Ergodic Optimization Gonzalo Contreras CIMAT Guanajuato, Mexico - PowerPoint PPT Presentation

Ergodic Optimization Gonzalo Contreras CIMAT Guanajuato, Mexico CIRM, Marseille. May, 2019. Ergodic Optimization

Expanding map example T : r 0 , 1 s Ñ r 0 , 1 s , T p x q “ 2 x mod 1. Each point has a neighborhood of fixed size where the inverse T ´ 1 has d “ 2 branches, and they are contractions ( λ “ 1 2 -Lipschitz). Ergodic Optimization

Expanding map X compact metric space. T : X Ñ X an expanding map i.e. T P C 0 , D d P Z ` , D 0 ă λ ă 1 , D e 0 ą 0 s.t. @ x P X the branches of T ´ 1 are λ -Lipschitz, i.e. @ x P X D S i : B p x , e 0 q Ñ X , i “ 1 , . . . , ℓ x ď d , ` ˘ d S i p y q , S i p z q ď λ d p y , z q , # T ˝ S i “ I B p x , e 0 q , S i ˝ T | B p S i p x q , λ e 0 q “ I B p S i p x q , λ e 0 q . Ergodic Optimization

Main Theorem X compact metric space. T : X Ñ X expanding map, F P Lip p X , R q . A maximizing measure is a T -invariant Borel probability µ on X such that ż ! ż ˇ ) F d µ “ max F d ν ˇ ν invariant Borel probability . ˇ Theorem If X is a compact metric space and T : X Ñ X is an expanding map then there is an open and dense set O Ă Lip p X , R q such that for all F P O there is a single F-maximizing measure and it is supported on a periodic orbit. Ergodic Optimization

Ground states Maximizing measures are called ground states because if F ě 0 and µ β is the invariant measure satisfying " * ż µ β “ argmax h ν p T q ` β F d ν ν inv. measure (the equilibrium state for β F ) ( β “ 1 τ “ the inverse of the temperature) then any limit β k Ñ`8 µ β k lim (a zero temperature limit) is a maximizing measure (a ground state). Ergodic Optimization

Bousch, Jenkinson: There is a residual set U Ă C 0 p X , R q s.t. F P U ù ñ F has a unique maximizing measure and it has full support. Yuan & Hunt: Generically periodic maximizing measures are stable. (i.e. same maximizing measures for perturbations of the potential in Hölder or Lipschitz topology.) Non-periodic maximizing measures are not stable in Hölder or Lipschitz topology. Contreras, Lopes, Thieullen: Generically in C α p X , R q there is a unique maximizing measure. If F P C α p X , R q , then F can be approximated in the C β topology β ă α by G with the maximizing measure supported on a periodic orbit. Ergodic Optimization

Bousch: Proves a similar result for Walters functions: @ ε ą 0 D δ ą 0 @ n P N , @ x , y P X , d n p x , y q ă δ ù ñ | S n F p x q ´ S n F p y q| ă ε. T i p x q , T i p y q ` ˘ d n p x , y q : “ sup d . i “ 0 ,..., n Quas & Siefken: prove a similar result for super-continuous functions. (functions whose local Lipschitz constant converges to 0 at a given rate: here X is a Cantor set or a shift space). Ergodic Optimization

For example in a subshift of finite type ( X is a Cantor set) locally constant functions have periodic maximizing measures. But those functions are not dense in C α p X , R q or Lip p X , R q . And they are not well adapted for applications to Lagrangian dynamics or twist maps with continuous phase space X . Ergodic Optimization

Write ż α : “ α p F q : “ ´ max F d µ. (Mañé’s critical value) µ P M p T q Set of maximizing measures ż ˇ ! ) M p F q : “ µ P M p T q F d µ “ ´ α p F q . ˇ ˇ Ergodic Optimization

Generic Uniqueness of minimizing measures Theorem (Contreras, Lopes, Thieullen) There is a generic set G in Lip p X , R q such that @ F P G # M p F q “ 1 . Moreover, for F P G , µ P M p F q supp µ is uniquely ergodic . Ergodic Optimization

Proof: Enough to prove O p ε q : “ t F P Lip p X , R q | diam M p F q ă ε u is open and dense. Because then take č O p 1 G : “ n q n P N ` will have G is generic and G Ă t F : # M p F q “ 1 u . Open = upper semicontinuity of M p F q . (limits of minimizing measures are minimizing) Ergodic Optimization

Density of O p ε q . Want to approximate any F 0 P Lip p X , R q by elements in O p ε q . Let F “ t f n u n P N ` be a dense set in Lip p X , R q X r} f } sup ď 1 s . We use f 0 “ ´ F 0 the original potential. ÿ 1 d p µ, ν q “ 2 n | µ p f n q ´ ν p f n q| . n P N is a metric on M p T q . Take a finite dimensional approximation of M p T q by projecting π N : M p T q Ñ R N ` 1 (integrals of test functions) ` ˘ π N p µ q : “ ´ µ p F 0 q , µ p f 1 q , . . . , µ p f N q . diam p π ´ 1 1 N t x uq ď ε N “ 2 N Ñ 0 . Ergodic Optimization

α p F 0 q “ argmin t µ p´ F 0 q | µ P M p T q u K N : “ π N p M p T qq is a convex subset in R N ` 1 and r x 0 “ α s is a supporting hyperplane for K N . Use your favourite argument to perturb the hyperplane so that it touches K N in a unique (exposed) point y . x 0 π N (M(T)) F grad G grad 0 x N Ergodic Optimization

The new supporting hyperplane has normal vector p 1 , z 1 , . . . , z N q . The touching (exposed) point is y : “ π N p argmin M p T q t´ G uq “ π N p M p G qq ´ G “ ´ F 0 ` ř N n “ 1 z n ¨ f n Then diam M p G q ď diam π ´ 1 1 N p y q ď ε N “ 2 N . So G P O p ε N q and G is very near to F 0 . l Ergodic Optimization

Sub-actions = Revelations A sub-action is a Lispchitz function u P Lip p X , R q such that F ` α ď u ˝ T ´ u . Writing G : “ F ` α ´ u ˝ T ` u we have G has the same maximizing measures as F . G ď 0. ş For µ P M max p F q we have G d µ “ 0. µ P M p F q ð ñ supp p µ q Ă r G “ 0 s . 6 If u exists: On the support of a maximizing measure G “ 0, i.e. F ` c is a coboundary. Ergodic Optimization

Generic unique ergodicity If we construct a sub-action µ P M p F q ð ñ supp p µ q Ă r G “ 0 s If M p F q “ t µ u then µ is the unique invariant measure in supp p µ q . Ergodic Optimization

One can construct sub-actions as “maximal profits” or “optimal values” along pre-orbits. For example ! n ´ 1 ( ˇ ) ÿ � F p T k y q ´ α ˇ T n p y q “ x , y P X u p x q “ sup ˇ k “ 0 will be a sub-action. Also Defining a “Mañé action potential”. Using methods from “Weak KAM Theory”. Ergodic Optimization

Lax Operator M p T q : “ t T -invariant Borel probabilities u F P Lip p X , R q , L F : Lip p X , R q Ñ Lip p X , R q : L F p u qp x q : “ y P T ´ 1 p x q t α ` F p x q ` u p x q u , max ż α : “ ´ max where F d µ. µ P M p T q Set of maximizing measures ż ˇ ! ) M p F q : “ µ P M p T q F d µ “ ´ α p F q . ˇ ˇ Ergodic Optimization

Calibrated sub-action Calibrated sub-action = Fixed point of Lax Operator = Solution to Bellman equation. L F p u q “ u write F : “ F ` α ` u ´ u ˝ T . REMARKS : ż ´ α p F q “ max F d µ “ 0. 1 µ P M p T q F ď 0. 2 ! ˇ ) M p F q “ M p F q “ µ P M p T q ˇ supp p µ q Ă r F “ 0 s . 3 ˇ Ergodic Optimization

Proposition If F is Lipschitz then there exists a Lipschitz calibrated sub-action. Proof. ` ˘ ` ˘ Prove that Lip L F p u q ď λ Lip p u q ` Lip p F q . 1 Then L F leaves invariant the space 2 ˇ Lip p u q ď λ Lip p F q ! ) ˇ E : “ u P Lip p X , R q . 1 ´ λ E { {constants} is compact & convex. 3 L F is continuous on E . Schauder Thm. ù ñ L F has a fixed pt. on E { {constants} . Prove it is a fixed point on E . 4 Ergodic Optimization

REMARKS If u is a calibrated sub-action: 1 Every point z P X has a calibrating pre-orbit p z k q k ď 0 s.t. # T i p z ´ i q “ z 0 “ z , @ i ě 0 ; u p z k ` 1 q “ u p z k q ` α ` F p z k q , @ k ď ´ 1 . Equivalently, since T p z k q “ z k ` 1 , F p z k q “ 0 @ k ď ´ 1 . Ergodic Optimization

α -limits Proposition If O p y q Ă X is a periodic orbit such that for every calibrated sub-action the α -limit of any calibrating pre-orbit is in O p y q then every maximizing measure has support in O p y q . Need: @ µ P M p F q supp p µ q Ă α -limit of calibrating pre-orbits Ă α -limit of orbits in r F “ 0 s enough : Ă α -limit of orbits in supp p µ q . For example: Extend T to an invertible dynamical system T : X Ñ X . Lift µ to a T -invariant µ . The set Y of recurrent points of T ´ 1 in supp p µ q has total µ -measure and projects onto a set Y “ π p Y q with total measure of points which are α -limits of pre-orbits in supp p µ q . Ergodic Optimization

Shadowing Lemma Definition p x n q n P N Ă X is a δ -pseudo-orbit if 1 ` ˘ d x n ` 1 , T p x n q ď δ , @ n P N . A point y P X ε -shadows a pseudo-orbit p x n q n P N if 2 T n p y q , x n ` ˘ d ă ε , @ n P N . Ergodic Optimization

Proposition (Shadowing Lemma) If p x k q k P N is a δ -pseudo-orbit ù ñ D y P X whose orbit ε -shadows p x k q δ with ε “ 1 ´ λ . If p x k q is periodic ù ñ y is a periodic point with the same period. Proof. λ δ a “ 1 ´ λ . t y u “ Ş 8 ` ˘ k “ 0 S 0 ˝ ¨ ¨ ¨ ˝ S k B p x k ` 1 , a q . where the inverse branch S k is chosen such that S k p T p x k qq “ x k . Ergodic Optimization

Zero entropy Theorem (Morris) Let X be a compact metric space and T : X Ñ X an expanding map. There is a residual set G Ă Lip p X , R q such that if F P G then there is a unique F-maximizing measure and it has zero metric entropy. Ergodic Optimization