Introduction to the dynamics of holomorphic endomorphisms of P k - PowerPoint PPT Presentation

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Introduction to the dynamics of holomorphic endomorphisms of P k Dimitra Tsigkari Postgraduate Conference in Complex Dynamics, London, 11-13 March 2015 Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Outline Definitions 1 Elements of Pluripotential Theory 2 The Green current of a holomorphic endomorphism 3 Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Outline Definitions 1 Elements of Pluripotential Theory 2 The Green current of a holomorphic endomorphism 3 Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Complex Projective Space CP k Let z , w ∈ C k + 1 . Consider the equivalence relation: z ∼ w if there is λ ∈ C ∗ such that z = λ w . Definition The projective space of dimension k is the quotient of C k + 1 \ { 0 } by this relation, i.e. P k := C k + 1 � { 0 } / ∼ . Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Complex Projective Space CP k Let z , w ∈ C k + 1 . Consider the equivalence relation: z ∼ w if there is λ ∈ C ∗ such that z = λ w . Definition The projective space of dimension k is the quotient of C k + 1 \ { 0 } by this relation, i.e. P k := C k + 1 � { 0 } / ∼ . In other words, P k is the parameter space of the complex lines passing through 0 in C k + 1 . We denote by [ z 0 : z 1 : . . . : z k ] the point of P k associated to the point ( z 0 , z 1 , . . . , z k ) of C k + 1 � { 0 } . Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Complex Projective Space CP k The space P k is: a compact complex manifold of dimension k . the holomorphic compactification of C k . Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Complex Projective Space CP k The space P k is: a compact complex manifold of dimension k . the holomorphic compactification of C k . We equip the space P k with the Fubini-Study metric. Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Holomorphic Endomorphisms of P k Theorem Let f : P k → P k be a holomorphic endomorphism. Then f is described by the coordinates [ f 0 : f 1 : . . . : f k ] where each f j is a homogeneous polynomial of degree d and the f j have no common zero except the origin. Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Holomorphic Endomorphisms of P k Theorem Let f : P k → P k be a holomorphic endomorphism. Then f is described by the coordinates [ f 0 : f 1 : . . . : f k ] where each f j is a homogeneous polynomial of degree d and the f j have no common zero except the origin. The space of holomorphic endomorphisms of degree d is denoted by H d . Examples: Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Holomorphic Endomorphisms of P k Theorem Let f : P k → P k be a holomorphic endomorphism. Then f is described by the coordinates [ f 0 : f 1 : . . . : f k ] where each f j is a homogeneous polynomial of degree d and the f j have no common zero except the origin. The space of holomorphic endomorphisms of degree d is denoted by H d . Examples: f : P k → P k , [ z 0 : z 1 : . . . : z k ] �→ [ z d 0 : z d 1 : . . . : z d k ] , d ≥ 2 . So f ∈ H d . Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Holomorphic Endomorphisms of P k Theorem Let f : P k → P k be a holomorphic endomorphism. Then f is described by the coordinates [ f 0 : f 1 : . . . : f k ] where each f j is a homogeneous polynomial of degree d and the f j have no common zero except the origin. The space of holomorphic endomorphisms of degree d is denoted by H d . Examples: f : P k → P k , [ z 0 : z 1 : . . . : z k ] �→ [ z d 0 : z d 1 : . . . : z d k ] , d ≥ 2 . So f ∈ H d . g : C 2 → C 2 , g ( z 0 , z 1 ) = ( z 0 + 1 , z d 0 + z 0 z d − 1 ) . 1 Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Holomorphic Endomorphisms of P k Theorem Let f : P k → P k be a holomorphic endomorphism. Then f is described by the coordinates [ f 0 : f 1 : . . . : f k ] where each f j is a homogeneous polynomial of degree d and the f j have no common zero except the origin. The space of holomorphic endomorphisms of degree d is denoted by H d . Examples: f : P k → P k , [ z 0 : z 1 : . . . : z k ] �→ [ z d 0 : z d 1 : . . . : z d k ] , d ≥ 2 . So f ∈ H d . g : C 2 → C 2 , g ( z 0 , z 1 ) = ( z 0 + 1 , z d 0 + z 0 z d − 1 ) . 1 We extend g to P 2 : g : [ z 0 : z 1 : z 2 ] �→ [ z 0 z d − 1 0 + z 0 z d − 1 + z d 2 : z d : z d ˜ 2 ] . Then ˜ g / ∈ H d . 2 1 Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Fatou and Julia sets As in dynamics in one complex variable, we define: Definition Let f ∈ H d ( P k ) . We define the Fatou set F of f as the largest open set where the family of iterates { f n } n = 1 , 2 ,... is locally equicontinuous. The Julia set J of f is defined by J := P k \ F . Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Outline Definitions 1 Elements of Pluripotential Theory 2 The Green current of a holomorphic endomorphism 3 Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Pluriharmonic Functions Definition Let Ω ⊂ C n be an open subset and u ∈ C 2 (Ω) be a real valued function. u is said to be pluriharmonic in Ω if, for every a , b ∈ C n , the function λ �→ u ( a + λ b ) is harmonic in { λ ∈ C | a + λ b ∈ Ω } . u is pluriharmonic in Ω if ∂ 2 u = 0 in Ω , where j , k = 1 , . . . , n . ∂ z j ∂ z k Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Plurisubharmonic Functions Definition Let Ω be an open subset of C n , and let u : Ω → [ −∞ , ∞ ) be an upper semicontinuous function which is not identically −∞ on any connected component of Ω . The function u is said to be plurisubharmonic if for each a ∈ Ω , b ∈ C n , the function λ �→ u ( a + λ b ) is subharmonic or identically −∞ on every component of the set { λ ∈ C | a + λ b ∈ Ω } . Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Plurisubharmonic Functions Definition Let Ω be an open subset of C n , and let u : Ω → [ −∞ , ∞ ) be an upper semicontinuous function which is not identically −∞ on any connected component of Ω . The function u is said to be plurisubharmonic if for each a ∈ Ω , b ∈ C n , the function λ �→ u ( a + λ b ) is subharmonic or identically −∞ on every component of the set { λ ∈ C | a + λ b ∈ Ω } . Example: If f : U → C is holomorphic in the open set U ⊂ C n and f �≡ 0 , then the function log | f | is plurisubharmonic in U . Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Differential Forms and Currents Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Differential Forms and Currents D p , q (Ω) : the space of differential forms of class C ∞ in Ω ⊂ C n with compact support and whose bidegree is ( p , q ) . If ϕ ∈ D p , q (Ω) , then ϕ = � ϕ IJ dz I ∧ dz J , where ϕ IJ ∈ C ∞ k (Ω) and ♯ I = p , ♯ J = q . Dynamics in Several Complex Variables

Definitions Elements of Pluripotential Theory The Green current of a holomorphic endomorphism Differential Forms and Currents D p , q (Ω) : the space of differential forms of class C ∞ in Ω ⊂ C n with compact support and whose bidegree is ( p , q ) . If ϕ ∈ D p , q (Ω) , then ϕ = � ϕ IJ dz I ∧ dz J , where ϕ IJ ∈ C ∞ k (Ω) and ♯ I = p , ♯ J = q . Definition The elements of the dual space ( D n − p , n − q (Ω)) ′ are called currents of bidegree ( p , q ) . Dynamics in Several Complex Variables