Teoria Erg odica Diferenci avel lecture 17: Weak mixing on torus - PowerPoint PPT Presentation

Smooth ergodic theory, lecture 17 M. Verbitsky Teoria Erg´ odica Diferenci´ avel lecture 17: Weak mixing on torus Instituto Nacional de Matem´ atica Pura e Aplicada Misha Verbitsky, November 10, 2017 1

Smooth ergodic theory, lecture 17 M. Verbitsky Convergence in density (reminder) DEFINITION: The (asymptotic) density of a subset J ⊂ Z � 1 is the limit | J ∩ [1 ,N ] | . A subset J ⊂ Z � 1 has density 1 if lim N | J ∩ [1 ,N ] | lim N = 1. N N DEFINITION: A sequence { a i } of real numbers converges to a in density if there exists a subset J ⊂ Z � 1 of density 1 such that lim i ∈ J a i = a . The convergence in density is denoted by Dlim i a i = a . PROPOSITION: (Koopman-von Neumann, 1932) Let { a i } be a se- quence of bounded non-negative numbers, a i ∈ [0 , C ]. Then convergence to 0 in density is equivalent to the convergence of Ces` aro sums: N 1 � Dlim i a i = 0 ⇔ lim a i = 0 N N i =1 2

Smooth ergodic theory, lecture 17 M. Verbitsky Mixing, weak mixing, ergodicity (reminder) DEFINITION: Let ( M, µ, T ) be a dynamic system, with µ a probability mea- sure. We say that � n − 1 (i) T is ergodic if lim n 1 i =0 µ ( T i ( A ) ∩ B ) = µ ( A ) µ ( B ) , for all measurable n sets A, B ⊂ M . i →∞ µ ( T i ( A ) ∩ B ) = µ ( A ) µ ( B ) . (ii) T is weak mixing if Dlim i →∞ µ ( T i ( A ) ∩ B ) = µ ( A ) µ ( B ) . (iii) T is mixing , or strongly mixing if lim REMARK: The first condition is equivalent to the usual definition of er- godicity by the previous remark. Indeed, from (usual) ergodicity it follows that lim n 1 � n − 1 i =0 ( T ∗ ) i ( χ A ) = µ ( A ), which gives lim n 1 � n − 1 i =0 ( T ∗ ) i ( χ A ) χ B = n n µ ( A ) χ ( B ) and the integral of this function is precisely µ ( A ) µ ( B ). Con- � ( T ∗ ) i ( χ A ) χ B depends only on the measure of B , the function versely, if lim n � ( T ∗ ) i ( χ A ) is constant, hence T is ergodic in the usual sense. lim n REMARK: Clearly, (iii) ⇒ (ii) ⇒ (i) (the last implication follows because the density convergence implies the Ces` aro convergence). 3

Smooth ergodic theory, lecture 17 M. Verbitsky Mixing and weak mixing on the product (reminder) DEFINITION: Let ( M, µ, T ) be a dynamical system. Consider the dynamical system ( M, µ, T ) 2 := ( M × M, µ × µ, T × T ), where µ × µ is the product measure on M × M , and T × T ( x, y ) = ( T ( x ) , T ( y )). THEOREM: Let ( M, µ, T ) be a dynamical system, and ( M, µ, T ) 2 its product with itself. Then ( M, µ, T ) 2 is (weak) mixing if and only ( M, µ, T ) is (weak) mixing. Proof. Step 1: To simplify the notation, assume µ ( M ) = 1. To see that (weak) mixing on ( M, µ, T ) 2 implies the (weak) mixing on ( M, µ, T ), we take the sets A 1 := A × M and B 1 := B × M . Then µ ( T i ( A 1 ) ∩ B 1 ) = µ ( T i ( A ) ∩ B ) and µ ( A 1 ) µ ( B 1 ) = µ ( A ) µ ( B ), hence i µ ( T i ( A 1 ) ∩ B 1 ) = µ ( A 1 ) µ ( B 1 ) lim implies i µ ( T i ( A ) ∩ B ) = µ ( A ) µ ( B ) . lim 4

Smooth ergodic theory, lecture 17 M. Verbitsky Mixing and weak mixing on the product 2 (reminder) THEOREM: Let ( M, µ, T ) be a dynamical system, and ( M, µ, T ) 2 its product with itself. Then ( M, µ, T ) 2 is (weak) mixing if and only ( M, µ, T ) is (weak) mixing. Step 2: Conversely, assume that ( M, µ, T ) is mixing. Since the subalgebra generated by cylindrical sets is dense in the algebra of measurable sets, it would suffice to show that lim i µ ( T i ( A 1 ) ∩ B 1 ) = µ ( A 1 ) µ ( B 1 ) where A 1 , B 1 ⊂ M 2 are cylindrical. Write A 1 = A × A ′ , B 1 = B × B ′ . Then µ ( T n A 1 ∩ B 1 ) = � �� � µ ( T n A ′ ∩ B ′ ) µ ( T n A ∩ B ) . The first of the terms in brackets converges to µ ( A ) µ ( B ), the second to µ ( A ′ ) µ ( B ′ ), giving i µ ( T i ( A 1 ) ∩ B 1 ) = µ ( A ) µ ( B ) µ ( A ′ ) µ ( B ′ ) = µ ( A 1 ) µ ( B 1 ) . lim REMARK: The same argument also proves that ergodicity of ( M, µ, T ) 2 implies ergodicity of ( M, µ, T ) . The converse implication is invalid even for a circle. 5

Smooth ergodic theory, lecture 17 M. Verbitsky Ergodic measures which are not mixing (reminder) → S 1 be a rotation with irrational angle α . In angle REMARK: Let L α : S 1 − coordinates on S 1 × S 1 , the rotation L α × L α acts as L α × L α ( x, y ) = ( x + α, y + α ). Therefore, the closure of the orbit of ( x, y ) is always contained in the closed set { ( a, b ) ∈ S 1 × S 1 | a − b = x − y } , and L α × L α has no dense orbits . This gives the claim. CLAIM: Irrational rotation of a circle is ergodic, but not weakly mixing. Proof: Otherwise, L α × L α would be weak mixing, and hence ergodic, on S 1 × S 1 . 6

Smooth ergodic theory, lecture 17 M. Verbitsky Weak mixing and non-constant eigenfunctions (reminder) I am going to prove the following theorem. Theorem 1: Let ( M, µ, T ) be a dynamical system. Then the following are equivalent. (i) ( M, µ, T ) is weakly mixing. (ii) The Koopman operator T : L 2 ( M, µ ) − → L 2 ( M, µ ) has no non-constant eigenvectors. (iii) ( M, µ, T ) 2 is ergodic. 7

Smooth ergodic theory, lecture 17 M. Verbitsky Tensor product DEFINITION: Let V, V ′ be vector spaces over k , and W a vector space freely generated by symbols v ⊗ v ′ , with v ∈ V, v ′ ∈ V ′ , and W 1 ⊂ W a subspace generated by combinations av ⊗ v ′ − v ⊗ av ′ , a ( v ⊗ v ′ ) − ( av ) ⊗ v ′ , ( v 1 + v 2 ) ⊗ v ′ − v 1 ⊗ v ′ − v 2 ⊗ v ′ and v ⊗ ( v ′ 1 + v ′ 2 ) − v ⊗ v ′ 1 − v ⊗ v ′ 2 , where a ∈ k . Define the tensor product V ⊗ k V ′ as a quotient vector space W/W 1 . PROPOSITION: (“Universal property of the tensor product”) For any vector spaces V, V ′ , R , there is a natural identification Hom( V ⊗ k V ′ , R ) = Bil( V × V ′ , R ) . DEFINITION: A basis in a vector space V is a subset { v α } ⊂ V which is linearly independent and generates V . CLAIM: Suppose that V, W are vector spaces (without topology), and { v α } , { w β } the bases (in Cauchy sense) in these spaces. Then { v α ⊗ w β } is a basis in V ⊗ W . Proof: The natural map � v α ⊗ w β � − → V ⊗ W is by construction surjective and invertible by the Universal Property of the tensor product. 8

Smooth ergodic theory, lecture 17 M. Verbitsky Tensor product and functions on a product THEOREM: Let C ( M ) be the space of functions f : → R , and C ( N ) M − the spave of functions f : N − → R . Consider the natural map Ψ : C ( M ) ⊗ C ( N ) − → C ( M × N ). Then Ψ is injective. Proof. Step 1: For N, M finite Ψ is an isomorphism. Indeed, for any m ∈ M and n ∈ N , the tensor product product χ m ⊗ χ n of atomic functions χ m and χ n is mapped to χ ( m,n ) , hence Ψ is surjective, and it is injective because dim C ( M ) ⊗ C ( N ) = | M || N | = dim C ( M × N ). Step 2: For any linearly independent set of k functions f 1 , ..., f k ∈ C ( M ), consider restriction of f 1 , ..., f k to a finite subset M 0 ⊂ M . If there is a linear � relation � � i λ i f i � M 0 for each finite subset, this linear relation is true on M . Therefore, linearly independent functions remain linearly independent if restricted on a sufficiently big finite subset. Step 3: Let { f α } be a basis in C ( M ), { g α } a basis in C ( N ). Then { f α ⊗ g α } is a basis in C ( M ) ⊗ C ( N ), indexed by α ∈ A , β ∈ B . Any vector x ∈ C ( M ) ⊗ C ( N ) takes form x = � i ∈ A 0 ,j ∈ B 0 x ij f i ⊗ g j , where A 0 ⊂ A , B 0 ⊂ B are finite subsets. � � Then x � M 0 × N 0 is non-zero for some finite subsets M 0 ⊂ M , N 0 ⊂ N (Step 2). � � This implies that Ψ( x ) � M 0 × N 0 is also non-zero (Step 1). 9

Smooth ergodic theory, lecture 17 M. Verbitsky Tensor product of Hilbert spaces DEFINITION: Let H, H ′ be two Hilbert spaces. The tensor product H ⊗ H ′ ⊗ H ′ is has a natural scalar product which is non-complete. Its completion H ˆ called completed tensor product of H and H ′ . ⊗ H ′ is all REMARK: Let { e i } , { e ′ i } be orthonormal bases in H, H ′ . Then H ˆ i,j | α ij | 2 < ∞ . series � i α ij e i ⊗ e ′ j with � CLAIM: Let ( M, µ ) and ( M ′ , µ ′ ) be metrizable spaces with Borel measure. Then L 2 ( M × M ′ , µ × µ ′ ) = L 2 ( M, µ )ˆ ⊗ L 2 ( M ′ , µ ′ ) . Proof: The product map L 2 ( M, µ ) ⊗ L 2 ( M ′ , µ ′ ) − → L 2 ( M × M ′ , µ × µ ′ ) is in- jective because it it is injective on all functions, as shown above. The tensor product C 0 ( M ) ⊗ C 0 ( M ′ ) is a dense (by Stone-Weierstrass) sub- ring in C 0 ( M × M ), the space L 2 ( M, µ ) ⊗ L 2 ( M ′ , µ ′ ) is its partial completion, and L 2 ( M, µ )ˆ ⊗ L 2 ( M ′ , µ ′ ) is its completion. Therefore, L 2 ( M, µ ) ⊗ L 2 ( M ′ , µ ′ ) ⊂ L 2 ( M × M ′ , µ × µ ′ ) is a dense subset. Therefore, both spaces L 2 ( M, µ )ˆ ⊗ L 2 ( M ′ , µ ′ ) and L 2 ( M × M ′ , µ × µ ′ ) are obtained as completions of L 2 ( M, µ ) ⊗ L 2 ( M ′ , µ ′ ). They are isomorphic because completion is unique. 10