Constructing Entire Functions (a summary) Kirill Lazebnik SUNY - PowerPoint PPT Presentation

Constructing Entire Functions (a summary) Kirill Lazebnik SUNY Stony Brook Kirill.Lazebnik@stonybrook.edu November 26, 2015 Kirill Lazebnik

Constructing Entire Functions By Quasiconformal Folding (a summary) Kirill Lazebnik SUNY Stony Brook Kirill.Lazebnik@stonybrook.edu November 26, 2015 Kirill Lazebnik

p ( z ) = z 3 2 − 3 z 2 Kirill Lazebnik

p ( z ) = z 3 2 − 3 z 2 p ′ ( z ) = 3 2 ( z − 1)( z + 1) Kirill Lazebnik

p ( z ) = z 3 2 − 3 z 2 (two critical values ± 1) p ′ ( z ) = 3 2 ( z − 1)( z + 1) Kirill Lazebnik

p ( z ) = z 3 2 − 3 z 2 (two critical values ± 1) p ′ ( z ) = 3 2 ( z − 1)( z + 1) Kirill Lazebnik

p ( z ) = z 4 4 − z 3 3 − z 2 2 + z Kirill Lazebnik

p ( z ) = z 4 4 − z 3 3 − z 2 2 + z p ′ ( z ) = ( z − 1) 2 ( z + 1) Kirill Lazebnik

p ( z ) = z 4 4 − z 3 3 − z 2 2 + z (two critical values 5 / 12 , − 11 / 12) p ′ ( z ) = ( z − 1) 2 ( z + 1) Kirill Lazebnik

p ( z ) = z 4 4 − z 3 3 − z 2 2 + z (two critical values 5 / 12 , − 11 / 12) p ′ ( z ) = ( z − 1) 2 ( z + 1) Kirill Lazebnik

Shabat polynomial - Kirill Lazebnik

Shabat polynomial - only has two critical values ± 1 Kirill Lazebnik

Shabat polynomial - only has two critical values ± 1 Proposition: For any Shabat polynomial p ( z ), it is true that p − 1 [ − 1 , 1] is a tree. Kirill Lazebnik

Shabat polynomial - only has two critical values ± 1 Proposition: For any Shabat polynomial p ( z ), it is true that p − 1 [ − 1 , 1] is a tree, with deg( p ) edges. Kirill Lazebnik

Shabat polynomial - only has two critical values ± 1 Proposition: For any Shabat polynomial p ( z ), it is true that p − 1 [ − 1 , 1] is a tree, with deg( p ) edges. Theorem (Grothendieck) : ALL combinatorial trees occur as p − 1 [ − 1 , 1] for some Shabat polynomial p ( z ). Kirill Lazebnik

Shabat polynomial - only has two critical values ± 1 Proposition: For any Shabat polynomial p ( z ), it is true that p − 1 [ − 1 , 1] is a tree, with deg( p ) edges. Theorem (Grothendieck) : ALL combinatorial trees occur as p − 1 [ − 1 , 1] for some Shabat polynomial p ( z ). Theorem (Bishop) : Any continua can be ǫ -approximated in the Hausdorff metric by some p − 1 [ − 1 , 1]. Kirill Lazebnik

⇐ ⇒ trees Shabat polynomials Kirill Lazebnik

trees ⇐ ⇒ Shabat polynomials infinite trees ⇐ ⇒ Transcendental Functions Kirill Lazebnik

trees ⇐ ⇒ Shabat polynomials infinite trees ⇐ ⇒ Subclass of Transcendental Functions Kirill Lazebnik

trees ⇐ ⇒ Shabat polynomials infinite trees ⇐ ⇒ Subclass of Transcendental Functions S 2 , 0 - transcendental functions with two critical values ± 1 and no asymptotic values Kirill Lazebnik

cosh( z ) := e z + e − z 2 Kirill Lazebnik

cosh( z ) := e z + e − z 2 cosh ′ ( z ) = e z − e − z 2 Kirill Lazebnik

cosh( z ) := e z + e − z 2 cosh ′ ( z ) = e z − e − z = 0 = ⇒ z = π i n : n ∈ Z 2 Kirill Lazebnik

cosh( z ) := e z + e − z 2 cosh ′ ( z ) = e z − e − z = 0 = ⇒ z = π i n : n ∈ Z (critical points) 2 Kirill Lazebnik

cosh( z ) := e z + e − z 2 cosh ′ ( z ) = e z − e − z = 0 = ⇒ z = π i n : n ∈ Z (critical points) 2 critical values: ± 1 Kirill Lazebnik

T - unbounded, locally finite tree Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T τ : ∪ Ω j → C - the map conformal on each Ω j to H r Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T τ : ∪ Ω j → C - the map conformal on each Ω j to H r V - the vertices of T . Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T . τ : ∪ Ω j → C - the map conformal on each Ω j to H r . V - the vertices of T . V j - the image of V under τ restricted to Ω j . Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T . τ : ∪ Ω j → C - the map conformal on each Ω j to H r . V - the vertices of T . V j - the image of V under τ restricted to Ω j . For r > 0, define T ( r ) = ∪ e ∈ T { z : dist ( z , e ) < r diam ( e ) } Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T . τ : ∪ Ω j → C - the map conformal on each Ω j to H r . V - the vertices of T . V j - the image of V under τ restricted to Ω j . For r > 0, define T ( r ) = ∪ e ∈ T { z : dist ( z , e ) < r diam ( e ) } The τ -size of edge e is the minimum length of the two images τ ( e ) Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T . τ : ∪ Ω j → C - the map conformal on each Ω j to H r . V - the vertices of T . V j - the image of V under τ restricted to Ω j . For r > 0, define T ( r ) = ∪ e ∈ T { z : dist ( z , e ) < r diam ( e ) } The τ -size of edge e is the minimum length of the two images τ ( e ) T has uniformly bounded geometry if: Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T . τ : ∪ Ω j → C - the map conformal on each Ω j to H r . V - the vertices of T . V j - the image of V under τ restricted to Ω j . For r > 0, define T ( r ) = ∪ e ∈ T { z : dist ( z , e ) < r diam ( e ) } The τ -size of edge e is the minimum length of the two images τ ( e ) T has uniformly bounded geometry if: (1) The edges of T are C 2 with uniform bounds. Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T . τ : ∪ Ω j → C - the map conformal on each Ω j to H r . V - the vertices of T . V j - the image of V under τ restricted to Ω j . For r > 0, define T ( r ) = ∪ e ∈ T { z : dist ( z , e ) < r diam ( e ) } The τ -size of edge e is the minimum length of the two images τ ( e ) T has uniformly bounded geometry if: (1) The edges of T are C 2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T . τ : ∪ Ω j → C - the map conformal on each Ω j to H r . V - the vertices of T . V j - the image of V under τ restricted to Ω j . For r > 0, define T ( r ) = ∪ e ∈ T { z : dist ( z , e ) < r diam ( e ) } The τ -size of edge e is the minimum length of the two images τ ( e ) T has uniformly bounded geometry if: (1) The edges of T are C 2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T . τ : ∪ Ω j → C - the map conformal on each Ω j to H r . V - the vertices of T . V j - the image of V under τ restricted to Ω j . For r > 0, define T ( r ) = ∪ e ∈ T { z : dist ( z , e ) < r diam ( e ) } The τ -size of edge e is the minimum length of the two images τ ( e ) T has uniformly bounded geometry if: (1) The edges of T are C 2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths (4) For non-adjacent edges e , f , we have diam ( e ) / dist ( e , f ) uniformly bounded. Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T . τ : ∪ Ω j → C - the map conformal on each Ω j to H r . V - the vertices of T . V j - the image of V under τ restricted to Ω j . For r > 0, define T ( r ) = ∪ e ∈ T { z : dist ( z , e ) < r diam ( e ) } The τ -size of edge e is the minimum length of the two images τ ( e ) T has uniformly bounded geometry if: (1) The edges of T are C 2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths (4) For non-adjacent edges e , f , we have diam ( e ) / dist ( e , f ) uniformly bounded. Theorem: Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ω j - components of C − T . τ : ∪ Ω j → C - the map conformal on each Ω j to H r . V - the vertices of T . V j - the image of V under τ restricted to Ω j . For r > 0, define T ( r ) = ∪ e ∈ T { z : dist ( z , e ) < r diam ( e ) } The τ -size of edge e is the minimum length of the two images τ ( e ) T has uniformly bounded geometry if: (1) The edges of T are C 2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths (4) For non-adjacent edges e , f , we have diam ( e ) / dist ( e , f ) uniformly bounded. Theorem: Suppose T has bounded geometry Kirill Lazebnik