Dynamics of composition operators on spaces of real analytic functions

Jos´eBonet

Instituto Universitario de Matem´aticaPura y Aplicada IUMPA

Universidad Polit´ecnica de Valencia

Operator Theory and Related Topics, Lille (France),

June 2010

Joint work with P. Doma´nski,Pozna´n(Poland) Statement of the Problem

Problem Study the behavior of iterates of a composition operator

on the space of real analytic functions A (Ω) .

We are mainly interested in the case when all the orbits are bounded.

Jos´eBonet Dynamics of composition operators Composition operator

Cϕ(f ) := f ◦ ϕ

ϕ : U → U holomorphic, U ⊂ Cd open set

Cϕ : H(U) → H(U)

ϕ :Ω → Ω real analytic, Ω ⊂ Rd open set

Cϕ : A (Ω) → A (Ω)

Iterates: n (Cϕ) := Cϕ ◦ Cϕ ◦ · · · ◦ Cϕ = Cϕn | {z } n times Universal functions for composition operators have been investigated by Bernal, Bonilla, Godefroy, Grosse-Erdmann, Le´on,Luh, Montes, Mortini, Shapiro and others.

Jos´eBonet Dynamics of composition operators Definitions

X is a Hausdorff locally convex space (lcs). L(X ) is the space of all continuous linear operators on X .

Power bounded operators m ∞ An operator T ∈ L(X ) is said to be power bounded if {T }m=1 is an equicontinuous subset of L(X ).

If X is a Fr´echetspace, an operator T is power bounded if and only if m ∞ the orbits {T (x)}m=1 of all the elements x ∈ X under T are bounded. This is a consequence of the uniform boundedness principle.

Hypercyclic operators An operator T ∈ L(X ) is called hypercyclic if there is x ∈ X such that its orbit is dense in X .

Jos´eBonet Dynamics of composition operators Definitions

Mean ergodic operators An operator T ∈ L(X ) is said to be mean ergodic if the limits

n 1 X Px := lim T mx, x ∈ X , (1) n→∞ n m=1 exist in X .

A power bounded operator T is mean ergodic precisely when

X = Ker(I − T ) ⊕ Im(I − T ), (2)

where I is the identity operator, Im(I − T) denotes the range of (I − T ) and the bar denotes the “closure in X ”.

Jos´eBonet Dynamics of composition operators Motivation

Trivial observation

λ ∈ C, |λ| ≤ 1.

1 λ := (λ + λ2 + ... + λn) [n] n

λ = 1 ⇒ λ[n] = 1, n = 1, 2, 3, ...

2 λ 6= 1 ⇒ |λ | ≤ → 0 [n] n|1 − λ|

as n → ∞.

Jos´eBonet Dynamics of composition operators Motivation

von Neumann, 1931 Let H be a Hilbert space and let T ∈ L(H) be unitary. Then there is a 1 Pn m projection P on H such that T[n] := n m=1 T converges to P in the strong operator topology.

Lorch, 1939 Let X be a reflexive Banach space. If T ∈ L(X ) is a power bounded operator, then there is a projection P on X such that T[n] converges to P in the strong operator topology; i.e. T is mean ergodic.

Hille, 1945

There exist mean ergodic operators T on L1([0, 1]) which are not power bounded.

Jos´eBonet Dynamics of composition operators Motivation

Problem attributed to Sucheston, 1976 Let X be a Banach space such that every power bounded operator T ∈ L(X ) is mean ergodic. Does it follow that X is reflexive?

YES if X is a Banach lattice (Emelyanov, 1997).

YES if X is a Banach space with basis (Fonf, Lin, Wojtaszczyk, J. Funct. Anal. 2001). This was a major breakthrough.

There are non reflexive Banach spaces in which all contractions are mean ergodic (Fonf, Lin, Wojtaszczyk, Preprint 2009).

Jos´eBonet Dynamics of composition operators Yosida’s mean ergodic Theorem

Barrelled locally convex spaces

Ls (X ), Lb(X )

Yosida, 1960 Let X be a barrelled lcs. The operator T ∈ L(X ) is mean ergodic if and 1 n only if it limn→∞ n T = 0, in Ls (X ) and

∞ 0 {T[n]x}n=1 is relatively σ(X , X )–compact, ∀x ∈ X. (3)

Setting P := τs -limn→∞ T[n], the operator P is a projection which commutes with T and satisfies Im(P) = Ker(I − T ) and Ker(P) = Im(I − T ).

Jos´eBonet Dynamics of composition operators Yosida’s mean ergodic Theorem

∞ If {T[n]}n=1 happens to be convergent in Lb(X ), then T is called uniformly mean ergodic.

Corollary If X is a reflexive lcs, then every power bounded operator on X is mean ergodic.

If X is a Montel space, then every power bounded operator on X is uniformly mean ergodic.

Jos´eBonet Dynamics of composition operators Results of Albanese, Bonet, Ricker, 2008

Main Results Let X be a complete barrelled lcs with a Schauder basis. Then X is reflexive if and only if every power bounded operator on X is mean ergodic.

Let X be a complete barrelled lcs with a Schauder basis. Then X is Montel if and only if every power bounded operator on X is uniformly mean ergodic.

Let X be a sequentially complete lcs which contains an isomorphic copy of the Banach space c0. Then there exists a power bounded operator on X which is not mean ergodic.

Jos´eBonet Dynamics of composition operators Results of Albanese, Bonet, Ricker, 2008

Results about Schauder basis A complete, barrelled lcs with a basis is reflexive if and only if every basis is shrinking if and only if every basis is boundedly complete. This extends a result of Zippin for Banach spaces and answers positively a problem of Kalton from 1970.

Every non-reflexive Fr´echetspace contains a non-reflexive, closed subspace with a basis. This is an extension of a result of A. Pelczynski for Banach spaces.

Bonet, de Pagter and Ricker (Preprint 2010) A Fr´echetlattice X is reflexive if and only if every power bounded operator on X is mean ergodic.

This is an extension of the result of Emelyanov.

Jos´eBonet Dynamics of composition operators Composition operators on spaces of analytic functions

Theorem Let U ⊆ Cd be a connected domain of holomorphy. The following assertions are equivalent:

(a) Cϕ : H(U) → H(U) is power bounded;

(b) Cϕ : H(U) → H(U) is uniformly mean ergodic;

(c) Cϕ : H(U) → H(U) is mean ergodic; (d) The map ϕ has stable orbits on U: n ∀ K b U ∃ L b U such that ϕ (K) ⊆ L for every n ∈ N; (e) There is a fundamental family of (connected) compact sets (Lj ) in U such that ϕ(Lj ) ⊆ Lj for every j ∈ N.

Jos´eBonet Dynamics of composition operators Composition operators on spaces of analytic functions

The result is valid for Stein manifolds U.

The equivalence (a)⇔(b)⇔(c) is true for arbitrary open connected U ⊆ Cd .

If U is hyperbolic, the conditions of the theorem are equivalent to the fact that all the orbits of ϕ are compact.

If U is complete hyperbolic, the conditions of the theorem are equivalent to the fact that there is a compact orbit of ϕ.

Jos´eBonet Dynamics of composition operators Composition operators on spaces of analytic functions

Theorem Let U be a connected open subset of C and let ϕ : U → U be a holomorphic self-map. If Cϕ is power bounded, then either ϕ is an automorphism of U or all orbits of ϕ tends to a constant u ∈ U such that u is a fixed point of ϕ.

If additionally U = D, then in the automorphism case ϕ has also a fixed point u and −1 ϕ = ϕu ◦ rθ ◦ ϕu, θ ∈ [0, 2), where u − z ϕ (z) := , r (z) = eiθπz u 1 − uz¯ θ

Jos´eBonet Dynamics of composition operators Theorem continued Moreover,

(i) If ϕ is not an automorphism then the projection P associated to Cϕ is given: P(f )(z) = f (u).

p (ii) If ϕ is an automorphism and θ = q is rational then 1 Pq−1 −1 P(f )(z) = f (ϕu (r jp (ϕu(z)))). q j=0 q

(iii) If ϕ is an automorphism and θ is irrational then 1 R 2 −1 P(f )(z) = 2 0 f (ϕu (rθ(ϕu(z))))dθ.

Jos´eBonet Dynamics of composition operators Composition operators on spaces of real analytic functions

Ω ⊆ Rd open connected set. The space of real analytic functions A (Ω) is equipped with the unique locally convex topology such that for any U ⊆ Cd open, Rd ∩ U = Ω, the restriction map R : H(U) −→ A (Ω) is continuous and for any compact set K ⊆ Ω the restriction map r : A (Ω) −→ H(K) is continuous. In fact,

(Ω) = proj H(K ) = proj ind H∞(U ). A N∈N N N∈N n∈N N,n

A (Ω) is complete, separable, barrelled and Montel.

Doma´nski,Vogt, 2000. The space A (Ω) has no Schauder basis.

Jos´eBonet Dynamics of composition operators Composition operators on spaces of real analytic functions

Theorem Let ϕ :Ω → Ω, Ω ⊆ Rd open connected, be a real analytic map. Then the following assertions are equivalent:

(a) Cϕ : A (Ω) → A (Ω) is power bounded;

(b) Cϕ : A (Ω) → A (Ω) is uniformly mean ergodic;

(c) Cϕ : A (Ω) → A (Ω) is mean ergodic; (d) ∀ K b Ω ∃ L b Ω ∀ U complex neighbourhood of L ∃ V complex neighbourhood of K:

n n ∀ n ∈ N ϕ is defined on V and ϕ (V ) ⊆ U; (e) For every complex neighbourhood U of Ω there is a complex (open!) neighbourhood V ⊆ U of Ω such that ϕ extends as a holomorphic function to V and ϕ(V ) ⊆ V .

Jos´eBonet Dynamics of composition operators Composition operators on spaces of real analytic functions

Corollary Let ϕ :Ω → Ω be a real analytic map, Ω ⊆ Rd open connected set.

If Cϕ : A (Ω) → A (Ω) is power bounded, then for every complex neighbourhood U of Ω there is a complex neighbourhood V ⊆ U of Ω and there is a fundamental system of compact sets (Lj ) in V such that ϕ(Lj ) ⊆ Lj .

In particular, ϕ(V ) ⊆ V and Cϕ : H(V ) → H(V ) is power bounded and (uniformly) mean ergodic.

Jos´eBonet Dynamics of composition operators Composition operators on spaces of real analytic functions

Corollary Let Ω be a real analytic connected manifold and let ϕ :Ω → Ω be a real analytic map. If Cϕ is hypercyclic, then (1) ϕ is injective, its derivative ϕ0(z) at arbitrary point z ∈ Ω is never a singular linear map and ϕ runs away, i.e., for every compact set K ⊆ Ω there is n ∈ N such that ϕn(K) ∩ K = ∅. (2) There is a complex neighbourhood U of Ω such that for every complex neighbourhood V ⊆ U of Ω the map ϕ does not extend as a holomorphic self map on V .

Jos´eBonet Dynamics of composition operators Composition operators on spaces of real analytic functions

Example (Why does the complex case not solve the real analytic case?): Even if ϕ :] − 1, 1[→] − 1, 1[ maps all bounded sets in ] − 1, 1[ in one compact set it does not follow that Cϕ : A (] − 1, 1[) → A (] − 1, 1[) is power bounded.

2 1 − iz  ϕ (z) := ln . α α · π 1 + iz

1 1
ϕα maps the unit disc onto the vertical strip with Re z ∈ − α , α . π·α If 4 < 1, the zero point is an attractive fixed point on the −1 −n imaginary line for ϕα . This implies that ϕα (i) → 0 as n → ∞. n+1 Since on this sequence ϕα is not defined, there is no common n neighbourhood of zero such that all ϕα are defined for all n ∈ N (α, π·α α < 1, 4 < 1). Thus Cϕ : A (] − 1, 1[) → A (] − 1, 1[) is not power bounded.

Jos´eBonet Dynamics of composition operators Composition operators on spaces of real analytic functions

Theorem Let a, b ∈ R¯ and let ϕ :]a, b[→]a, b[ be real analytic. The following are equivalent:

(a) Cϕ : A (]a, b[) −→ A (]a, b[) is power bounded; (b) there exists a complex neighbourhood U of ]a, b[ such that ϕ(U) ⊆ U, C \ U contains at least two points, and ϕ has a (real) fixed point u, or equivalently, there is a fundamental family of such neighbourhoods of ]a, b[; (c) ϕ is one of the following forms:

1 ϕ = id ; 2 2 ϕ = id ; n 3 As n → ∞ the sequence ϕ tends to a constant function ≡ u ∈]a, b[ in A (]a, b[).

Jos´eBonet Dynamics of composition operators Composition operators on spaces of real analytic functions

Theorem continued If u is the fixed point of ϕ then the above cases in (c) correspond to: 0 1 ϕ (u) = 1; 0 2 ϕ (u) = −1; 0 3 |ϕ (u)| < 1.

Moreover, Cϕ is uniformly mean ergodic and the projection 1 N P n P := limN∈N N n=1 Cϕ is of the following form: 1 P = id ;

2 f +f ◦ϕ P(f ) = 2 , im P = {f : f = f ◦ ϕ}, ker P = {f : f = −f ◦ ϕ}; 3 P(f ) = f (u), im P = the set of constant functions, ker P = {f : f (u) = 0}.

Jos´eBonet Dynamics of composition operators Composition operators on spaces of real analytic functions

Proposition Let ϕ be a self map on the unit disc D which is holomorphic on a neighbourhood of D and satisfies ϕ(] − 1, 1[) ⊂] − 1, 1[. Assume that ϕ is injective and its derivative does not vanish on ] − 1, 1[. If ϕ runs away on ] − 1, 1[, then Cϕ is hypercyclic on A (] − 1, 1[).

Jos´eBonet Dynamics of composition operators Summary

Characterization of holomorphic ϕ such that Cϕ power bounded — form of projection P. Characterization of hypercyclicity already known, at least for one complex variable, by the work of Bernal, Montes, Grosse-Erdamnn, Mortini and others.

Characterization of real analytic ϕ such that Cϕ power bounded — form of projection P. Some consequences about hypercyclic composition operators.

The power bounded case is the only one where real analytic iterates of a selfmap can be fully understood through iterates of a holomorphic selfmap.

Jos´eBonet Dynamics of composition operators References

1 A. A. Albanese, J. Bonet, W. J. Ricker, Mean Ergodic Operators in Fr´echetSpaces, Anal. Acad. Math. Sci. Fenn. Math. 34 (2009), 401-436. 2 J. Bonet, P. Doma´nski, Mean ergodic composition operators on spaces of holomorphic functions, RACSAM, to appear in 2011. 3 J. Bonet, P. Doma´nski, Power bounded composition operators on spaces of analytic functions, Preprint, 2010.

Jos´eBonet Dynamics of composition operators