HOLOMORPHIC INTERPOLATION:FROM DYNAMICS

TO ANALYSIS

Warsaw - September 2015

István Prause

TOPICS

variation of dimension of Julia sets

quasiconvexity and Burkholder’s functional

rotational estimates for bilipschitz maps

dimension of quasicircles

TOPICS

variation of dimension of Julia sets

quasiconvexity and Burkholder’s functional

rotational estimates for bilipschitz maps

dimension of quasicircles

holomorphic interpolation

VARIATION OF DIM OF JULIA SETSRansford

( ) , ∈ analytic family of rational maps, �

Corollary:

D simply connected, hyperbolic

/dim( )−

/dim( )−� exp ( , )

/dim = inf∈H

( )Theorem:harmonic functions

H a collection of

Pf: − �Harnack

dim � ,( )−

( )−� exp ( , )

Example:= +

∈ Ddim � + | |

THERMODYNAMICSvariational principle (Ruelle, Bowen)

dim( ) = sup∈M

( )´

log | � |

: →

= ◦ ◦ −

dim= inf

∈M

´

log | � ◦ |

( )

holomorphic motion

Mañé-Sad-Sullivan

harmonic

variation of dimension Riesz-Thorin

holomorphic motions holomorphic exponent

dimension norm

variational principle duality

apriori bounds endpoint estimates

Harnack’s inequality Hadamard’s three lines theorem

HOLOMORPHIC INTERPOLATION

U

10

!! !!

non-vanishing ( ) �=

analytic family,

⇒

< , � ∞, ∈ ( , )

∈ = { > }( )

� � �

� � � �− ·

�=

�−+

INTERPOLATION LEMMA

� � �

Astala-Iwaniec-Prause-Saksman

Hadamard Harnack

change p freeze p

subharmonic harmonic

duality log-convexity

!

!

!

!θ

� � � · +

cf. RIESZ-THORIN

Burkholder

MARTINGALE INEQUALITY

subordinated martingales

⇒ �!"�# � (#− !) �$"�#.

!"(#,$) =�

|#| − ("− !) |$|�

·�

|#| + |$|�"−!

E !"(#$, %$) � E !"(#$−!, %$−!) � . . . � "

!" ≺ #"

| | − ( − ) | | � ( , )

| −−

| ≤ | −−

| a.s.

Morrey

! ∈ "+ #∞! (",R$)

�!

E(!") �

�!

E(#) = E(#)|!|!"#$ ! = %

E : R!×!

→ R

⇐

! � ! Šverák

! = !

�

? Faraco-Székelyhidi: “localization”

local

cal

global

!" (#) =� "

!det # + ("−

"

!)�

�#�

�

!�

· |#|"−!

Burkholder: rank-one concave!"(#$) = !"($%, $%̄)

�→ E( + )

(lower semicontinuity)(ellipticity of Euler-Lagrange)

Rank-one convexity vs Quasiconvexity

Quasiconvexity result

Theorem:

! � !!"(#,$) =�

|#| − ("− !) |$|�

·�

|#| + |$|�"−!

!"(#$) = !"($%, $%̄)

!(") ∈ "+ #∞! ("), $%(&!) � !, " ∈ "�( ) �

�( ) = | |

full quasiconvexity� � (C) = −

Astala-Iwaniec-Prause-Saksman

( ) = −��C

( )

( − )( )

STRETCHING vs ROTATION

stretching rotation

quasiconformal bilipschitz

Grötzsch problem John’s problem

Hölder exponent rate of spiralling

log J(z,f) ∈ BMO arg f ∈ BMO

higher integrability exponential integrability

multifractal spifractal spectrum

z

harmonic dependence “conjugate harmonic”

MULTIFRACTAL SPECTRA

dim { ∈ C : ( ) = } ≤ + −| − |

1/K K1 L-1/L-(L-1/L) 0

dim { : ( ) = } ≤ −−

| |

0

2

K-quasiconformal L-bilipschitz

( ) = lim| − |= →

log | ( )− ( )|

log | − |( ) = lim

→

( ( + )− ( ))

log | ( + )− ( )|

: C → C

Astala-Iwaniec-Prause-Saksman

scaling:

Courtesy of D. Marshall

multifractal spectrum:

What about conformal maps?

(−, +, ) ≤

− ( + )(

−

++

)

−+

− +

?

multifractality of ω

( ) = dimF

F ( , ) ≈

≈

Makarov: dim =

+ | |

DIMENSION OF QUASICIRCLES

Example:

dim ( ) = +log

�

| |�

+ · · ·Ruelle:

= +

Smirnov:

=+

+ ¯

� +

�

| |�

+ . . .

dim (S ) � +

k-quasiconformal,: C → C � �∞ �

( ) = −

� �

�,¯

=

�

· −

SHARPNESS ?

1-k²

1+k1-k

existence of quasicircle with dim=1+k²

compressing/expanding

compr

conformal map

lower bound formultifractal spectrum

( ) � −

: D → D

( , ) = −( , )

λµ

ηµ̄

(·, ) = ( , ·) =

dim =

∂�∂�̄ (�, �̄)|�= = ⇒ (�, ) =

BLASCHKE PRODUCTS

=

Blaschke products of degree dwith an attracting fixed point

�

D

Example: ∼= D∈ D

Julia set = S¹

quasisymmetrically conjugate to each other

( ) =+

+ ¯

MATING

d=2

F rational map

Jordan curve

Example: =+

+ ¯

cf. Bers’ simultaneous uniformization

¯

| + ∼=

|−

∼=¯

= [ , ]

WEIL-PETERSSON METRICMcMullen

=

�

�

�

=

= [ , ]

∈

: D∗

→ C

conformal conjugacy

( ) =

,= Lebesgue m.

= =−�

�

�

=

dim(,)

�

�

�

=

dim( ( )) ( �)

ASYMPTOTIC VARIANCE

( ) = lim sup→ | log( − )|

�| |=

| |

∈ B

�

� � �B

�

Example: ( ) =

∞�

=

=log

� �B = sup∈D

( − | | )| �( )| < ∞

Ruelle

WP METRIC at zd

∈ D

( ) = ( ) + ( ) ( ) =

( ) = −

∞�

=

−( − )

, | | > .

∈ D∗

( ) =

∞�

=

( ), + ( ) =−

( )

= +

= ˙

( ) = ( ) +−

( ), ( ) = − −

WP METRIC at zd

�( ) =( − )

·�

≥

−( − ) + , ∈ B ∗

( �) =( − )

log

( ) is a quasicircle ( has |t|-qc extension)

quadratic terms in dimension expansion for k-quasicircles

( ) = + ( �)| | +O(| | )

= . . . . . . . .

= . . . . . . . .

= . . . . . . . .

= . . . . . . . .

C ( ) =

�C

( )

−

( )

Find

pull-back

Lemma:

∗( ) =�

( )∗ )�

( ) = ( )¯ −

−

Proof: ∂̄take and → ∞

EXPLICIT REPRESENTATION

, ⊂ D, = C � �∞

C

�

( )∗�

( ) =�

−�

�

C ( )− C ( )�

, ∈ C

Astala-Ivrii-Perälä-Prause

Building block: ( ) :=�

/| |�−

( , )

C ( ) =−

�

− − −�

−, | | >

C =

+ :=∗, C + ( ) =

−( ) = + ( )

optimize over r

:=

∞�

=

, C =

optimize over d (d=20)

−−

=

/( −)

�

=

disjoint spt

EXPLICIT REPRESENTATION

( ) = . . . .

∈ D= +

: D∗ → C conformal conjugacy

λ-lemma: |t|-qc extension

Thm: qc extension

Corollary: 1+0.879 k²

QUASICONFORMAL EXTENSION

/( − )

| |+O(| | )

( ) is a k-quasicircle with dimension >

for k small (d=20)

Astala-Ivrii-Perälä-Prause

∈ D= +

: D∗ → C conformal conjugacy

λ-lemma: |t|-qc extension

Thm: qc extension

Corollary: 1+0.879 k²

QUASICONFORMAL EXTENSION

/( − )

| |+O(| | )

( ) is a k-quasicircle with dimension >

for k small (d=20)

Astala-Ivrii-Perälä-Prause

Oleg Ivrii’s talk:No k-quasicircles with

dimension 1+ k²