Hypercyclicity versus disjoint hypercyclicity

Transcription

1 Hypercyclicity versus disjoint hypercyclicity J. Bès 1, Ö. Martin 2, A. Peris 3, R. Sanders 4, and S. Shkarin 5 Istanbul Analysis Seminars November 1, Bowling Green State University 2 Miami University 3 Universidad Politécnica de Valencia 4 University of Milwakee 5 Queen s University Belfast

2 Introduction Throughout, X is a separable, infinite dimensional Fréchet space. Definition T : X X is hypercyclic provided f X so that {f, Tf, T 2 f,... } = X. We call f a hypercyclic vector for T. HC(T ) = {hyp. vectors for T } Definition (Bernal 07, Bès, Peris, 07) T 1, T 2 : X X are disjoint-hypercyclic (d-hypercyclic) provided f X : {(T1 nf, T 2 n f ) : n = 0, 1,... } = X X. (i.e., T 1 T 2 : X X X X has a hypercyclic vector of the form (f, f )). We call f a d-hyp. vector for T 1, T 2. d-hc(t 1, T 2) = {d-hyp. vectors for T 1, T 2} Goal: To compare these two notions; look for properties they share or not.

3 Introduction Throughout, X is a separable, infinite dimensional Fréchet space. Definition T : X X is hypercyclic provided f X so that {f, Tf, T 2 f,... } = X. We call f a hypercyclic vector for T. HC(T ) = {hyp. vectors for T } Definition (Bernal 07, Bès, Peris, 07) T 1, T 2 : X X are disjoint-hypercyclic (d-hypercyclic) provided f X : {(T1 nf, T 2 n f ) : n = 0, 1,... } = X X. (i.e., T 1 T 2 : X X X X has a hypercyclic vector of the form (f, f )). We call f a d-hyp. vector for T 1, T 2. d-hc(t 1, T 2) = {d-hyp. vectors for T 1, T 2} Goal: To compare these two notions; look for properties they share or not.

4 Introduction Throughout, X is a separable, infinite dimensional Fréchet space. Definition T : X X is hypercyclic provided f X so that {f, Tf, T 2 f,... } = X. We call f a hypercyclic vector for T. HC(T ) = {hyp. vectors for T } Definition (Bernal 07, Bès, Peris, 07) T 1, T 2 : X X are disjoint-hypercyclic (d-hypercyclic) provided f X : {(T1 nf, T 2 n f ) : n = 0, 1,... } = X X. (i.e., T 1 T 2 : X X X X has a hypercyclic vector of the form (f, f )). We call f a d-hyp. vector for T 1, T 2. d-hc(t 1, T 2) = {d-hyp. vectors for T 1, T 2} Goal: To compare these two notions; look for properties they share or not.

5 Introduction The three simplest conditions to show an operator is hypercyclic are: 1 Topological transitivity. 2 The Hypercyclicity Criterion. 3 The Blow-up/collapse property. Each such condition has a disjoint version, with d-hypercyclic consequences: 1 d-topological transitivity: Translation operators on H(C) 2 The d-hypercyclicity Criterion: { Composition operators on H(D) or H 2 (D) Powers of weighted shift operators on l p (1 p < ) 3 The d-blow-up/collapse property: Weighted shift operators.

6 Introduction The three simplest conditions to show an operator is hypercyclic are: 1 Topological transitivity. 2 The Hypercyclicity Criterion. 3 The Blow-up/collapse property. Each such condition has a disjoint version, with d-hypercyclic consequences: 1 d-topological transitivity: Translation operators on H(C) 2 The d-hypercyclicity Criterion: { Composition operators on H(D) or H 2 (D) Powers of weighted shift operators on l p (1 p < ) 3 The d-blow-up/collapse property: Weighted shift operators.

7 Introduction The three simplest conditions to show an operator is hypercyclic are: 1 Topological transitivity. 2 The Hypercyclicity Criterion. 3 The Blow-up/collapse property. Each such condition has a disjoint version, with d-hypercyclic consequences: 1 d-topological transitivity: Translation operators on H(C) 2 The d-hypercyclicity Criterion: { Composition operators on H(D) or H 2 (D) Powers of weighted shift operators on l p (1 p < ) 3 The d-blow-up/collapse property: Weighted shift operators.

8 Definition T : X X is topologically transitive provided for each non-empty open subsets U, V of X there exits n so that T 1, T 2 : X X are d-topologically transitive provided for each V 0, V 1, V 2 there exists n so that T n (U) V. (1) T is mixing if (1) always holds for all but finitely many positive integers n. Facts: TFAE: 1 T is top. transitive. 2 HC(T ) is residual. 3 HC(T ) is dense. T mixing T r top. transitive. V 0 T n 1 (V 1) T n 2 (V 2). (2) T 1, T 2 d-mixing if (2) always holds for all but finitely many positive integers n. TFAE: 1 T 1, T 2 d-top. transitive. 2 d-hc(t 1, T 2 ) is residual. 3 d-hc(t 1, T 2 ) is dense. T 1, T 2 d-mixing T r 1, T r 2 d-top. transitive

9 Let H(C) = {entire functions}, with compact-open topology. τ a : H(C) H(C), τ a(f )(z) = f (z + a) (z, a C). Hypercyclic translations on H(C) (Birkhoff, 29) τ a is hypercyclic a C \ {0}. (Dujos-Ruiz 83) τ 1 has hypercyclic vectors of arbitrary small order (> 0). (Chan, Shapiro 91) τ 1 is hypercyclic on Hilbert subspaces E 2,γ H(C) of entire functions growth order one and exponential type zero. d-hypercyclic translations on H(C) (Bernal 07 / Bès, Peris, 07) τ a1, τ a2 d-hypercyclic a 1, a 2 C \ {0} are distinct. (Bès, M., Peris, Shkarin, 12) If a 1, a 2 C \ {0} are distinct, then τ a1, τ a2 are d-hypercyclic on E 2,γ.

10 Powers and rotations of hypercyclic and of mixing operators (Ansari 95) HC(T ) = HC(T r ) (r N). (León-Müller 04) HC(T ) = HC(e iθ T ). If T is mixing, so is T r (r N). If T is mixing, so is e iθ T (θ R). Powers and rotations of d-hypercyclic and of d-mixing operators (Ansari 95) d-hc(t 1,..., T N ) = d-hc(t r 1,..., T r N) (r N). d-hc(t 1,..., T N ) = d-hc(e iθ 1 T 1,..., e iθ N T N ) (θ j [0, 2π)). Indeed, HC(T 1 T N ) = HC(e iθ 1 T 1 e iθ N T N ) (Shkarin 08) T 1,..., T N d-mixing T r 1 1,..., T r N N d-hypercyclic. T 1,..., T N d-mixing e iθ 1 T 1,..., e iθ N T N d-mixing.

11 Hypercyclicity Criterion (Kitai, Gethner-Shapiro) Let T : X X so that there exist integers 1 < n 1 < n 2 <... and dense subsets X 0, Y 0 of X and mappings S k : Y 0 X (k N) so that 1 T n k x 0 (x X 0) 2 S k x 0 (x Y 0), and 3 T n k S k x x (x Y 0). Then T is hypercyclic. D-Hypercyclicity Criterion. (Bès, Peris, 07) Let T 1, T 2 : X X so that there exist 1 < n 1 < n 2 <..., dense subsets X 0, X 1, X 2 of X, and mappings S l,k : X l X (k N, l = 1, 2) satisfying 1 T n k l x 0 (x X0), 2 S l,k x 0 (x X l ), and 3 (T n k i S l,k δ i,l Id Xl )x 0 (x X l ), (1 i N). Then T 1, T 2 are d-hypercyclic.

12 Hypercyclicity Criterion (Kitai, Gethner-Shapiro) Let T : X X so that there exist integers 1 < n 1 < n 2 <... and dense subsets X 0, Y 0 of X and mappings S k : Y 0 X (k N) so that 1 T n k x 0 (x X 0) 2 S k x 0 (x Y 0), and 3 T n k S k x x (x Y 0). Then T is hypercyclic. D-Hypercyclicity Criterion. (Bès, Peris, 07) Let T 1, T 2 : X X so that there exist 1 < n 1 < n 2 <..., dense subsets X 0, X 1, X 2 of X, and mappings S l,k : X l X (k N, l = 1, 2) satisfying 1 T n k l x 0 (x X0), 2 S l,k x 0 (x X l ), and 3 (T n k i S l,k δ i,l Id Xl )x 0 (x X l ), (1 i N). Then T 1, T 2 are d-hypercyclic.

13 Bès, Peris 99 TFAE: 1 T satisfies the Hyp. Criterion. 2 r: r j=1t is hypercyclic on X r. 3 T is hereditarily topologically transitive. Bès, Peris, 07 TFAE: 1 T 1, T 2 satisfies the d-hyp. Criterion. 2 r: r j=1t 1, r j=1t 2 are d-hypercyclic on X r. 3 T 1, T 2 are hereditarily d-topologically transitive. In particular, mixing Hyp. Criterion. In particular, d-mixing d-hyp. Criterion.

14 Bès, Peris 99 TFAE: 1 T satisfies the Hyp. Criterion. 2 r: r j=1t is hypercyclic on X r. 3 T is hereditarily topologically transitive. Bès, Peris, 07 TFAE: 1 T 1, T 2 satisfies the d-hyp. Criterion. 2 r: r j=1t 1, r j=1t 2 are d-hypercyclic on X r. 3 T 1, T 2 are hereditarily d-topologically transitive. In particular, mixing Hyp. Criterion. In particular, d-mixing d-hyp. Criterion.

15 Bès, Peris 99 TFAE: 1 T satisfies the Hyp. Criterion. 2 r: r j=1t is hypercyclic on X r. 3 T is hereditarily topologically transitive. Bès, Peris, 07 TFAE: 1 T 1, T 2 satisfies the d-hyp. Criterion. 2 r: r j=1t 1, r j=1t 2 are d-hypercyclic on X r. 3 T 1, T 2 are hereditarily d-topologically transitive. In particular, mixing Hyp. Criterion. In particular, d-mixing d-hyp. Criterion.

16 Linear fractional d-hypercyclicity Notation: D := {z C : z < 1} Ĉ := C { } az + b LFT (Ĉ) := {ϕ(z) = : a, b, c, d C, ad bc 0} cz + d LFT (D) := {ϕ LFT (Ĉ) : ϕ(d) D} Aut(D) := {ϕ LFT (Ĉ) : ϕ(d) = D} Classification of LFT (Ĉ) ϕ LFT (Ĉ) is called parabolic if it has (exactly) one fixed point α. If so, ψ = σ ϕ σ 1 = z + b, for some 0 b C, where σ(z) = 1. z α If ϕ has (exactly) two fixed points α and β, then taking σ(z) = z β it is z α conjugate to ψ = σ ϕ σ 1 = λz, for some λ 1. Then ϕ is called: 1 Elliptic if λ = 1, 2 Hyperbolic if λ > 0, and 3 Loxodromic if ϕ is neither elliptic nor hyperbolic.

17 Linear fractional d-hypercyclicity Fixed point configuration of members of LFT (D) (a) (b) (c) Parabolic members of LFT (D) have their fixed point on D. Hyperbolic members of LFT (D) must have their attractive fixed point in D, with the other fixed point outside D, and lying on D if and only if the map is an automorphism. Loxodromic and elliptic members of LFT (D) have a fixed point in D and a fixed point outside D. The elliptic ones are precisely the automorphisms in LFT (D) with this fixed point configuration.

18 Linear fractional d-hypercyclicity (Shapiro 01) Let Ω = D or C, and ϕ : Ω Ω holomorphic. TFAE 1 C ϕ hypercyclic on H(Ω) 2 C ϕ mixing on H(Ω). 3 ϕ univalent, without fixed points on Ω. (Bourdon, Shapiro, 97) Let ϕ LFT (D), H 2 = {f = n 0 anz n H(D) : f 2 = n 0 an 2 < }. TFAE: 1 C ϕ is hypercyclic on H 2 2 C ϕ is mixing on H 2 3 ϕ is a parabolic automorphism or hyperbolic without fixed points on D.

19 Linear fractional d-hypercyclicity For v R, let S v := {f (z) = n a nz n H(D) : f 2 v := n a n 2 (n + 1) 2v < } If v 1 < v 2, then S v1 S v2. S v is the Bergman, Hardy, and Dirichlet space, when v = 1 2, 0 and 1 2, respectively. Gallardo, Montes, 05 Let ϕ LFT (D). Then 1 C ϕ hypercyclic on S v C ϕ hypercyclic on S 0 = H 2 (D), and v < On S v with v < 1, Cϕ hypercyclic Cϕ supercyclic. 2 (T : X X is supercyclic provided there is some f X for which {λt n f : λ C, n = 0, 1,... } = X )

20 Linear fractional d-hypercyclicity For v R, let S v := {f (z) = n a nz n H(D) : f 2 v := n a n 2 (n + 1) 2v < } If v 1 < v 2, then S v1 S v2. S v is the Bergman, Hardy, and Dirichlet space, when v = 1 2, 0 and 1 2, respectively. Gallardo, Montes, 05 Let ϕ LFT (D). Then 1 C ϕ hypercyclic on S v C ϕ hypercyclic on S 0 = H 2 (D), and v < On S v with v < 1, Cϕ hypercyclic Cϕ supercyclic. 2 (T : X X is supercyclic provided there is some f X for which {λt n f : λ C, n = 0, 1,... } = X )

21 Linear fractional d-hypercyclicity For v R, let S v := {f (z) = n a nz n H(D) : f 2 v := n a n 2 (n + 1) 2v < } If v 1 < v 2, then S v1 S v2. S v is the Bergman, Hardy, and Dirichlet space, when v = 1 2, 0 and 1 2, respectively. Gallardo, Montes, 05 Let ϕ LFT (D). Then 1 C ϕ hypercyclic on S v C ϕ hypercyclic on S 0 = H 2 (D), and v < On S v with v < 1, Cϕ hypercyclic Cϕ supercyclic. 2 (T : X X is supercyclic provided there is some f X for which {λt n f : λ C, n = 0, 1,... } = X )

22 Linear fractional d-hypercyclicity (Bès, M., Peris, 12) Theorem 1. Let ϕ 1, ϕ 2 LFT (D) without fixed points in D. Then T.F.A.E.: 1 C ϕ1, C ϕ2 are d-supercyclic on H(D). 2 λ 1C ϕ1, λ 2C ϕ2 are d-hypercyclic on H(D), for any λ 1, λ 2 C \ {0}. 3 If ϕ 1 and ϕ 2 have the same attractive fixed point α, then ϕ 1(α) = ϕ 2(α) < 1 does not occur. Theorem 2. Let v < 1 2. For j = 1, 2, let ϕ j LFT (D) be either a parabolic automorphism or hyperbolic, with no fixed point in D. Then T.F.A.E.: 1 C ϕ1, C ϕ2 are d-supercyclic on S v. 2 C ϕ1, C ϕ2 are d-hypercyclic on S v. 3 If ϕ 1 and ϕ 2 have the same attractive fixed point α, then ϕ 1(α) = ϕ 2(α) < 1 does not occur.

23 Linear fractional d-hypercyclicity (Bès, M., Peris, 12) Theorem 1. Let ϕ 1, ϕ 2 LFT (D) without fixed points in D. Then T.F.A.E.: 1 C ϕ1, C ϕ2 are d-supercyclic on H(D). 2 λ 1C ϕ1, λ 2C ϕ2 are d-hypercyclic on H(D), for any λ 1, λ 2 C \ {0}. 3 If ϕ 1 and ϕ 2 have the same attractive fixed point α, then ϕ 1(α) = ϕ 2(α) < 1 does not occur. Theorem 2. Let v < 1 2. For j = 1, 2, let ϕ j LFT (D) be either a parabolic automorphism or hyperbolic, with no fixed point in D. Then T.F.A.E.: 1 C ϕ1, C ϕ2 are d-supercyclic on S v. 2 C ϕ1, C ϕ2 are d-hypercyclic on S v. 3 If ϕ 1 and ϕ 2 have the same attractive fixed point α, then ϕ 1(α) = ϕ 2(α) < 1 does not occur.

24 Linear fractional d-hypercyclicity Fact: For T invertible, T hypercyclic T 1 hypercyclic. In contrast, T 1, T 2 d-mixing and invertible T 1 1, T 1 2 d-supercyclic. The hyperbolic maps ϕ j Aut(D) (j = 1, 2) given by ϕ 1(z) = (3 + i)z 1 i ( 1 + i)z + 3 i and ϕ 2 = (3 + 2i)z 1 2i ( 1 + 2i)z + 3 2i have the attractive fixed points i and 3 4 i, respectively, and have the same 5 5 repellent fixed point 1. So C ϕ1, C ϕ2 are d-hypercyclic on H(D), by Theorem 1. But Cϕ 1 1 = C ϕ 1, Cϕ 1 2 = C 1 ϕ 1 satisfy that ϕ 1 1, ϕ 1 2 have the same attractive 2 fixed point at 1, with By Theorem 1, C 1 ϕ 1, C 1 ϕ 2 (ϕ 1 1 ) (1) = (ϕ 1 2 ) (1) = 1 2 < 1. (= C ϕ 1 1, C ϕ 1) are not d-supercyclic. 2

25 Linear fractional d-hypercyclicity Fact: For T invertible, T hypercyclic T 1 hypercyclic. In contrast, T 1, T 2 d-mixing and invertible T 1 1, T 1 2 d-supercyclic. The hyperbolic maps ϕ j Aut(D) (j = 1, 2) given by ϕ 1(z) = (3 + i)z 1 i ( 1 + i)z + 3 i and ϕ 2 = (3 + 2i)z 1 2i ( 1 + 2i)z + 3 2i have the attractive fixed points i and 3 4 i, respectively, and have the same 5 5 repellent fixed point 1. So C ϕ1, C ϕ2 are d-hypercyclic on H(D), by Theorem 1. But Cϕ 1 1 = C ϕ 1, Cϕ 1 2 = C 1 ϕ 1 satisfy that ϕ 1 1, ϕ 1 2 have the same attractive 2 fixed point at 1, with By Theorem 1, C 1 ϕ 1, C 1 ϕ 2 (ϕ 1 1 ) (1) = (ϕ 1 2 ) (1) = 1 2 < 1. (= C ϕ 1 1, C ϕ 1) are not d-supercyclic. 2

26 Linear fractional d-hypercyclicity Fact: For T invertible, T hypercyclic T 1 hypercyclic. In contrast, T 1, T 2 d-mixing and invertible T 1 1, T 1 2 d-supercyclic. The hyperbolic maps ϕ j Aut(D) (j = 1, 2) given by ϕ 1(z) = (3 + i)z 1 i ( 1 + i)z + 3 i and ϕ 2 = (3 + 2i)z 1 2i ( 1 + 2i)z + 3 2i have the attractive fixed points i and 3 4 i, respectively, and have the same 5 5 repellent fixed point 1. So C ϕ1, C ϕ2 are d-hypercyclic on H(D), by Theorem 1. But Cϕ 1 1 = C ϕ 1, Cϕ 1 2 = C 1 ϕ 1 satisfy that ϕ 1 1, ϕ 1 2 have the same attractive 2 fixed point at 1, with By Theorem 1, C 1 ϕ 1, C 1 ϕ 2 (ϕ 1 1 ) (1) = (ϕ 1 2 ) (1) = 1 2 < 1. (= C ϕ 1 1, C ϕ 1) are not d-supercyclic. 2

27 Linear fractional d-hypercyclicity Fact: For T invertible, T hypercyclic T 1 hypercyclic. In contrast, T 1, T 2 d-mixing and invertible T 1 1, T 1 2 d-supercyclic. The hyperbolic maps ϕ j Aut(D) (j = 1, 2) given by ϕ 1(z) = (3 + i)z 1 i ( 1 + i)z + 3 i and ϕ 2 = (3 + 2i)z 1 2i ( 1 + 2i)z + 3 2i have the attractive fixed points i and 3 4 i, respectively, and have the same 5 5 repellent fixed point 1. So C ϕ1, C ϕ2 are d-hypercyclic on H(D), by Theorem 1. But Cϕ 1 1 = C ϕ 1, Cϕ 1 2 = C 1 ϕ 1 satisfy that ϕ 1 1, ϕ 1 2 have the same attractive 2 fixed point at 1, with By Theorem 1, C 1 ϕ 1, C 1 ϕ 2 (ϕ 1 1 ) (1) = (ϕ 1 2 ) (1) = 1 2 < 1. (= C ϕ 1 1, C ϕ 1) are not d-supercyclic. 2

28 Linear fractional d-hypercyclicity Fact: For T invertible, T hypercyclic T 1 hypercyclic. In contrast, T 1, T 2 d-mixing and invertible T 1 1, T 1 2 d-supercyclic. The hyperbolic maps ϕ j Aut(D) (j = 1, 2) given by ϕ 1(z) = (3 + i)z 1 i ( 1 + i)z + 3 i and ϕ 2 = (3 + 2i)z 1 2i ( 1 + 2i)z + 3 2i have the attractive fixed points i and 3 4 i, respectively, and have the same 5 5 repellent fixed point 1. So C ϕ1, C ϕ2 are d-hypercyclic on H(D), by Theorem 1. But Cϕ 1 1 = C ϕ 1, Cϕ 1 2 = C 1 ϕ 1 satisfy that ϕ 1 1, ϕ 1 2 have the same attractive 2 fixed point at 1, with By Theorem 1, C 1 ϕ 1, C 1 ϕ 2 (ϕ 1 1 ) (1) = (ϕ 1 2 ) (1) = 1 2 < 1. (= C ϕ 1 1, C ϕ 1) are not d-supercyclic. 2

29 d-hypercyclicity of powers of shift operators Bès, Peris, 07 Let λ 1, λ 2 C \ {0}, and let 1 r 1 r 2 be integers. Then 1 λ 1D r 1, λ 2D r 2 are d-hypercyclic on H(D) r 1 < r 2. 2 Let B : l 2 l 2, (x 0, x 1,... ) (x 1, x 2,... ). Then λ 1B r 1, λ 2B r 2 are d-hypercyclic on l 2 r 1 < r 2 and 1 < λ 1 < λ 2. (e.g., 2B, 3B 2 are d-hypercyclic on l 2, while 3B, 2B 2 are not) 3 Let B 1, B 2 unilateral (or bilateral) weighted backward shifts on l 2 (l 2(Z)). If r 1 < r 2, then B r 1 1, Br 2 2 d-hypercyclic B r 1 1, Br 2 2 satisfy the d-hyp. Criterion.

30 d-hypercyclicity of powers of shift operators Bès, Peris, 07 Let λ 1, λ 2 C \ {0}, and let 1 r 1 r 2 be integers. Then 1 λ 1D r 1, λ 2D r 2 are d-hypercyclic on H(D) r 1 < r 2. 2 Let B : l 2 l 2, (x 0, x 1,... ) (x 1, x 2,... ). Then λ 1B r 1, λ 2B r 2 are d-hypercyclic on l 2 r 1 < r 2 and 1 < λ 1 < λ 2. (e.g., 2B, 3B 2 are d-hypercyclic on l 2, while 3B, 2B 2 are not) 3 Let B 1, B 2 unilateral (or bilateral) weighted backward shifts on l 2 (l 2(Z)). If r 1 < r 2, then B r 1 1, Br 2 2 d-hypercyclic B r 1 1, Br 2 2 satisfy the d-hyp. Criterion.

31 d-hypercyclicity of powers of shift operators Bès, Peris, 07 Let λ 1, λ 2 C \ {0}, and let 1 r 1 r 2 be integers. Then 1 λ 1D r 1, λ 2D r 2 are d-hypercyclic on H(D) r 1 < r 2. 2 Let B : l 2 l 2, (x 0, x 1,... ) (x 1, x 2,... ). Then λ 1B r 1, λ 2B r 2 are d-hypercyclic on l 2 r 1 < r 2 and 1 < λ 1 < λ 2. (e.g., 2B, 3B 2 are d-hypercyclic on l 2, while 3B, 2B 2 are not) 3 Let B 1, B 2 unilateral (or bilateral) weighted backward shifts on l 2 (l 2(Z)). If r 1 < r 2, then B r 1 1, Br 2 2 d-hypercyclic B r 1 1, Br 2 2 satisfy the d-hyp. Criterion.

32 d-hypercyclicity of powers of shift operators Corollary of d-hypercyclic LFT theorem: Let T = C ϕ be hypercyclic on X = H(D) or S v, where ϕ LFT (D) and 1 r 1 < r 2. 1 Then T r 1, T r 2 are d-hypercyclic on X. 2 If in addition ϕ Aut(D), then T r 1, T r 2 are d-hypercyclic on X for any non-zero integers r 1 < r 2. Corollary of d-hypercyclic powers of shifts theorem: Let T be a hypercyclic weighted shift on l 2, and let 1 r 1 < r 2 be integers. Then T r 1, T r 2 are d-hypercyclic on l 2 the direct sum operator T r 1 T r 2 T r 2 r 1 is hypercyclic on l 2 l 2 l 2.

33 d-hypercyclicity of powers of shift operators Corollary of d-hypercyclic LFT theorem: Let T = C ϕ be hypercyclic on X = H(D) or S v, where ϕ LFT (D) and 1 r 1 < r 2. 1 Then T r 1, T r 2 are d-hypercyclic on X. 2 If in addition ϕ Aut(D), then T r 1, T r 2 are d-hypercyclic on X for any non-zero integers r 1 < r 2. Corollary of d-hypercyclic powers of shifts theorem: Let T be a hypercyclic weighted shift on l 2, and let 1 r 1 < r 2 be integers. Then T r 1, T r 2 are d-hypercyclic on l 2 the direct sum operator T r 1 T r 2 T r 2 r 1 is hypercyclic on l 2 l 2 l 2.

34 d-hypercyclicity for shift operators Let U(X ) = { = V X open}, and U 0(X ) = {W U(X ) : 0 W }. T : X X... is weakly mixing provided T T : X X X X is top. transitive. satisfies the Blow-up/Collapse property provided U, V U(X ) and W U 0(X ), n so that W T n (V ) U T n (W ). Definition. T 1, T 2 : X X... are d-weakly mixing when T 1 T 1, T 2 T 2 are d-topologically transitive on X X. satisfy the d-blow up/collapse property provided for each V 0, V 1, V 2 U and W U 0, n : W T n 1 (V 1) T n 2 (V 2) = V 0 T n 1 (W ) T n 2 (W ).

35 d-hypercyclicity for shift operators Bès, Peris 07 d-blow-up/collapse d-topologically transitive. For N = 2, 3,..., there exist unilateral weighted backward shifts B 1,..., B N satisfying the d-blow-up/collapse property on l 2. Problem: When are B 1,..., B N given weighted shifts d-hypercyclic? If so, do they satisfy the d-hypercyclicity Criterion? Recall: (Bès, Peris 99, Bernal, Grosse-Erdmann 03) For T : X X, Hyp. Criterion Hered. top. transitive Weak mixing Blow-up/Collapse. (León, Montes 97) Every hypercyclic shift on l 2 is weakly mixing. (De la Rosa, Read 08) T hypercyclic, not weakly mixing (Hard!).

36 d-hypercyclicity for shift operators Theorem 1. Bès, M., Sanders (JOT +) Let B l : l 2(Z) l 2(Z), B l e j = w (l) j e j 1, l = 1, 2,... N. Then TFAE: (1) B 1, B 2,..., B N are d-hypercyclic. (2) B 1, B 2,..., B N satisfy the d-blow-up/collapse property. (3) There exist integers 1 < n 1 < n 2 <... so that (a) for each i k: w (1) i+1 w (1) i+2 w (1) i+n k k w (l) i n k +1 w (l) i 1 w (l) i 0 l = 1,..., N. k (b) the set A = {(..., α (2) 1,n k, α (3) 1,n k,..., α (N) 1,n k, α (2) 0,n k,..., α (N) 0,n k,... ) : k = 0, 1,... } is dense in (K Z, product topology), where α (l) nj=1 (l) i,n = w i+j nj=1 w (1) i+j

37 d-hypercyclicity for shift operators Fact: If B 1, B 2 are bilateral shifts on l 2(Z), they are not d-weakly mixing! Proof: Let f g l 2(Z) l 2(Z) be d-hypercyclic for B 1 B 1, B 2 B 2. Get n so that (B 1 B 1) n (f g) ( e 0 e 0) < 1 4 (B 2 B 2) n (f g) (e 0 e 0) < 1 4. (3) Then there exist ɛ j < 1 (j = 1,..., 4) so that 4 n f, e n w (1) j = B1 n f, e 0 = 1 + ɛ 1 g, e n f, e n g, e n j=1 n j=1 n j=1 n j=1 w (1) j = B n 1 g, e 0 = 1 + ɛ 2 w (2) j = B n 2 f, e 0 = 1 + ɛ 3 w (2) j = B n 2 g, e 0 = 1 + ɛ 4, a contradiction. (4)

38 d-hypercyclicity for shift operators Fact: If B 1, B 2 are bilateral shifts on l 2(Z), they are not d-weakly mixing! Proof: Let f g l 2(Z) l 2(Z) be d-hypercyclic for B 1 B 1, B 2 B 2. Get n so that (B 1 B 1) n (f g) ( e 0 e 0) < 1 4 (B 2 B 2) n (f g) (e 0 e 0) < 1 4. (3) Then there exist ɛ j < 1 (j = 1,..., 4) so that 4 n f, e n w (1) j = B1 n f, e 0 = 1 + ɛ 1 g, e n f, e n g, e n j=1 n j=1 n j=1 n j=1 w (1) j = B n 1 g, e 0 = 1 + ɛ 2 w (2) j = B n 2 f, e 0 = 1 + ɛ 3 w (2) j = B n 2 g, e 0 = 1 + ɛ 4, a contradiction. (4)

39 d-hypercyclicity for shift operators Fact: If B 1, B 2 are bilateral shifts on l 2(Z), they are not d-weakly mixing! Proof: Let f g l 2(Z) l 2(Z) be d-hypercyclic for B 1 B 1, B 2 B 2. Get n so that (B 1 B 1) n (f g) ( e 0 e 0) < 1 4 (B 2 B 2) n (f g) (e 0 e 0) < 1 4. (3) Then there exist ɛ j < 1 (j = 1,..., 4) so that 4 n f, e n w (1) j = B1 n f, e 0 = 1 + ɛ 1 g, e n f, e n g, e n j=1 n j=1 n j=1 n j=1 w (1) j = B n 1 g, e 0 = 1 + ɛ 2 w (2) j = B n 2 f, e 0 = 1 + ɛ 3 w (2) j = B n 2 g, e 0 = 1 + ɛ 4, a contradiction. (4)

40 d-hypercyclicity for shift operators Bès, M., Sanders d-weakly-mixing d-blow-up/collapse. For N = 2, 3,..., there exist bilateral weighted backward shifts B 1,..., B N satisfying the d-blow-up/collapse property. In consequence, hereditarily d-topologically transitive d-hypercyclicity Criterion d-weakly mixing d-blow-up/collapse Problem 1: T 1, T 2 d-weakly mixing? T 1, T 2 satisfy the d-hyp. Criterion?

41 Problem 2: T 1, T 2 d-hypercyclic? T 1, T 2 d-topologically transitive? Equivalently, must d-hc(t 1, T 2) be either empty or dense in X? True for weighted shifts and their powers, linear fractional composition operators, certain differentiation operators, operators satisfying the d-blow-up/collapse property.... Problem 3: T 1, T 2 d-hypercyclic? T 1, T 2 support a d-hypercyclic manifold? True whenever T 1, T 2 satisfy the d-hypercyclicity Criterion (Bès, Peris 07). True in some other cases (Salas + ) What if T 1, T 2 are d-topologically transitive? Problem 4: T T 2 hypercyclic on X X? T, T 2 are d-hypercyclic? True if T = B w unilateral/bilateral, or if T = C ϕ with ϕ LFT (D). T T 2 mixing on X X T, T 2 d-mixing. (Shkarin 12)

42 Existence of d-hypercyciic operators and c 0 -semigroups Let X be a separable, infinite dimensional Fréchet space. Existence of hypercyclic operators and c 0-semigroups 1 X supports a mixing operator with a hypercyclic subspace. (Grivaux 05, Bonet, Peris 98, León, Montes 01, Bernal 99, Ansari 97) 2 If X is Banach with separable dual X, it supports a dual hypercyclic operator T (Salas 07). 3 If X ω supports a mixing c 0-semigroup (Bermúdez, Bonilla, Conejero, Peris 05). Existence of d-hypercyclic operators and c 0-semigroups 1 For each k 2, T 1,..., T k d-mixing on X, commuting, with a d-hypercyclic subspace (Salas 11, Shkarin 10, Bès, M., Peris, Shkarin 12, Bès, M., Sanders +). 2 If X Banach with X separable and k 2, T 1,..., T k dual-hypercyclic and commuting. (Salas 11, Bès, M., Peris, Shkarin 12) 3 If X ω and k 2, {T (1) t } t 0,..., {T (k) t } t 0 d-mixing and commuting c 0-semigroups. (Bès, M., Peris, Shkarin 12)