 2 q , r 2 ` 1 q cos p θ 2 ` q p ω 2 ` r 2 qqq 12/13

T : “ Φ f b g ˝ p F ˆ G 0 q : M 1 ˆ M 2 ý satisfies z ˚ , Φ g ˝ G q T q p z ˚ , z 2 q “ ` ˘ 0 p z 2 q for all z 2 P M 2 . ψ : “ Φ g ˝ G q 0 : M 2 ý appears as a subsystem of T q : M 1 ˆ M 2 ý To prove PROP: sin p 2 πθ 2 q Use g p r 2 , θ 2 q “ ´ 1 , so ψ = rescaled standard map 2 π q ψ p θ 2 , r 2 q “ p θ 2 ` q p ω 2 ` r 2 q , r 2 ` 1 q cos p θ 2 ` q p ω 2 ` r 2 qqq not close to integrable! Drift will take place 12/13