Introduction Dynamics of Hypercyclic Operators Results Dynamics of Disjoint Hypercyclic Operators: Hypercyclicity vs. Disjoint Hypercyclicity Rebecca Sanders Department of Mathematics, Statistics, and Computer Sci. Marquette University April 11-13, 2014 Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Throughout this talk, let • X = separable, ∞ -dimensional Banach space • B ( X ) = algebra of operators T : X − → X Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Throughout this talk, let • X = separable, ∞ -dimensional Banach space • B ( X ) = algebra of operators T : X − → X Definition. An operator T ∈ B ( X ) is hypercyclic if there is a vector x ∈ X for which its orbit Orb ( T, x ) = { T n x : n ≥ 0 } is dense in X . Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Throughout this talk, let • X = separable, ∞ -dimensional Banach space • B ( X ) = algebra of operators T : X − → X Definition. An operator T ∈ B ( X ) is hypercyclic if there is a vector x ∈ X for which its orbit Orb ( T, x ) = { T n x : n ≥ 0 } is dense in X . HC ( T ) = set of hypercyclic vectors for T Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Definition. Operators T 1 , T 2 , . . . , T N ∈ B ( X ) with N ≥ 2 are disjoint hypercyclic or d-hypercyclic if there is a vector x ∈ X for which ( x, x, . . . , x ) ∈ HC ( T 1 ⊕ T 2 ⊕ · · · ⊕ T N ) Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Definition. Operators T 1 , T 2 , . . . , T N ∈ B ( X ) with N ≥ 2 are disjoint hypercyclic or d-hypercyclic if there is a vector x ∈ X for which ( x, x, . . . , x ) ∈ HC ( T 1 ⊕ T 2 ⊕ · · · ⊕ T N ) d- HC ( T 1 , T 2 , . . . , T N ) = set d-hypercyclic vectors for T 1 , . . . , T N Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Definition. Operators T 1 , T 2 , . . . , T N ∈ B ( X ) with N ≥ 2 are disjoint hypercyclic or d-hypercyclic if there is a vector x ∈ X for which ( x, x, . . . , x ) ∈ HC ( T 1 ⊕ T 2 ⊕ · · · ⊕ T N ) d- HC ( T 1 , T 2 , . . . , T N ) = set d-hypercyclic vectors for T 1 , . . . , T N Definition. The operators T 1 , T 2 , . . . , T N are densely d-hypercyclic if the set d- HC ( T 1 , T 2 , . . . , T N ) is dense in X . Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Goal Show several of the standard dynamical properties of hypercyclic operators fail to hold true for disjoint hypercyclic operators. Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Goal Show several of the standard dynamical properties of hypercyclic operators fail to hold true for disjoint hypercyclic operators. Remark. Restrict our attention for d-hypercyclicity to two operators T 1 , T 2 . Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Dynamics of Hypercyclic Operators Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Dynamics of Hypercyclic Operators Fact 1. The following statements are equivalent: (1) T is hypercyclic. (2) HC ( T ) is a dense G δ set. (3) T is topologically transitive; that is, for any nonempty, open sets U, V , there is n ≥ 1 such that U ∩ T − n ( V ) � = ∅ . Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Fact 2. The following statements are equivalent: (1) T satisfies the Hypercyclicity Criterion. (2) T is weakly mixing; that is, T ⊕ T is topologically transitive. (3) � M i =1 T is topologically transitive for each M ≥ 1 . Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Fact 2. The following statements are equivalent: (1) T satisfies the Hypercyclicity Criterion. (2) T is weakly mixing; that is, T ⊕ T is topologically transitive. (3) � M i =1 T is topologically transitive for each M ≥ 1 . (4) T satisfies the Blow-up/Collapse Property; that is, for any nonempty, open sets U, V, W with 0 ∈ W , there is n ≥ 1 such that U ∩ T − n ( W ) � = ∅ and W ∩ T − n ( V ) � = ∅ . Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Fact 2. The following statements are equivalent: (1) T satisfies the Hypercyclicity Criterion. (2) T is weakly mixing; that is, T ⊕ T is topologically transitive. (3) � M i =1 T is topologically transitive for each M ≥ 1 . (4) T satisfies the Blow-up/Collapse Property; that is, for any nonempty, open sets U, V, W with 0 ∈ W , there is n ≥ 1 such that U ∩ T − n ( W ) � = ∅ and W ∩ T − n ( V ) � = ∅ . Remark. If � M i =1 T with M ≥ 2 is hypercyclic, then � M +1 i =1 T is also hypercyclic. Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Defintion. Operators T 1 , T 2 ∈ B ( X ) are d-topologically transitive if for any nonempty, open sets U, V 1 , V 2 , there is n ≥ 1 such that U ∩ T − n ( V 1 ) ∩ T − n ( V 2 ) � = ∅ . 1 2 Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Defintion. Operators T 1 , T 2 ∈ B ( X ) are d-topologically transitive if for any nonempty, open sets U, V 1 , V 2 , there is n ≥ 1 such that U ∩ T − n ( V 1 ) ∩ T − n ( V 2 ) � = ∅ . 1 2 Theorem A. (B` es, Peris) The following statements are equivalent: (1) T 1 , T 2 are densely d-hypercyclic. (2) d- HC ( T 1 , T 2 ) is a dense G δ set. (3) T 1 , T 2 are d-topologically transitive. Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Questions 1. Does d-hypercyclicity imply dense d-hypercyclicity? Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Definition. Operators T 1 , T 2 ∈ B ( X ) are d-weakly mixing if T 1 ⊕ T 1 , T 2 ⊕ T 2 are d-topologically transitive. Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Definition. Operators T 1 , T 2 ∈ B ( X ) are d-weakly mixing if T 1 ⊕ T 1 , T 2 ⊕ T 2 are d-topologically transitive. Disjoint Hypercyclicity Criterion. Operators T 1 , T 2 ∈ B ( X ) satisfy the d-Hypercyclicity Criterion if there exist sequence ( n k ) ∞ k =1 , dense sets X 0 , X 1 , X 2 of X , and maps S m,k : X m − → X ( m = 1 , 2 ) such that T n k m − → 0 pointwise on X 0 for m = 1 , 2 , S m,k − → 0 pointwise on X m for m = 1 , 2 , T n k m S i,k − → δ i,m I pointwise on X i for i, m = 1 , 2 . Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Disjoint Blow-up/Collapse Property. Operators T 1 , T 2 ∈ B ( X ) satisfy the Disjoint Blow-up/Collapse Property if for any nonempty, open sets U, V 1 , V 2 , W with 0 ∈ W , there is n ≥ 1 for which U ∩ T − n ( W ) ∩ T − n ( W ) � = ∅ and 1 2 W ∩ T − n ( V 1 ) ∩ T − n ( V 2 ) � = ∅ . 1 2 Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Theorem B. (B` es, Peris) • The two statements are equivalent: (1) T 1 , T 2 satisfy d-Hypercyclicity Criterion (2) � M i =1 T 1 , � M i =1 T 2 are d-topologically transitive for each M ≥ 1 . Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Theorem B. (B` es, Peris) • The two statements are equivalent: (1) T 1 , T 2 satisfy d-Hypercyclicity Criterion (2) � M i =1 T 1 , � M i =1 T 2 are d-topologically transitive for each M ≥ 1 . • If T 1 , T 2 satisfy the Disjoint Blow-up/Collapse Property, then T 1 , T 2 are d-topologically transitive and so densely d-hypercyclic. Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Theorem B. (B` es, Peris) • The two statements are equivalent: (1) T 1 , T 2 satisfy d-Hypercyclicity Criterion (2) � M i =1 T 1 , � M i =1 T 2 are d-topologically transitive for each M ≥ 1 . • If T 1 , T 2 satisfy the Disjoint Blow-up/Collapse Property, then T 1 , T 2 are d-topologically transitive and so densely d-hypercyclic. • If T 1 , T 2 are d-weakly mixing, then T 1 , T 2 satisfy the Disjoint Blow-up/Collapse Property. Hence, d-Hypercyclicity Criterion implies the Disjoint Blow-up/Collapse Property. Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Questions 1. Does d-hypercyclicity imply dense d-hypercyclicity? Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Questions 1. Does d-hypercyclicity imply dense d-hypercyclicity? 2. Can we add “ T 1 , T 2 are d-weakly mixing” to the first statement in Theorem B as in Fact 2? Sanders Disjoint Hypercyclic Operators
Introduction Dynamics of Hypercyclic Operators Results Questions 1. Does d-hypercyclicity imply dense d-hypercyclicity? 2. Can we add “ T 1 , T 2 are d-weakly mixing” to the first statement in Theorem B as in Fact 2? 3. Are the d-Hypercyclicity Criterion and the Disjoint Blow-up/Collapse Property equivalent as in Fact 2? Sanders Disjoint Hypercyclic Operators
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