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Dynamics of weighted composition operators on function spaces defined by local properties Thomas Kalmes Faculty of Mathematics Chemnitz Technical University Pawe Domaski Memorial Conference Bdlewo July 1 - 7, 2018 T. Kalmes (TU

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Dynamics of weighted composition operators on function spaces defined by local properties

Thomas Kalmes Faculty of Mathematics Chemnitz Technical University Paweł Domański Memorial Conference Będlewo July 1 - 7, 2018

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 1 / 16

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Continuous linear operator T on a lcs E is called (topologically) transitive :⇔ ∀ U, V ⊆ E open, non-empty ∃ m ∈ N0 : T m(U) ∩ V = ∅ (E separable Fréchet, equivalent to T hypercyclic, i.e. there is x ∈ E s.th. {T mx; m ∈ N0} is dense in E.)

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 2 / 16

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Continuous linear operator T on a lcs E is called (topologically) transitive :⇔ ∀ U, V ⊆ E open, non-empty ∃ m ∈ N0 : T m(U) ∩ V = ∅ weakly mixing :⇔ T ⊕ T transitive on E ⊕ E, i.e. ∀ Uj, Vj ⊆ E open, non-empty ∃ m ∈ N0 : T m(Uj) ∩ Vj = ∅ (j = 1, 2)

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 2 / 16

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Continuous linear operator T on a lcs E is called (topologically) transitive :⇔ ∀ U, V ⊆ E open, non-empty ∃ m ∈ N0 : T m(U) ∩ V = ∅ weakly mixing :⇔ T ⊕ T transitive on E ⊕ E, i.e. ∀ Uj, Vj ⊆ E open, non-empty ∃ m ∈ N0 : T m(Uj) ∩ Vj = ∅ (j = 1, 2) power bounded :⇔ {T m; m ∈ N0} is equicontinuous, i.e. ∀ p ∈ cs(E) ∃ q ∈ cs(E) ∀ m ∈ N0, x ∈ E : p(T mx) ≤ q(x) Albanese, Bonet, Ricker ’09: E Fréchet-Montel, T power bounded ⇒ T uniformly mean ergodic, i.e. ∀ x ∈ E ∃ limn→∞ 1

n

n−1

m=0 T mx

and convergence is uniform on bounded subsets of E.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 2 / 16

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Continuous linear operator T on a lcs E is called (topologically) transitive :⇔ ∀ U, V ⊆ E open, non-empty ∃ m ∈ N0 : T m(U) ∩ V = ∅ weakly mixing :⇔ T ⊕ T transitive on E ⊕ E, i.e. ∀ Uj, Vj ⊆ E open, non-empty ∃ m ∈ N0 : T m(Uj) ∩ Vj = ∅ (j = 1, 2) power bounded :⇔ {T m; m ∈ N0} is equicontinuous, i.e. ∀ p ∈ cs(E) ∃ q ∈ cs(E) ∀ m ∈ N0, x ∈ E : p(T mx) ≤ q(x) Albanese, Bonet, Ricker ’09: E Fréchet-Montel, T power bounded ⇒ T uniformly mean ergodic, i.e. ∀ x ∈ E ∃ limn→∞ 1

n

n−1

m=0 T mx

and convergence is uniform on bounded subsets of E. Folklore (see e.g. Yoshida, 1980): E sequentially complete lcs, T continuous, linear, power bounded ⇒ (exp(sT))s≥0 C0-semigroup, where exp(sT)x = ∞

m=0 sm m! T mx

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 2 / 16

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Continuous linear operator T on a lcs E is called (topologically) transitive :⇔ ∀ U, V ⊆ E open, non-empty ∃ m ∈ N0 : T m(U) ∩ V = ∅ weakly mixing :⇔ T ⊕ T transitive on E ⊕ E, i.e. ∀ Uj, Vj ⊆ E open, non-empty ∃ m ∈ N0 : T m(Uj) ∩ Vj = ∅ (j = 1, 2) power bounded :⇔ {T m; m ∈ N0} is equicontinuous, i.e. ∀ p ∈ cs(E) ∃ q ∈ cs(E) ∀ m ∈ N0, x ∈ E : p(T mx) ≤ q(x) Several authors investigated these properties for weighted composition

  • perators Cw,ψ(f) = w · (f ◦ ψ) on various function spaces, e.g.

Große-Erdmann, Mortini ’09; Zając ’16; Bonet, Domański ’12; Przestacki ’17; Beltrán-Meneu, Gómez-Callado, Jordá, Jornet ’16;...

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 2 / 16

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Continuous linear operator T on a lcs E is called (topologically) transitive :⇔ ∀ U, V ⊆ E open, non-empty ∃ m ∈ N0 : T m(U) ∩ V = ∅ weakly mixing :⇔ T ⊕ T transitive on E ⊕ E, i.e. ∀ Uj, Vj ⊆ E open, non-empty ∃ m ∈ N0 : T m(Uj) ∩ Vj = ∅ (j = 1, 2) power bounded :⇔ {T m; m ∈ N0} is equicontinuous, i.e. ∀ p ∈ cs(E) ∃ q ∈ cs(E) ∀ m ∈ N0, x ∈ E : p(T mx) ≤ q(x) Several authors investigated these properties for weighted composition

  • perators Cw,ψ(f) = w · (f ◦ ψ) on various function spaces, e.g.

Große-Erdmann, Mortini ’09; Zając ’16; Bonet, Domański ’12; Przestacki ’17; Beltrán-Meneu, Gómez-Callado, Jordá, Jornet ’16;... Objective: study these dynamical properties for weighted composition

  • perators Cw,ψ(f) = w · (f ◦ ψ) on function spaces "in a general framework".
  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 2 / 16

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Ω locally compact, σ-compact, non-compact Hausdorff space, F a sheaf of K-valued functions on Ω, i.e.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 3 / 16

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Ω locally compact, σ-compact, non-compact Hausdorff space, F a sheaf of K-valued functions on Ω, i.e. ∀ X ⊆ Ω open: F(X) is a K-vector space of K-valued functions on X s.th. ∀ Y ⊆ X ⊆ Ω open: rY

X : F(X) → F(Y ), f → f|Y well-defined

(Gluing) ∀ open cover (Xι)ι∈I of an open set X ⊆ Ω ∀ (fι)ι∈I ∈

ι∈I F(Xι) with fι|Xι∩Xκ = fκ|Xι∩Xκ (ι, κ ∈ I) there is

f ∈ F(X) with f|Xι = fι (ι ∈ I).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 3 / 16

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Ω locally compact, σ-compact, non-compact Hausdorff space, F a sheaf of K-valued functions on Ω, i.e. ∀ X ⊆ Ω open: F(X) is a K-vector space of K-valued functions on X s.th. ∀ Y ⊆ X ⊆ Ω open: rY

X : F(X) → F(Y ), f → f|Y well-defined

(Gluing) ∀ open cover (Xι)ι∈I of an open set X ⊆ Ω ∀ (fι)ι∈I ∈

ι∈I F(Xι) with fι|Xι∩Xκ = fκ|Xι∩Xκ (ι, κ ∈ I) there is

f ∈ F(X) with f|Xι = fι (ι ∈ I). ⇒ ∀ X ⊆ Ω open ∀ (Xn)n∈N0 open, relatively compact exhaustion of X: F(X) ∼ = proj(F(Xn+1), rXn

Xn+1)n∈N0

= {(fn)n∈N0 ∈

  • n

F(Xn); ∀ n ∈ N : fn|Xn−1 = fn−1} via f → (f|Xn)n∈N0

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 3 / 16

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Ω locally compact, σ-compact, non-compact Hausdorff space, F a sheaf of K-valued functions on Ω, i.e. ∀ X ⊆ Ω open: F(X) is a K-vector space of K-valued functions on X s.th. ∀ Y ⊆ X ⊆ Ω open: rY

X : F(X) → F(Y ), f → f|Y well-defined

(Gluing) ∀ open cover (Xι)ι∈I of an open set X ⊆ Ω ∀ (fι)ι∈I ∈

ι∈I F(Xι) with fι|Xι∩Xκ = fκ|Xι∩Xκ (ι, κ ∈ I) there is

f ∈ F(X) with f|Xι = fι (ι ∈ I). ⇒ ∀ X ⊆ Ω open ∀ (Xn)n∈N0 open, relatively compact exhaustion of X: F(X) ∼ = proj(F(Xn+1), rXn

Xn+1)n∈N0

= {(fn)n∈N0 ∈

  • n

F(Xn); ∀ n ∈ N : fn|Xn−1 = fn−1} via f → (f|Xn)n∈N0 Examples: Ω = Rd, F(X) = C∞(X), F(X) = C(X), F(X) = A (X), or for Ω = C, F(X) = H (X). Lp(X) is not a sheaf.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 3 / 16

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We define the following properties for a sheaf of functions F on Ω: (F1) ∀ X ⊆ Ω: F(X) is a webbed, ultrabornological Hausdorff lcs, F(X) ⊆ C(X) with ∀ x ∈ X : δx ∈ F(X)′,

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 4 / 16

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We define the following properties for a sheaf of functions F on Ω: (F1) ∀ X ⊆ Ω: F(X) is a webbed, ultrabornological Hausdorff lcs, F(X) ⊆ C(X) with ∀ x ∈ X : δx ∈ F(X)′, and ∀ open, rel. comp. exh. (Xn)n∈N0 of X: F(X) ∼ = proj(F(Xn+1), rXn

Xn+1)n∈N0 topologically

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 4 / 16

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We define the following properties for a sheaf of functions F on Ω: (F1) ∀ X ⊆ Ω: F(X) is a webbed, ultrabornological Hausdorff lcs, F(X) ⊆ C(X) with ∀ x ∈ X : δx ∈ F(X)′, and ∀ open, rel. comp. exh. (Xn)n∈N0 of X: F(X) ∼ = proj(F(Xn+1), rXn

Xn+1)n∈N0 topologically

(F2) ∀ K ⋐ Ω ∃ fK ∈ F(Ω) ∀ x ∈ K : fK(x) = 0

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 4 / 16

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We define the following properties for a sheaf of functions F on Ω: (F1) ∀ X ⊆ Ω: F(X) is a webbed, ultrabornological Hausdorff lcs, F(X) ⊆ C(X) with ∀ x ∈ X : δx ∈ F(X)′, and ∀ open, rel. comp. exh. (Xn)n∈N0 of X: F(X) ∼ = proj(F(Xn+1), rXn

Xn+1)n∈N0 topologically

(F2) ∀ K ⋐ Ω ∃ fK ∈ F(Ω) ∀ x ∈ K : fK(x) = 0 (F3) ∀ x, y ∈ Ω, x = y ∃ f ∈ F(Ω) : f(x) = 0, f(y) = 1

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 4 / 16

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We define the following properties for a sheaf of functions F on Ω: (F1) ∀ X ⊆ Ω: F(X) is a webbed, ultrabornological Hausdorff lcs, F(X) ⊆ C(X) with ∀ x ∈ X : δx ∈ F(X)′, and ∀ open, rel. comp. exh. (Xn)n∈N0 of X: F(X) ∼ = proj(F(Xn+1), rXn

Xn+1)n∈N0 topologically

(F2) ∀ K ⋐ Ω ∃ fK ∈ F(Ω) ∀ x ∈ K : fK(x) = 0 (F3) ∀ x, y ∈ Ω, x = y ∃ f ∈ F(Ω) : f(x) = 0, f(y) = 1 Examples: F = Cr (r ∈ N0 ∪ {∞}) on Ω = Rd satisfies (F1) − (F3) when equipped with the seminorms fl,K := max|α|≤l,x∈K |∂αf(x)| (l < r + 1, K ⋐ X).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 4 / 16

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We define the following properties for a sheaf of functions F on Ω: (F1) ∀ X ⊆ Ω: F(X) is a webbed, ultrabornological Hausdorff lcs, F(X) ⊆ C(X) with ∀ x ∈ X : δx ∈ F(X)′, and ∀ open, rel. comp. exh. (Xn)n∈N0 of X: F(X) ∼ = proj(F(Xn+1), rXn

Xn+1)n∈N0 topologically

(F2) ∀ K ⋐ Ω ∃ fK ∈ F(Ω) ∀ x ∈ K : fK(x) = 0 (F3) ∀ x, y ∈ Ω, x = y ∃ f ∈ F(Ω) : f(x) = 0, f(y) = 1 Examples: F = Cr (r ∈ N0 ∪ {∞}) on Ω = Rd satisfies (F1) − (F3) when equipped with the seminorms fl,K := max|α|≤l,x∈K |∂αf(x)| (l < r + 1, K ⋐ X). F = H on Ω = Cd equipped with compact-open top. satisfies (F1) − (F3).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 4 / 16

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We define the following properties for a sheaf of functions F on Ω: (F1) ∀ X ⊆ Ω: F(X) is a webbed, ultrabornological Hausdorff lcs, F(X) ⊆ C(X) with ∀ x ∈ X : δx ∈ F(X)′, and ∀ open, rel. comp. exh. (Xn)n∈N0 of X: F(X) ∼ = proj(F(Xn+1), rXn

Xn+1)n∈N0 topologically

(F2) ∀ K ⋐ Ω ∃ fK ∈ F(Ω) ∀ x ∈ K : fK(x) = 0 (F3) ∀ x, y ∈ Ω, x = y ∃ f ∈ F(Ω) : f(x) = 0, f(y) = 1 Examples: F = Cr (r ∈ N0 ∪ {∞}) on Ω = Rd satisfies (F1) − (F3) when equipped with the seminorms fl,K := max|α|≤l,x∈K |∂αf(x)| (l < r + 1, K ⋐ X). F = H on Ω = Cd equipped with compact-open top. satisfies (F1) − (F3). F = A on Ω = Rd with the canonical locally convex top. satisfies (F1) − (F3).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 4 / 16

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We define the following properties for a sheaf of functions F on Ω: (F1) ∀ X ⊆ Ω: F(X) is a webbed, ultrabornological Hausdorff lcs, F(X) ⊆ C(X) with ∀ x ∈ X : δx ∈ F(X)′, and ∀ open, rel. comp. exh. (Xn)n∈N0 of X: F(X) ∼ = proj(F(Xn+1), rXn

Xn+1)n∈N0 topologically

(F2) ∀ K ⋐ Ω ∃ fK ∈ F(Ω) ∀ x ∈ K : fK(x) = 0 (F3) ∀ x, y ∈ Ω, x = y ∃ f ∈ F(Ω) : f(x) = 0, f(y) = 1 Examples: F = Cr (r ∈ N0 ∪ {∞}) on Ω = Rd satisfies (F1) − (F3) when equipped with the seminorms fl,K := max|α|≤l,x∈K |∂αf(x)| (l < r + 1, K ⋐ X). F = H on Ω = Cd equipped with compact-open top. satisfies (F1) − (F3). F = A on Ω = Rd with the canonical locally convex top. satisfies (F1) − (F3). F ∈ {E{ω}, E(ω)} on Ω = Rd with the canonical locally convex top. satisfies (F1) − (F3).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 4 / 16

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Additionally, in case of (F1), Ω = Rd and F(X) ⊆ C1(X) we define (F4) ∀ X ⊆ Rd open, 1 ≤ j ≤ d, x ∈ X : f → ∂jf(x) ∈ F(X)′ and for each h ∈ Rd\{0}, λ ∈ K: kern

  • f → λf(x) −

d

  • j=1

hj∂jf(x)

  • = F(X)
  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 5 / 16

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Additionally, in case of (F1), Ω = Rd and F(X) ⊆ C1(X) we define (F4) ∀ X ⊆ Rd open, 1 ≤ j ≤ d, x ∈ X : f → ∂jf(x) ∈ F(X)′ and for each h ∈ Rd\{0}, λ ∈ K: kern

  • f → λf(x) −

d

  • j=1

hj∂jf(x)

  • = F(X)

Let Ω and F with (F1) be given, fix X ⊆ Ω open and let w ∈ C(X) and ψ : X → X be continuous such that Cw,ψ : F(X) → F(X), f →

  • x → w(x)f(ψ(x))
  • is well-defined ((F1) ⇒ Cw,ψ continuous).
  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 5 / 16

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Additionally, in case of (F1), Ω = Rd and F(X) ⊆ C1(X) we define (F4) ∀ X ⊆ Rd open, 1 ≤ j ≤ d, x ∈ X : f → ∂jf(x) ∈ F(X)′ and for each h ∈ Rd\{0}, λ ∈ K: kern

  • f → λf(x) −

d

  • j=1

hj∂jf(x)

  • = F(X)

Let Ω and F with (F1) be given, fix X ⊆ Ω open and let w ∈ C(X) and ψ : X → X be continuous such that Cw,ψ : F(X) → F(X), f →

  • x → w(x)f(ψ(x))
  • is well-defined ((F1) ⇒ Cw,ψ continuous).

Cw,ψ acts locally on F(X) :⇔ ∀ Y ⊆ ψ(X) open : Cw,ψ,Y : F(Y ) → F(ψ−1(Y )), f →

  • x → w(x)f(ψ(x))
  • is well-defined ((F1) ⇒ Cw,ψ,Y is continuous)
  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 5 / 16

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Additionally, in case of (F1), Ω = Rd and F(X) ⊆ C1(X) we define (F4) ∀ X ⊆ Rd open, 1 ≤ j ≤ d, x ∈ X : f → ∂jf(x) ∈ F(X)′ and for each h ∈ Rd\{0}, λ ∈ K: kern

  • f → λf(x) −

d

  • j=1

hj∂jf(x)

  • = F(X)

Let Ω and F with (F1) be given, fix X ⊆ Ω open and let w ∈ C(X) and ψ : X → X be continuous such that Cw,ψ : F(X) → F(X), f →

  • x → w(x)f(ψ(x))
  • is well-defined ((F1) ⇒ Cw,ψ continuous).

Cw,ψ acts locally on F(X) :⇔ ∀ Y ⊆ ψ(X) open : Cw,ψ,Y : F(Y ) → F(ψ−1(Y )), f →

  • x → w(x)f(ψ(x))
  • is well-defined ((F1) ⇒ Cw,ψ,Y is continuous)

ψ run-away :⇔ ∀ K ⋐ X∃ m ∈ N : ψm(K) ∩ K = ∅

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 5 / 16

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Additionally, in case of (F1), Ω = Rd and F(X) ⊆ C1(X) we define (F4) ∀ X ⊆ Rd open, 1 ≤ j ≤ d, x ∈ X : f → ∂jf(x) ∈ F(X)′ and for each h ∈ Rd\{0}, λ ∈ K: kern

  • f → λf(x) −

d

  • j=1

hj∂jf(x)

  • = F(X)

Let Ω and F with (F1) be given, fix X ⊆ Ω open and let w ∈ C(X) and ψ : X → X be continuous such that Cw,ψ : F(X) → F(X), f →

  • x → w(x)f(ψ(x))
  • is well-defined ((F1) ⇒ Cw,ψ continuous).

Cw,ψ acts locally on F(X) :⇔ ∀ Y ⊆ ψ(X) open : Cw,ψ,Y : F(Y ) → F(ψ−1(Y )), f →

  • x → w(x)f(ψ(x))
  • is well-defined ((F1) ⇒ Cw,ψ,Y is continuous)

ψ run-away :⇔ ∀ K ⋐ X∃ m ∈ N : ψm(K) ∩ K = ∅ (Xn)n∈N0 open, rel. comp. exh. of X

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 5 / 16

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An almost characterisation of weak mixing

Assume F satisfies (F1)−(F3). If Cw,ψ acts locally on F(X), i) ⇒ ii) ⇒ iv): i) a) ∀ m ∈ N0, Y ⊆ ψm(X) open, rel. comp. r(ψm)−1(Y )

X

(F(X)) ⊆

  • Cw,ψ,(ψ(m−1))−1(Y ) ◦ . . . ◦ Cw,ψ,Y
  • (F(Y ))

F

  • (ψm)−1(Y )
  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 6 / 16

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An almost characterisation of weak mixing

Assume F satisfies (F1)−(F3). If Cw,ψ acts locally on F(X), i) ⇒ ii) ⇒ iv): i) a) ∀ m ∈ N0, Y ⊆ ψm(X) open, rel. comp. r(ψm)−1(Y )

X

(F(X)) ⊆

  • Cw,ψ,(ψ(m−1))−1(Y ) ◦ . . . ◦ Cw,ψ,Y
  • (F(Y ))

F

  • (ψm)−1(Y )
  • b) ∃ (Xn)n∈N0 ∀ n ∈ N0 ∃ m ∈ N :

b1) Xn ∩ ψm(Xn) = ∅, b2) ψm(Xn) is open, b3) rXn∪ψm(Xn)

X

has dense range.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 6 / 16

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An almost characterisation of weak mixing

Assume F satisfies (F1)−(F3). If Cw,ψ acts locally on F(X), i) ⇒ ii) ⇒ iv): i) a) ∀ m ∈ N0, Y ⊆ ψm(X) open, rel. comp. r(ψm)−1(Y )

X

(F(X)) ⊆

  • Cw,ψ,(ψ(m−1))−1(Y ) ◦ . . . ◦ Cw,ψ,Y
  • (F(Y ))

F

  • (ψm)−1(Y )
  • b) ∃ (Xn)n∈N0 ∀ n ∈ N0 ∃ m ∈ N :

b1) Xn ∩ ψm(Xn) = ∅, b2) ψm(Xn) is open, b3) rXn∪ψm(Xn)

X

has dense range.

ii) Cw,ψ is weakly mixing on F(X). iv) a) from i) holds, w has no zeros, ψ is injective and run-away [and in case of (F4) and ψ C1, additionally detJψ(x) = 0, x ∈ X].

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 6 / 16

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An almost characterisation of weak mixing/transitivity

Assume F satisfies (F1)−(F3). If Cw,ψ acts locally on F(X), i) ⇒ ii) ⇒ iv): i) a) ∀ m ∈ N0, Y ⊆ ψm(X) open, rel. comp. r(ψm)−1(Y )

X

(F(X)) ⊆

  • Cw,ψ,(ψ(m−1))−1(Y ) ◦ . . . ◦ Cw,ψ,Y
  • (F(Y ))

F

  • (ψm)−1(Y )
  • b) ∃ (Xn)n∈N0 ∀ n ∈ N0 ∃ m ∈ N :

b1) Xn ∩ ψm(Xn) = ∅, b2) ψm(Xn) is open, b3) rXn∪ψm(Xn)

X

has dense range.

ii) Cw,ψ is weakly mixing on F(X). iii) Cw,ψ is transitive on F(X). iv) a) from i) holds, w has no zeros, ψ is injective and run-away [and in case of (F4) and ψ C1, additionally detJψ(x) = 0, x ∈ X]. Additionally, if F(Ω) is dense in C(Ω) or if |w(x)| ≤ 1, x ∈ X, then it holds i) ⇒ ii) ⇒ iii) ⇒ iv).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 6 / 16

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SLIDE 29

Some thoughts on the condition a) ∀ m ∈ N0, Y ⊆ ψm(X) open, rel. comp. r(ψm)−1(Y )

X

(F(X)) ⊆

  • Cw,ψ,(ψ(m−1))−1(Y ) ◦ . . . ◦ Cw,ψ,Y
  • (F(Y ))

F((ψm)−1(Y ))

for injective, open ψ (with detJψ(x) = 0) and w without zeros:

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 7 / 16

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SLIDE 30

Some thoughts on the condition a) ∀ m ∈ N0, Y ⊆ ψm(X) open, rel. comp. r(ψm)−1(Y )

X

(F(X)) ⊆

  • Cw,ψ,(ψ(m−1))−1(Y ) ◦ . . . ◦ Cw,ψ,Y
  • (F(Y ))

F((ψm)−1(Y ))

for injective, open ψ (with detJψ(x) = 0) and w without zeros: For f ∈ F((ψm)−1(Y )), ˜ f : Y → K, y →

  • f

m−1

j=0 w(ψj(·))

  • (ψm)−1(y)

is well-defined and continuous. In case of ˜ f ∈ F(Y ), we have

  • Cw,ψ,(ψ(m−1))−1(Y ) ◦ . . . ◦ Cw,ψ,Y
  • ( ˜

f) = f.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 7 / 16

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SLIDE 31

Some thoughts on the condition a) ∀ m ∈ N0, Y ⊆ ψm(X) open, rel. comp. r(ψm)−1(Y )

X

(F(X)) ⊆

  • Cw,ψ,(ψ(m−1))−1(Y ) ◦ . . . ◦ Cw,ψ,Y
  • (F(Y ))

F((ψm)−1(Y ))

for injective, open ψ (with detJψ(x) = 0) and w without zeros: For f ∈ F((ψm)−1(Y )), ˜ f : Y → K, y →

  • f

m−1

j=0 w(ψj(·))

  • (ψm)−1(y)

is well-defined and continuous. In case of ˜ f ∈ F(Y ), we have

  • Cw,ψ,(ψ(m−1))−1(Y ) ◦ . . . ◦ Cw,ψ,Y
  • ( ˜

f) = f. ˜ f ∈ F(Y ) in many concrete cases for F whenever w has no zeros and ψ is injective (and open); for example: F = continuous (X ⊆ Rd open), smooth, holomorphic, or real analytic functions

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 7 / 16

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SLIDE 32

Some thoughts on the condition a) ∀ m ∈ N0, Y ⊆ ψm(X) open, rel. comp. r(ψm)−1(Y )

X

(F(X)) ⊆

  • Cw,ψ,(ψ(m−1))−1(Y ) ◦ . . . ◦ Cw,ψ,Y
  • (F(Y ))

F((ψm)−1(Y ))

for injective, open ψ (with detJψ(x) = 0) and w without zeros: For f ∈ F((ψm)−1(Y )), ˜ f : Y → K, y →

  • f

m−1

j=0 w(ψj(·))

  • (ψm)−1(y)

is well-defined and continuous. In case of ˜ f ∈ F(Y ), we have

  • Cw,ψ,(ψ(m−1))−1(Y ) ◦ . . . ◦ Cw,ψ,Y
  • ( ˜

f) = f. ˜ f ∈ F(Y ) in many concrete cases for F whenever w has no zeros and ψ is injective (and open); for example: F = continuous (X ⊆ Rd open), smooth, holomorphic, or real analytic functions Moreover: Cw,ψ has dense range ⇒ condition satisfied (without any restrictions

  • n w and ψ)
  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 7 / 16

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SLIDE 33

ψ : X → X has stable orbits :⇔ ∀ K ⋐ X ∃ L ⋐ X ∀ m ∈ N0 : ψm(K) ⊆ L

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 8 / 16

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SLIDE 34

ψ : X → X has stable orbits :⇔ ∀ K ⋐ X ∃ L ⋐ X ∀ m ∈ N0 : ψm(K) ⊆ L

Power boundedness

Assume that F satisfies (F1). Moreover, assume a) ∀ x ∈ X : kern δx = F(X). b) ∃ (Xn)n∈N0 ∀ n, x ∈ X\Xn, W ⊆ X\Xn nbh. of x ∃ U ⊆ W open nbh. x: rXn∪U

X

has dense range. c) ∀ m ∈ N0 : {x ∈ X; w(ψm(x)) = 0} is dense in X. If w ∈ F(X), i) ⇒ ii): i) Cw,ψ is power bounded on F(X). ii) ψ has stable orbits and {Cm

w,ψ(w); m ∈ N0} is bounded in F(X).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 8 / 16

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SLIDE 35

ψ : X → X has stable orbits :⇔ ∀ K ⋐ X ∃ L ⋐ X ∀ m ∈ N0 : ψm(K) ⊆ L

Power boundedness

Assume that F satisfies (F1). Moreover, assume a) ∀ x ∈ X : kern δx = F(X). b) ∃ (Xn)n∈N0 ∀ n, x ∈ X\Xn, W ⊆ X\Xn nbh. of x ∃ U ⊆ W open nbh. x: rXn∪U

X

has dense range. c) ∀ m ∈ N0 : {x ∈ X; w(ψm(x)) = 0} is dense in X. If w ∈ F(X), i) ⇒ ii): i) Cw,ψ is power bounded on F(X). ii) ψ has stable orbits and {Cm

w,ψ(w); m ∈ N0} is bounded in F(X).

In case F is equipped with the compact open topology the above are equivalent.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 8 / 16

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SLIDE 36

ψ : X → X has stable orbits :⇔ ∀ K ⋐ X ∃ L ⋐ X ∀ m ∈ N0 : ψm(K) ⊆ L

Power boundedness

Assume that F satisfies (F1). Moreover, assume a) ∀ x ∈ X : kern δx = F(X). b) ∃ (Xn)n∈N0 ∀ n, x ∈ X\Xn, W ⊆ X\Xn nbh. of x ∃ U ⊆ W open nbh. x: rXn∪U

X

has dense range. c) ∀ m ∈ N0 : {x ∈ X; w(ψm(x)) = 0} is dense in X. If w ∈ F(X), i) ⇒ ii): i) Cw,ψ is power bounded on F(X). ii) ψ has stable orbits and {Cm

w,ψ(w); m ∈ N0} is bounded in F(X).

In case F is equipped with the compact open topology the above are equivalent. c) satisfied if w−1(K\{0}) = X and ∀ x ∃ open nbh. Ux : ψ|Ux injective, open.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 8 / 16

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SLIDE 37

Apply (almost) characterisations of transitivity/ (weak) mixing and power boundedness to kernels of differential operators: Let P ∈ C[X1, . . . , Xd], P(ξ) =

|α|≤m aαξα be non-constant, X ⊆ Rd open

EP (X) := {f ∈ C∞(X); P(∂)f = 0 in X}, where P(∂)f =

|α|≤m aα∂αf.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 9 / 16

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SLIDE 38

Apply (almost) characterisations of transitivity/ (weak) mixing and power boundedness to kernels of differential operators: Let P ∈ C[X1, . . . , Xd], P(ξ) =

|α|≤m aαξα be non-constant, X ⊆ Rd open

EP (X) := {f ∈ C∞(X); P(∂)f = 0 in X}, where P(∂)f =

|α|≤m aα∂αf.

EP (X) is a closed subspace of C∞(X), hence a separable nuclear Fréchet space (thus Montel), and F = EP defines sheaf on Rd satisfying (F1) and (F2).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 9 / 16

slide-39
SLIDE 39

Apply (almost) characterisations of transitivity/ (weak) mixing and power boundedness to kernels of differential operators: Let P ∈ C[X1, . . . , Xd], P(ξ) =

|α|≤m aαξα be non-constant, X ⊆ Rd open

EP (X) := {f ∈ C∞(X); P(∂)f = 0 in X}, where P(∂)f =

|α|≤m aα∂αf.

EP (X) is a closed subspace of C∞(X), hence a separable nuclear Fréchet space (thus Montel), and F = EP defines sheaf on Rd satisfying (F1) and (F2). (F3) need not be satisfied: d = 2, P(ξ1, ξ2) = ξ1 ⇒ EP does not separate points in R2.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 9 / 16

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SLIDE 40

Concrete examples: R2 = C and P(ξ1, ξ2) = 1

2(ξ1 + iξ2) ⇒ P(∂) = ∂ Cauchy-Riemann operator

and EP (X) = H (X) P(ξ1, . . . , ξd) = d

j=1 ξ2 j ⇒ P(∂) = △ Laplace operator,

EP (X) = {f; f harmonic in X} P(ξ0, . . . , ξd) = ξ0 − d

j=1 ξ2 j → P(∂) = ∂ ∂ξ0 − △ heat operator

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 10 / 16

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SLIDE 41

Concrete examples: R2 = C and P(ξ1, ξ2) = 1

2(ξ1 + iξ2) ⇒ P(∂) = ∂ Cauchy-Riemann operator

and EP (X) = H (X) P(ξ1, . . . , ξd) = d

j=1 ξ2 j ⇒ P(∂) = △ Laplace operator,

EP (X) = {f; f harmonic in X} P(ξ0, . . . , ξd) = ξ0 − d

j=1 ξ2 j → P(∂) = ∂ ∂ξ0 − △ heat operator

P(ξ) =

|α|≤m aαξα elliptic :⇔ ∀ ξ ∈ Rd\{0} : 0 = Pm(ξ) := |α|=m aαξα

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 10 / 16

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SLIDE 42

Concrete examples: R2 = C and P(ξ1, ξ2) = 1

2(ξ1 + iξ2) ⇒ P(∂) = ∂ Cauchy-Riemann operator

and EP (X) = H (X) P(ξ1, . . . , ξd) = d

j=1 ξ2 j ⇒ P(∂) = △ Laplace operator,

EP (X) = {f; f harmonic in X} P(ξ0, . . . , ξd) = ξ0 − d

j=1 ξ2 j → P(∂) = ∂ ∂ξ0 − △ heat operator

P(ξ) =

|α|≤m aαξα elliptic :⇔ ∀ ξ ∈ Rd\{0} : 0 = Pm(ξ) := |α|=m aαξα

P hypoelliptic :⇔ ∀ u ∈ D′(X) :

  • P(∂)u = 0 ⇒ u ∈ C∞(X)
  • Then EP (X) = {u ∈ D′(X); P(∂)u = 0}, the relative top. from D′(X) on

EP (X) coincides with the original top. on EP (X), and therefore: EP (X) is a nuclear Fréchet space when equipped with the compact open topology. P elliptic ⇒ P hypoelliptic ⇒ EP satisfies (F1) − (F4)

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 10 / 16

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SLIDE 43

Objective: Apply (almost) characterisation of transitivity/weak mixing and power boundedness of Cw,ψ to EP .

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 11 / 16

slide-44
SLIDE 44

Objective: Apply (almost) characterisation of transitivity/weak mixing and power boundedness of Cw,ψ to EP . To apply both (almost) characterisations we need particular open, rel. comp. exh. (Xn)n∈N0 of open X ⊆ Rd open such that rXn∪B

X

has dense range for all n and suitable B ⊆ X\Xn open.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 11 / 16

slide-45
SLIDE 45

Objective: Apply (almost) characterisation of transitivity/weak mixing and power boundedness of Cw,ψ to EP . To apply both (almost) characterisations we need particular open, rel. comp. exh. (Xn)n∈N0 of open X ⊆ Rd open such that rXn∪B

X

has dense range for all n and suitable B ⊆ X\Xn open. This implies that necessarily P(∂) : C∞(X) → C∞(X) is surjective (⇔: X is P-convex).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 11 / 16

slide-46
SLIDE 46

Objective: Apply (almost) characterisation of transitivity/weak mixing and power boundedness of Cw,ψ to EP . To apply both (almost) characterisations we need particular open, rel. comp. exh. (Xn)n∈N0 of open X ⊆ Rd open such that rXn∪B

X

has dense range for all n and suitable B ⊆ X\Xn open. This implies that necessarily P(∂) : C∞(X) → C∞(X) is surjective (⇔: X is P-convex). True for elliptic operators and arbitrary open X ⊆ Rd but not true for general (hypoelliptic) operators and arbitrary X.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 11 / 16

slide-47
SLIDE 47

Theorem (Power boundedness on EP)

Let X ⊆ Rd be open, w ∈ C∞(X), ψ : X → X smooth s.th. w−1(C\{0}) is dense in X and ψ is locally injective, P hypoelliptic with P(0) = 0. Then in any

  • f the two cases

a) P is elliptic, b) d ≥ 3, {ξ ∈ Rd; Pm(ξ) = 0} is a one-dimensional subspace, X is P-convex, tfae: i) Cw,ψ is power bounded on EP (X). ii) ψ has stable orbits and {Cm

w,ψ(w); m ∈ N} is bounded in C(X).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 12 / 16

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SLIDE 48

Theorem (Power boundedness on EP)

Let X ⊆ Rd be open, w ∈ C∞(X), ψ : X → X smooth s.th. w−1(C\{0}) is dense in X and ψ is locally injective, P hypoelliptic with P(0) = 0. Then in any

  • f the two cases

a) P is elliptic, b) d ≥ 3, {ξ ∈ Rd; Pm(ξ) = 0} is a one-dimensional subspace, X is P-convex, tfae: i) Cw,ψ is power bounded on EP (X). ii) ψ has stable orbits and {Cm

w,ψ(w); m ∈ N} is bounded in C(X).

If w = 1 the following are equivalent to i), ii). iii) Cw,ψ is (uniformly) mean ergodic on EP (X).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 12 / 16

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SLIDE 49

Theorem (Power boundedness on EP)

Let X ⊆ Rd be open, w ∈ C∞(X), ψ : X → X smooth s.th. w−1(C\{0}) is dense in X and ψ is locally injective, P hypoelliptic with P(0) = 0. Then in any

  • f the two cases

a) P is elliptic, b) d ≥ 3, {ξ ∈ Rd; Pm(ξ) = 0} is a one-dimensional subspace, X is P-convex, tfae: i) Cw,ψ is power bounded on EP (X). ii) ψ has stable orbits and {Cm

w,ψ(w); m ∈ N} is bounded in C(X).

If w = 1 the following are equivalent to i), ii). iii) Cw,ψ is (uniformly) mean ergodic on EP (X). b) applicable to non-degenerate parabolic operators like the heat operator

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 12 / 16

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SLIDE 50

Theorem (Weak mixing on EP)

Let X ⊆ Rd be homeomorphic to Rd and let P be elliptic. If Cw,ψ acts locally on EP (X), tfae: i) Cw,ψ is weakly mixing on EP (X). ii) Cw,ψ has dense range, w has no zeros, and ψ is injective and run-away. Moreover, detJψ(x) = 0 for all x ∈ X can be added to ii).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 13 / 16

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SLIDE 51

Theorem (Weak mixing on EP)

Let X ⊆ Rd be homeomorphic to Rd and let P be elliptic. If Cw,ψ acts locally on EP (X), tfae: i) Cw,ψ is weakly mixing on EP (X). ii) Cw,ψ has dense range, w has no zeros, and ψ is injective and run-away. Moreover, detJψ(x) = 0 for all x ∈ X can be added to ii). Additionally, if |w| ≤ 1 the above are equivalent to iii) Cw,ψ is hypercyclic on EP (X).

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 13 / 16

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SLIDE 52

Consider the elliptic operator P(∂) = △ − λ, λ ∈ C.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 14 / 16

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SLIDE 53

Consider the elliptic operator P(∂) = △ − λ, λ ∈ C. Given X ⊆ Rd open and C2-functions w, ψ. Cw,ψ is well-defined on EP (X) if and

  • nly if

∀ 1 ≤ j = k ≤ d : w|∇ψj|2 = w|∇ψk|2, w∇ψj, ∇ψk = 0 ∀ 1 ≤ j ≤ d : w ∆ψj + 2∇w, ∇ψj = 0 ∆w − λw = −λw|∇ψ1|2 (⇒ Cw,ψ acts locally on EP (X))

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 14 / 16

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SLIDE 54

Corollary

Let X ⊆ Rd be open and homeomorphic to Rd, P(∂) = △ − λ, λ ∈ C, w, ψ be C2 such that Cw,ψ is well-defined on EP (X). Tfae i) Cw,ψ is hypercyclic on EP (X). ii) Cw,ψ is weakly mixing on EP (X). iii) w has no zeros and ψ is injective as well as run-away.

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 15 / 16

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SLIDE 55

Corollary

Let X ⊆ Rd be open and homeomorphic to Rd, P(∂) = △ − λ, λ ∈ C, w, ψ be C2 such that Cw,ψ is well-defined on EP (X). Tfae i) Cw,ψ is hypercyclic on EP (X). ii) Cw,ψ is weakly mixing on EP (X). iii) w has no zeros and ψ is injective as well as run-away. Question: When is an arbitrary (injective) ψ satisfying ∀ 1 ≤ j = k ≤ d : w|∇ψj|2 = w|∇ψk|2, w∇ψj, ∇ψk = 0 ∀ 1 ≤ j ≤ d : w △ψj + 2∇w, ∇ψj = 0 △w − λw = λw|∇ψ1|2 for a zero-free w run-away, resp. when does it have stable orbits?

  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 15 / 16

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SLIDE 56

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  • A. Albanese, J. Bonet, W. Ricker, Mean ergodic operators in Fréchet spaces, Ann.
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M.J. Beltrán-Meneu, M.C. Gómez-Callado, E. Jordá, D. Jornet, Mean ergodicity of weighted composition operators on spaces of holomorphic functions, J. Math. Anal.

  • Appl. 444 (2016), no. 2, 1640–1651.
  • J. Bonet, P. Domański, Hypercyclic composition operators on spaces of real

analytic functions, Math. Proc. Camb. Phil. Soc. 153 (2012), no. 3, 489–503. K.G. Große-Erdmann, R. Mortini, Universal functions for composition operators with non-automorphic symbol, J. Math. Anal. 107 (2009), 355–376.

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by local properties, to appear in Stud. Math., arXiv-Preprint 1712.06110, 2017.

  • A. Przestacki, Dynamical properties of weighted composition operators on the

space of smooth functions, J. Math. Anal. Appl. 445 (2017), no. 1, 1097-1113.

  • K. Yosida, Functional Analysis, Springer Verlag, Berlin, 1980.
  • S. Zając, Hypercyclicity of composition operators in Stein manifolds, Proc. Amer.
  • Math. Soc. 144 (2016), no. 9, 3991–4000.
  • T. Kalmes (TU Chemnitz)

Dynamics of weighted composition operators PDMC, Będlewo, July 4, 2018 16 / 16