MUTIFRACTAL ANALYSIS FOR POINTWISE HOLDER EXPONENTS OF THE COMPLEX TAKAGI FUNCTIONS IN RANDOM COMPLEX DYNAMICS (The Theory of Random Dynamical Systems and Its Applications)

MULT[FRACTAL ANALYSIS FOR POINTWISE

HÖLDER

EXPONENTS OF THE COMPLEX TAKAGI FUNCTIONS IN RANDOM COMPLEX DYNAMICS

JOHANNES JAERISCH AND HIROKI SUMI

ABSTRACT. We consider_hyperbolicrandom_{complex dynamical}systemsonthe Riemann_{spherewith sep‐}

aratingcondition and_multiplermnimalsets. We_investigatethe Hölder_regularityofthe function T of the

probabilityof_tendingtooneminimal set, thepartialderivatives of T with_respecttothe_probabilityparameters,

whichcanbe_regardedas_{complexanalogues}of the_Takagifunction,and the_{higher partial}derivatives C of T.

Our main result_givesa_{dynamical description}of the_pointwiseHölderexponentsof Tand C,which allowsus

todeterminethespectrumof_pointwiseHölderexponentsby employingthe multifractal formalism inergodic theory. Also,we_{prove that the bottom of thespectrum}$\alpha$_{-}is_strictlyless than1,which allowsustoshow that the_averagedsystemacts_chaoticallyon_{the Banach space}C^{ $\alpha$}of $\alpha$‐Hölder continuous functions for every $\alpha$\in($\alpha$_{-}, 1),thoughtheaveragedsystembehaves verymildly (e.g. wehavespectral gaps)onC^{ $\beta$}for small

$\beta$>0.

1. MAINRESULTS

Thisnoteis_{the summary of the results from paper}

[JS16].

Wedonot

_give

_any

_proofs

of them in this

note. Forthe

_proofs

of theresults, see

_[JS16].

In_{this paper,} weconsider random

_dynamical

systems

of rational mapsonthe Riemann

_sphere

\hat{\mathbb{C}}

_{:=\mathbb{C}\cup\{\infty\}\cong S^{2}}

. The

study

ofrandom

complex

dynamics

wasinitiated

by

J.E. Fomaess and N.

Sibony

_([FS91]).

There are_manynew

_interesting

_phenomena

in random

_dynamical

_systems,socalled randomness‐induced

_phenomena

ornoise‐induced

_phenomena,

which cannothold in the deterministic iteration

dynamics.

For the motivations andrecentresearch of random

complex

dynamical

systemsfocusedontherandomness‐induced

_phenomena,

seethe second author’s works

[Sumlla,

Sum13,

\mathrm{S}\mathrm{u}\mathrm{m}\mathrm{i}\mathrm{l}5\mathrm{a},

\mathrm{S}\mathrm{u}\mathrm{m}\mathrm{l}5\mathrm{b}_]. Inthese papers itwasshown that fora

_generic

random

_dynamical

systemof

_complex

_polynomials,

thesystemacts_very

_mildly

onthe space of continuous functionson

\hat{\mathbb{C}}

and onthe space

C^{ $\alpha$}(\hat{\mathbb{C}})

for small

$\alpha$\in(0,1)

,where

C^{ $\alpha$}(\hat{\mathbb{C}})

denotes the Banach space of $\alpha$‐Hölder continuous

functionson

\hat{\mathbb{C}}

endowed with $\alpha$‐Höldernonn,but under certain conditions thesystemstillacts

_chaoticaly

on_{the space}

C^{ $\beta$}(\hat{\mathbb{C}})

forsome

$\beta$\in(0,1)

closeto 1. Thus, we

_investigate

the

_gradation

between chaos and order in random

_(complex)

_dynamical

_systems.

In orderto_{show the main ideas of the paper, let Rat denote the}setof allnon‐constantrational mapson

\hat{\mathbb{C}}

.This isa

semigroup

whose

semigroup operation

is the

composition

of maps.

Throughout

the paper, let

s\geq 1and let

_(fl,

\cdots_,

f_{s+1}

)

\in(\mathrm{R}\mathrm{a}\mathrm{t})^{s+1}

with

\deg(f_{i})\geq 2,i=1

, s+1.Let

\mathrm{p}=(p_{1}, \ldots,p_{s})\in(0,1)^{s}

with

\displaystyle \sum_{i=1}^{s}p_{i}<1

and let_{p_{s+1}}

_{:=1-\displaystyle \sum_{i=1}^{s}p_{i}}

. We consider the

(i.i.

\mathrm{d}.

)

random

dynamical

system

on

\hat{\mathbb{C}}

such that

at_everystepwechoose

f_{ $\iota$}

’with

probability

p_{i}.This definesaMarkov chain withstatespace

\hat{\mathbb{C}}

such that

for each

x\in\hat{\mathbb{C}}

and for each Borel measurable subset A of

\hat{\mathbb{C}}

,the transition

probability

p(x,A)

fromxtoA

Date:8th_May2016. MSC2010:37\mathrm{H}10,37\mathrm{F}15.

Keywords_{andphrases. Complex dynamical}systems,rational_semigroups,random_{complex dynamics,}multifractalformalism,Julia

set, random iteration.

Johannes Jaerisch

Departmentof_{Mathematics, Faculty}of Science and_Engineering,ShimaneUniversity,Nishikawatsu 1060Matsue,Shimane 690‐

8504,_JapanE‐‐mail:_{jaerisch@nko.shimane‐u.ac.jp}Web:_{http://www.math.shimane‐u.ac.jp/}\simjaerisch/

Hiroki Sumi_{(corresponding author)}

DepartmentofMathematics,Graduate School ofScience,Osaka_University,1‐1_{Machikaneyama, Toyonaka,}Osaka, 560‐0043,Japan

is

_equal

to

\displaystyle \sum_{i=1}^{s+1}p_{i}1_{A}(f_{i}(x))

,where

1_{A}

denotes the characteristic function of A. Let

G=(f_{1},\ldots,f_{s},f_{s+1})

be the rational

_semigroup

_(i.e.,

_subsemigroup

of

_{Rat) generated Uy}

_{\{f_{1}, f_{s+1}\}}

. More

precisely,

G=

{f_{0\}_{l}}\circ\cdots\circ f_{$\varpi$_{1}}

:

_{n\in \mathbb{N},}

_{t)_{1}},\cdots,

$\omega$_{n}\in\{1

,. s+1 We denote

by

F(G)

the maximal open subset of

\hat{\mathbb{C}}

on which G is

_{equicontinuous}

with_respecttothe

_spherical

distanceon

\hat{\mathbb{C}}

.Theset

F(G)

iscalled the Fatou

setof G,and theset

J(G)

:=\hat{\mathbb{C}}\backslash F(G)

iscalled the Juliasetof G. We remark that in orderto

investigate

randomcomplex

dynamical

systems,it is very

important

to

_investigate

the

_dynamics

of associated rational

semigroups.

The first

_study

of

_dynamics

of rational

_semigroups

wasconducted

_by

A. Hinkkanenand G.

J. Margin

_([HM96]),

whowereinterested in the role of

_polynomial

_semigroups

(i.e.,

semigroups

ofnon‐

constant

_polynomial

_maps)

while

_studying

various

_{one‐complex‐dimensional}

_{moduli spaces for discrete} groups, and

by

F. Ren’s group

([GR96]),

who studied such

semigroups

from the

_perspective

of random

dynamical

systems. For the

_interplay

of random

_{complex dynamics}

and

_dynamics

of rational

_semigroups,

see

_{[\mathrm{S}\mathrm{u}\mathrm{m}00]-[\mathrm{S}\mathrm{u}\mathrm{m}\mathrm{l}5\mathrm{b}]}

,

[SSII

,SU13, \mathrm{J}\mathrm{S}15\mathrm{a},\mathrm{J}\mathrm{S}15\mathrm{b}

].

Throughout

the paper,we assumethe

following.

(1)

G is

_hyperbolic,

_i.e.,wehave

P(G)\subset F(G)

,where

P(\mathrm{G})

:=\overline{\cup g(\bigcup_{i=1}^{s+1}\{\mathrm{c}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}1}

values of

f_{i}:\hat{\mathbb{C}}\rightarrow\hat{\mathbb{C}}

_}).

Here,the closure is taken in

\hat{\mathbb{C}}.

g\in G\cup\{\mathrm{i}\mathrm{d}\}

(2)

(fl,

\cdots_,

f_{s+1}

)

satisfies theseparaung

condition,

i.e.,

f_{i}^{-1}(J(G))\cap f_{j}^{-1}(J(G))=\emptyset

whenever

i,j\in

\{1, s+1\},i\neq j.

(3)

There existatleasttwominimalsetsof G. Here,anon‐empty compactsubsetKof

\hat{\mathbb{C}}

is calleda minimalsetof G if

K=\overline{\bigcup_{g\in G}\{g(z)\}}

for eachz\in K.

Note that

_by

_{assumption (2), [Sum97,}

Lemma 1,

1.4]

and

[Sumlla,

Theorem

3.15],

wehave that there exist

at most

_finitely

_{many minimal}setsof G. Moreover,

denoting by

S_{G}the union of minimalsetsof Gand

semngI

:=\{1, s+1\}

,wehave that for each

z\in\hat{\mathbb{C}}

there existsaBorel subset

A_{z}

of

I^{\mathrm{N}}

with

\tilde{p}_{\mathrm{p}}(A_{z})=1

such that

_{d(f_{\mathfrak{B}_{l}}\cdots f_{0)_{1}}(z),S_{G})\rightarrow 0}

as n\rightarrow\inftyfor all

_{a=($\omega$_{\dot{7}})_{i=1}^{\infty}\in A_{z}}

,where

\overline{p}_{\mathrm{p}} :=\otimes_{n=1}^{\infty}p_{\mathrm{p}}

denotes the

product

measure on

I^{\mathrm{N}}

_given

_by

_{p_{\mathrm{p}}}

:=\displaystyle \sum_{i=1}^{s+1}p_{i}$\delta$_{7}

.with

$\delta$_{i} denoting

the Diracmeasureconcentratedati\in I.

Throughout,

wefix aminimalsetLof G

_(e.g.

_L=\{\infty\}

when G is a

polynomial

_semigroup).

Denote

by

T_{\mathrm{p}}(z)

the

_probability

of

_tending

toLof the processon

\hat{\mathbb{C}}

which startsin

z\in\hat{\mathbb{C}}

and which is

_given

by drawing independently

with

_probability

p_{i}the map

f_{i}

. More

precisely,

T_{\mathrm{p}}(\mathrm{z}) :=\tilde{p}_{\mathrm{p}}(\{ $\omega$=($\omega$_{i})_{i=1}^{\infty}\in

I^{\mathrm{N}}:d(f_{$\omega$_{n}}\circ\cdots\circ f_{0J_{1}}(z),L)\rightarrow 0

as n\rightarrow\infty Itwasshown

by

the second author in

[Sum13] that,

foreach

\mathrm{p}=(p_{1}, \ldots,p_{s})

there exists

_{$\alpha$\in(0,1)}

such that

_{\mathrm{x}=(x_{1}, \ldots,x_{s})\mapsto T_{(xx_{S},1-$\Sigma$_{i=1^{X_{i)}}}^{s}}1,\ldots,\in C^{ $\alpha$}(\hat{\mathbb{C}})}

is

_{real‐analytic}

ina

neighbourhood

_{of \mathrm{p}}, where

C^{ $\alpha$}(\hat{\mathbb{C}})

denoted

the \mathbb{C}‐Banach space of $\alpha$‐Hölder continuous \mathbb{C}‐valued

functionson

\hat{\mathbb{C}}

endowed with $\alpha$‐Höldernorm

\Vert\cdot\Vert_{ $\alpha$}

(Remark 1.17).

Thus it is very natural and

_important

to

consider the

_following.

For

_{\mathrm{N}_{0}}

_{:=\mathbb{N}\cup\{0\}}

and

\mathrm{n}=(n_{1}, \ldots,n_{S})\in \mathbb{N}_{0}^{s}

wedenote

_by

C_{\mathrm{n}}\in C^{ $\alpha$}(\hat{\mathbb{C}})

the

_higher

order

_partial

derivative of

_{T_{\mathrm{p}}}

oforder

_{|\displaystyle \mathrm{n}|:=\sum_{i=1}^{s}n_{i}}

withrespecttothe

_probability

_parameters

_given

_Uy

\displaystyle \mathcal{C}_{\mathrm{n}}(z):=\frac{\partial^{|\mathrm{n}|}T_{1}(x\ldots jx_{s},1-.$\Sigma$_{i--1}^{s}x_{i})(z)}{\partial x_{1^{1}'}^{n}\partial x_{2^{2}}^{n}\cdot\cdot\partial x_{s}^{n_{s}}}|_{\mathrm{x}=\mathrm{p}}, z\in\hat{\mathbb{C}}.

These functionsareintroduced in

[Sum13]

_by

the second author. We introduce the \mathbb{C}‐vector_space

\mathscr{C}:= span

\{\mathrm{C}_{\mathrm{n}}|\mathrm{n}\in \mathrm{N}_{0}^{s}\}\subset C^{a}(\hat{\mathbb{C}})

,

which consists of all the finite

_complex

linear combinations of elements from

_{\{C_{\mathrm{n}}|\mathrm{n}\in \mathrm{N}_{0}^{s}\}}

.The first order derivativesarecalled

_{complex analogues}

ofthe

_Takagi

function in

[Sum13].

Note that

_{C_{0}=T_{\mathrm{p}}.}

ForanelementC\in \mathscr{C}and

z\in\hat{\mathbb{C}}

the Holder_exponentHöl

_(C,z)

is

_given

_by

Höl

_(C,z)

:=\displaystyle \sup\{ $\alpha$\in[0,\infty)

:

where d denotes the

_spherical

distanceon

\hat{\mathbb{C}}

.Itwasshown in

[\mathrm{J}\mathrm{S}15\mathrm{a}]

that the levelsets

H(C_{0}, $\alpha$)

:=

{

z\in\hat{\mathbb{C}}

:

Höl(Co, z)= $\alpha$},

$\alpha$\in \mathbb{R},

satisfy

the multifractal formalism. In

_particular,

there existsaninterval ofparameters

_{($\alpha$_{-}, $\alpha$_{+})}

such that

the Hausdorff dimension of

_{H(C_{0}, $\alpha$)}

is

_positive

and varies real

_analytically

_(see

Theorem 1.2

_below).

The first main result of this paper

gives

a

_dynamical

_description

ofthe

_pointwise

Hölderexponentsforan

arbitrary

C\in \mathscr{C}.Wesay that

C=\displaystyle \sum_{\mathrm{n}\in \mathrm{N}_{0}^{s}}$\beta$_{\mathrm{n}}C_{\mathrm{n}}\in \mathscr{C}

is non‐trivial if there exists

\mathrm{n}\in \mathrm{N}_{0}^{s}

with

$\beta$_{\mathrm{n}}\neq 0

.Ittums

out_{in Theorem 1.1 below that every non‐trivial}C\in \mathscr{C}has thesame

_pointwise

Hölder_exponents.Tostate theresult,wedefine the skew

_product

_map

_(associated

with

(f_{i})_{i\in}

_{) (see [SumOO])}

\tilde{f}:I^{\mathrm{N}}\times\hat{\mathbb{C}}\rightarrow I^{\mathrm{N}}\times\hat{\mathbb{C}}, \tilde{f}( $\omega$,z):=( $\sigma$( $\omega$),f_{0)\mathrm{l}}(z))

,

where

$\sigma$:I^{\mathrm{N}}\rightarrow I^{\mathrm{N}}

denotes the shift map

given by

$\sigma$($\omega$_{1},

$\varpi$_{2},

:=($\omega$_{2},

$\omega$_{3}, for

$\omega$=(\mathrm{t}0_{1},

w,

\in I^{\mathrm{N}}.

For every

$\omega$=($\omega$_{j})_{j\in \mathbb{N}}\in I^{\mathrm{N}}

andn\in \mathrm{N},let

f_{w|_{n}}

:=f_{$\omega$_{n}}\circ

\circ f_{$\omega$_{1}}

andwedenote

by

F_{ $\varpi$}the maximal open

subset of

\hat{\mathbb{C}}

onwhich

\{f_{c\mathrm{o}|_{n}}\}_{n\in \mathrm{N}}

is

_{equicontinuous}

withrespecttod.Let

J_{ $\omega$}

:=\hat{\mathbb{C}}\backslash F_{a\mathrm{J}}

.The Juliasetof

\overline{f}

is

given

by

J(f $\gamma$=\displaystyle \bigcup_{ $\omega$\in I^{\mathrm{N}}}\{ $\omega$\}\times J_{0)}

where the closure is taken in

I^{\mathrm{N}}\times\hat{\mathbb{C}}

.Notethat

denoting

by

$\pi$:I^{\mathrm{N}}\times\hat{\mathbb{C}}\rightarrow\hat{\mathbb{C}}

the canonical

_projection,

_{$\pi$:J(\tilde{f})\rightarrow J(G)}

isa

_{homeomorphism}

_([Sumlla,

Lemma

4.5], [Sum97,

Lemma

1.1.4]

and

_{assumption (2))}

and

_{$\pi$\circ\overline{f}= $\sigma$\circ $\pi$}

.Weintroduce the

potentials \overline{ $\varphi$},

\overline{ $\psi$}:J(\tilde{f})\rightarrow \mathbb{R}

given

by

\tilde{ $\varphi$}( $\omega$,z):=-\log\Vert f_{0 $\eta$}'(z)\Vert

, $\iota$

ỹ(oJ,

z

) :=\log p_{$\omega$_{1}},

where

_\Vert

.

|

denotes thenormofthe derivative with_respecttothe

_spherical

metricon

\hat{\mathbb{C}}

.Notethat

\tilde{f}^{-1}(J(\tilde{f}))=

J(\tilde{f})=\tilde{f}(J(\overline{f}))

([Sum00]).

We denote

_by

S_{n}the

_ergodic

sumof the

_dynamical

system

(J(\hat{f}),f $\gamma$.

Theorem 1.1. For_everynon‐trivial

_{C=\displaystyle \sum_{\mathrm{n}\in \mathrm{N}_{0}^{s}}$\beta$_{\mathrm{n}}C_{\mathrm{n}}\in \mathscr{C}}

wehave

(1.1)

Höl(C,z)

=\displaystyle \lim_{k\rightarrow}\inf_{\infty}\frac{S_{k}\tilde{ $\psi$}( $\omega$,z)}{S_{k}\overline{ $\varphi$}( $\omega$,z)}

,

forall

( $\omega$,z)\in J(\tilde{f})

.

Combining

Theorem 1.1 withourresults from_[\mathrm{J}\mathrm{S}15\mathrm{a},Theorem

1.2]

onthe multifractalformalism, we

estabhsh the multifractal formalism for the

_pointwise

Hölder_exponentsofan

_arbitrary

non‐trivialC\in \mathscr{C}.

Tostatetheresults,for any non‐trivialC\in(\mathscr{E}and $\alpha$\in \mathbb{R}wedenote

by

H(C, $\alpha$):= {

y\in\hat{\mathbb{C}}

:Höl

_{(C,y)= $\alpha$}}

the levelsetof

_prescribed

Hölder_exponent $\alpha$.The range of the multifractalspectrumis

given

by

$\alpha$_{-}:=\displaystyle \inf\{ $\alpha$\in \mathbb{R}:H(C, $\alpha$)\neq\emptyset\}\in \mathbb{R}

and

_{$\alpha$_{+}:=\displaystyle \sup\{ $\alpha$\in \mathbb{R}:H(C, $\alpha$)\neq\emptyset\}\in \mathbb{R}.}

By

Theorem 1.1,thesets

_{H(C, $\alpha$)}

coincide for all non‐trivialC\in \mathscr{C}. Thus, $\alpha$_{-}and $\alpha$_{+} donot

depend

on the choice ofanon‐trivialC\in \mathscr{C}.Also,$\alpha$_{-}>0

([Sum98,

Theorem2.6],seealso

Corollary

1.11).

Theorem 1.2. All

_of

the

_following

hold.

(1)

LetC\in \mathscr{C}be non‐trivial.

_{If $\alpha$_{-}<$\alpha$_{+}then}

the

_{Hausdolffdimensionfunction}

_{$\alpha$\mapsto\dim_{H}(H(C, $\alpha$))}

,

$\alpha$\in($\alpha$_{-}, $\alpha$_{+})

,

defines

areal

analytic

and

strictly

concave

positivefunction

on

($\alpha$_{-}, $\alpha$_{+})

withmax‐

imum value

_{\dim_{H}(J(G))}

.

If

$\alpha$_{-}=$\alpha$_{+},thenwehave

H(C, $\alpha$_{-})=J(G)

.

(2)

Wehave

_{a_{-}= $\alpha$+if}

and

_{only if}

there existan

_automorphism

$\theta$\in \mathrm{A}\mathrm{u}\mathrm{t}(\hat{\mathbb{C}})

,

complex

numbers

(a_{i})_{i\in I}

and $\lambda$\in \mathbb{R}such

_thatfor

all i\in I and

z\in\hat{\mathbb{C}}_{J}

$\theta$\circ f\circ$\theta$^{-1}(z)=a_{i}z^{\pm\deg(f)}

and

_logdeg

_{(f_{i})= $\lambda$\log p_{i}.}

Theorem 1.3. For_everynon‐trivialC\in \mathscr{C}

_andfor

_every $\alpha$<$\alpha$_{-},

thefunction

Cis

$\alpha$-Ho7der

continuous

on

\hat{\mathbb{C}}

.Moreover,

C_{0}

is$\alpha$_{-}‐Hölder continuouson

\hat{\mathbb{C}}.

To prove Theorem 1.3we

_develop

someideas from

[KS08,

JKPS09]for interval maps. The relation between the Hölder

_continuity

of

_singular

measuresand their multifractal_spectrahas been first observed in_[KS08],

where itwasshown that the Hölder

_continuity

ofthe Minkowski’s

question

mark function coincides with the bottom of the

_Lyapunov

spectrumof the

_Farey

_{map. In}[JKPS09]asimilar result hasbeen obtained for

expanding

interval maps.

JJ] the

_following

Theorem 1.4we_{prove that $\alpha$_{-}<1}. This result allowsusto

give

a

complete

answerto

two

_{important problems}

raised in

[Sum13],

which

_{greatly improves}

the

_{previous partial}

results in

[Sumlla,

Sum13,

\mathrm{J}\mathrm{S}15\mathrm{a}]

.The first

implication

is

that,

under the

assumptions

ofourpaper, every non‐trivial C\in \mathscr{C} is

notdifferentiableat_every

_point

ofaBorel dense subset A of

_J(G)

with

_{\dim_{H}(A)>0}

.

Secondly,

weobtain in Theorem 1.5 that the

averaged

systemstillacts

_chaotically

on_{the space}

C^{ $\alpha$}(\hat{\mathbb{C}})

_{for any}

$\alpha$\in($\alpha$_{-}, 1)

,

although

the

_averaged

_systemacts_very

_mildly

on_{the Banach space}

C(\hat{\mathbb{C}})

of\mathbb{C}‐valued continuous functions

on

\hat{\mathbb{C}}

_{endowed with the supremum}normandonthe Banach space

C^{ $\alpha$}(\hat{\mathbb{C}})

for small $\alpha$>0

_{(see [Sum97,}

Lemma

1.1.4],

[Sumlla,Theorem

3.15]

and_[Sum13,Theorem

1.10]).

We recall that if

_Höl(C,

z

)

<1 then

Cisnotdifferentiableatz.If

Höl(C,

z

)

>1 then C is differentiableatzand the derivative of Cat \mathrm{z}iszero. Theorem 1.4. We have a_{-}<1. Moreover,

for

every

$\alpha$\in($\alpha$_{-}, 1)

there existsaBorel dense subsetA

ofJ(G)

with

_{\dim_{H}(A)>0}

such

_thatfor

_everynon‐trivialC\in \mathscr{C}

_andfor

_{every z\in A}, wehave

Höl(C,

z

)

= $\alpha$<1

and C isnot

_{differentiable}

at \mathrm{z}.

In the

_proof,

wecombine the result that

_{C_{0}}

is$\alpha$_{-}‐Hölder continuouson

\hat{\mathbb{C}}

_{(Theorem 1.3),}

the multifractal

analysis

onthe

_pointwise

Hölder_exponentsof

C_{0} (Theorems

I.2),

an_argumenton

Lipschitz

functionson

\mathbb{C} and the fact that

_{\dim_{H}(J(G))<2}

,which follows fromour

assumptions (1)

and

(2)

([Sum98]).

TostateTheorem 1.5,let M:

C(\hat{\mathbb{C}})\rightarrow C(\hat{\mathbb{C}})

be the transition_operatorofthe _systemwhich is defined

_Uy

M( $\phi$)(z)=\displaystyle \sum_{j=1}^{s+1}p_{j} $\phi$(f_{j}(z))

,where

$\psi$\in C(\hat{\mathbb{C}}),z\in\hat{\mathbb{C}}

.Note that

M(C^{ $\alpha$}(\hat{\mathbb{C}}))\subset C^{ $\alpha$}(\hat{\mathbb{C}})

for any

$\alpha$\in(0,1].

Theorem 1.5. Let

_{$\alpha$\in($\alpha$_{-}, 1)}

and let

$\phi$\in C^{ $\alpha$}(\hat{\mathbb{C}})

such that

_{$\phi$|_{L}=1}

and

_{$\phi$|_{L'}=0}

_for

_everyminimalset L'

_of

G with

_{L'\neq L}

. Then

\Vert M^{n}( $\phi$)\Vert_{ $\alpha$}\rightarrow\infty

as n\rightarrow\infty.In

particular, for

every

$\xi$\in C^{ $\alpha$}(\hat{\mathbb{C}})

and

for

every

a\in \mathbb{C}\backslash \{0\}

,wehave

\Vert M^{n}( $\xi$+a $\phi$)-M^{n}( $\xi$)\Vert_{ $\alpha$}\rightarrow\infty

as n\rightarrow\infty.

Wenow_presentthe corollaries ofourmain results. The firstone_{establishes that every non‐trivial}C\in \mathscr{C}

varies

_precisely

onthe Juliaset

_J(G)

. This follows

immediately

from Theorem 1.1 because the

right‐hand

side of

(1.1)

is

_always

finite

([Sum98,

Theorem

2.6],

seealso

Corollary

_1.11).

This

_generalises

a

_previous

result from

[Sumlla]

for

_{C_{0}=T_{\mathrm{p}}}

anda

_partial

result for the

_higher

order

_pamal

denvatives fiiom

[Sum13].

Corollary

1.6.

_Every

non‐trivialC\in \mathscr{C}varies

_precisely

on

J(G)

, i.e.,

J(G)

is

equal

to theset

ofpoints

Z0\in\hat{\mathbb{C}}

such that C isnot constant_{in any}

_{neighborhood of}

z\mathrm{o}in

\hat{\mathbb{C}}

.In

particular, thefunctions

C_{\mathrm{n}},\mathrm{n}\in \mathrm{N}_{0}^{s},

are

_{linearly independent}

over\mathbb{C} and (g hasa

_{representation}

as adirectsum

of

vectorspaces

given by

\displaystyle \mathscr{C}=\bigoplus_{\mathrm{n}\in \mathrm{N}_{0}^{s}}\mathbb{C}C_{\mathrm{n}}.

Weremark

_again

that

_{0<\dim_{H}(J(G))<2}

_([Sum98]).

By combining

Theorem 1.1 with Birkhoff’s

_ergodic

theoremweobtain the

_following

extension

of[Sum13,

Theorem 3.40

(2)].

Recall thataBorel

_probability

measurev on

_J(f)

is called

_\overline{f}

‐invmant if

_{v(\overline{f}^{-1}(A))=}

Corollary

1.7. Letvbean

\overline{f}

‐invanant

_ergodic

Borel

_probability

measure on

J(\hat{f})

. Let

$\pi$:I^{\mathrm{N}}\times\hat{\mathbb{C}}\rightarrow\hat{\mathbb{C}}

denote the canonical

_projection

onto

\hat{\mathbb{C}}

Then there existsaBorel subset A

of

J(G)

with

($\pi$_{*}(v))(A)=1

such that

_for

_everynon‐trivialC\in \mathscr{C}and

_for

_{every z\in A},wehave

Höl

(C,z)=\displaystyle \frac{-\int\log p_{$\omega$_{1}}dv(0),x)}{\int\log\Vert f_{$\omega$_{1}}(x)||dv( $\omega$,x)}

, where

$\omega$=(0)_{1},

$\Phi$,

)

\in I^{\mathrm{N}}.

By

combining Corollary

1,7 with

_[Sumlla,

Theorem

3.82]

in which the

_potential

_theory

was_used, we

obtain the

_following

result

_(Corollary

1.8)

onthe

pointwise

Hölder_exponentsand the

non‐differentiauility

of elements of\mathscr{C}. Tostatetheresult,when G isa

polynomial semigroup,

wedenote

by

\tilde{ $\mu$}_{\mathrm{p}}

the maximal relative _entropymeasure on

J(f)

for

\tilde{f}

with_respectto

_{( $\sigma$,\tilde{ $\rho$}_{\mathrm{p}})}

_{(see [Sum00], [Sumlla,}

Remark

3.79]).

Notethat

_{\tilde{ $\mu$}_{\mathrm{p}}}

is

_\tilde{f}

‐invanant and

_{ergodic ([Sum00]).}

Let

_{$\mu$_{\mathrm{p}}=$\pi$_{*}(\overline{ $\mu$}_{\mathrm{p}})}

.For any

( $\omega$,z)\in I^{\mathrm{N}}\times\hat{\mathbb{C}}

,let

\mathscr{G}_{ $\varpi$}(z)

:=

\mathrm{h}\mathrm{m}_{n\rightarrow\infty}(1/\deg(f_{ $\omega$|_{n}}))\log^{+}|f_{o\mathrm{J}|_{n}}(z)|

, where

\log^{+}(a)

:=\displaystyle \max\{\log a,0\}

for every a>0.

By

theargumentin

[SesOl],

wehave that

\mathscr{G}_{ $\omega$}(y)

_{exists for every}

( $\omega$,z)\in I^{\mathrm{N}}\times \mathbb{C}, ( $\omega$,z)\in I^{\mathrm{N}}\times \mathbb{C}\mapsto \mathscr{G}_{ $\varpi$}(z)

is continuouson

I^{\mathrm{N}}\times

\mathbb{C},

g_{ $\omega$}

is subharmonicon\mathbb{C} and

_{y_{0\mathrm{J}}}

restrictedtothe intersection of \mathbb{C} and the basin

_{A_{\infty, $\omega$}}

of\infty_for

\{f_{w|_{n}}\}_{n=1}^{\infty}

isthe Green’s functionon

_{A_{\infty, $\omega$}}

with

_pole

at\infty.Let

$\Lambda$( $\omega$)=\displaystyle \sum_{c}\mathscr{G}_{ $\omega$}(c)

,wherecruns overall critical

points

of

_{f_{o\mathrm{J}_{1}}}

in

_{A_{\infty, $\omega$}}

,

counting multiplicities.

Note that

$\mu$_{\mathrm{p}}=\displaystyle \int_{I^{\mathrm{N}}}dd^{c}\mathscr{G}_{0)}d\tilde{p}_{\mathrm{p}}(\mathrm{t}0)

where

d^{C}=(\sqrt{-1}/2 $\pi$)(\overline{\partial}-\partial)

([Sumlla,

Lemma

_5.51]),

_supppp

_=J(G)

and_{$\mu$_{\mathrm{p}}}is non‐atomic

_([Sum00]).

_Also,wehave

\dim_{H}($\mu$_{\mathrm{p}})=

(

\displaystyle \sum_{i\in I}p_{i}\log\deg f-\sum_{i\in J}

pilog

p_{j}

) / (

_{\displaystyle \sum_{i\in l}p_{i}}

logdeg

f_{i}+\displaystyle \int_{I^{\mathrm{N}}} $\Lambda$( $\omega$)d\tilde{ $\rho$}_{\mathrm{p}}(0)

))

>0

([Sumlla,

Proof of Theorem

3.82]).

Here,

_{\dim_{H}($\mu$_{\mathrm{p}})}

:=\displaystyle \inf\{\dim_{H}(A)\}

where the infimum is takenoverall Borel subsets A of

J(G)

with

$\mu$_{\mathrm{p}}(A)=1.

Corollary

1.8.

(1) Suppose

that

_{f_{1}}

,\cdots_,

f_{s+1}

are

polynomials.

Then thereexistsaBorel dense subsetA

of

J(G)

with

_{$\mu$_{\mathrm{p}}(A)=l}

and

_{\dim_{H}(A)\geq (}

_{$\Sigma$_{i\in I}p_{i}}

_logdeg

_{f_{i}-\displaystyle \sum_{i\in I}}

_pilog

p_{i}

) / (\displaystyle \sum_{i\in I}p_{i}

logdeg

f_{i}+\displaystyle \int_{I^{\mathrm{N}}} $\Lambda$( $\omega$)d\tilde{ $\rho$}_{\mathrm{p}}( $\omega$)

)

>0 such

thatfor

every non‐trivialC\in \mathscr{C}

andfor

everyz\in A,wehave

Höl

(C,z)=\displaystyle \frac{-$\Sigma$_{i\in I}p_{i}\log p_{i}}{$\Sigma$_{i\in I}p_{i}\log\deg f_{i}+\int_{I^{\mathrm{N}}} $\Lambda$( $\omega$)d\overline{p}_{\mathrm{p}}( $\omega$)}\prime.

(2) Suppose

that

_{f_{1}}

,\cdots,

f_{s+1}

are

polynomials satisfying

atleastone

of

the

following

conditions:

(a)

\displaystyle \sum_{i\in IPi}\log(Pi\log f_{i})>0.

(b)

G=(f_{1},

\ldots,f_{s+1}\}

is

_{postcntically}

bounded,i.e.

_{P(G)\backslash \{\infty\}}

isboundedin\mathbb{C}.

\prime(c)s=1.

Then there existsaBorel dense subsetA

of

J(G)

with

$\mu$_{\mathrm{p}}(A)=1

such that

_for

_everynon‐trivialC\in \mathscr{C}

andfor

everyz\in A, wehave Höl

(C,z)<1

. In

particular,

everynon‐trivialC\in \mathscr{C}is

non‐differentiable

$\mu$_{\mathrm{p}}‐almostey

erywhere

on

J(G)

.

Note that ifwe assumethat every

_{f_{i}}

isa

_polynomial

and

_{P(G)\backslash \{\infty\}}

isbounded in \mathbb{C},then

$\Lambda$( $\omega$)=0

for

every0)

\in I^{\mathrm{N}}

,thus

Corollary

1,8

implies

that there existsaBorel dense subset A of

J(G)

with

$\mu$_{\mathrm{p}}(A)=1, \displaystyle \dim_{H}(A)\geq 1+\frac{-\sum_{i\in I}p_{i}\log p_{i}}{\sum_{i\in I}p_{i}\log\deg(f_{i})}>1

such that for every non‐trivialC\in \mathscr{C}and for every

point

z\in A,wehave

Höl(C,

z

)

=\displaystyle \frac{-\sum_{i\in I}p_{i}\log p_{i}}{\sum_{i\in I}p_{i}\log\deg(f_{i})}<1.

The

_following

is one of the other

important applications

of

_Corollary

1.7. In order to state the res‐

ult,

let $\delta$

_{:=\dim_{H}(J(G))}

and let

H^{ $\delta$}

denote the $\delta$‐dimensional Hausdorffmeasure on

\hat{\mathbb{C}}

. Notethat

by

[Sum05],

wehave

_{0<H^{ $\delta$}(J(G))<\infty}

.Let

C(J(G))

be the space of all continuous \mathbb{C}‐valued functions

\displaystyle \sum_{i\in I}\sum_{f $\iota$(y)=z} $\phi$(y)\Vert f_{i}'(y)\Vert^{- $\delta$}

where

_{$\phi$\in C(J(G)),z\in J(G)}

.

By [Sum05] again,

wehave that

$\gamma$=\displaystyle \lim_{n\rightarrow\infty}L^{n}(1)

\in C(J(G))

exists,where 1 denotes theconstantfunctionon

J(G)

_taking

its value_1,the function

_{$\gamma$ \mathrm{i}\mathrm{s}}

pos‐

itiveon

J(G)

, and there exists an

\tilde{f}

‐invanant

ergodic probability

measure\tilde{v}on

J(f)

such that

$\pi$_{*}(\overline{v})=

$\gamma$ H^{ $\delta$}/H^{ $\delta$}(J(G))

and

_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}$\pi$_{*}(v)=J(G)}

.

By Corollary

1.7 and

[Sumlla,

Theorem 3.84

(5)],

weobtain the

following,

Corollary

1.9. Under the abovenotations, there existsaBorel dense subset A

of

J(G)

with

H^{ $\delta$}(A)=

H^{ $\delta$}(J(G))>0

such

_thatfor

_{every non‐trivial}C\in\subset \mathscr{E}

_andfor

everyz\in A,wehave

Höl(C,

z

)

=\displaystyle \frac{-\sum_{i\in I}\log p_{i}\int_{f_{i}^{-1}(J(G))} $\gamma$(y)dH^{ $\delta$}(y)}{\sum_{i\in I}\int_{f_{i}^{-1}(J(G))} $\gamma$(y)\log||f_{i}(\mathrm{y})\Vert dH^{ $\delta$}(y)}.

Remark 1.10. We remark thata non‐trivial_{C\in \mathscr{C} may possess}

_points

of

_{differentiability.}

In _fact,

_by

choosing

oneofthe

_probability

_parameters

_sufficiently

_small,we candeduce from

_Corollary

1.9 that for every non‐trivialC\in \mathscr{C}and for

H^{ $\delta$}

‐almost every

z\in J(G)

,wehave Höl

(C,z)>1,

C is differentiableat

zand the derivative of Cat ziszero. Note thatevenunder the abovecondition,Theorem 1.4

_implies

that

there existan $\alpha$<1 andadense subset A of

J(G)

with

\dim_{H}(A)>0

such that for every non‐trivialC\in \mathbb{C}

and for everyz\in A,wehave

Höl(C,

z

)

= $\alpha$<1 and C isnotdifferentiableatz.Jn

particular,

in thiscase,

wehave

_{$\alpha$_{-}<1<$\alpha$_{+}}

andwehaveadifferent kind of

_{phenomenon regarding}

the

_(complex)

_analogues

of the

_Takagi

function,whereas the

_{original Takagi}

function doesnothave this_property.

We also have the

_following

_corollary

of Theorem 1.1. Tostatethe_result,

_by

_[Sum98,Theorem

2.6]

there exists

_{k0\in \mathbb{N}}

_{such that for every}

_{k\geq k0}

_{and for every}

_{$\omega$=($\omega$_{\mathrm{i}})_{i=1}^{k}\in l^{k}}

,wehave

\displaystyle \min_{z\in f_{\text{の}}^{-1}(J(G))}\Vert f_{ $\omega$}'(z)\Vert>1,

where

_{f\text{の}=f_{$\omega$_{k}}\circ\cdots\circ f_{o\mathrm{J}_{1}}}

.Letp_{ $\omega$}:=p_{$\omega$_{k}}\cdots p_{$\omega$_{1}}for

$\omega$=(0 $\lambda$)_{i=1}^{k}\in I^{k}.

Corollary

1.11. For_every

_{k\geq k_{0}}

,wehave

0<\displaystyle \min_{ $\omega$\in J^{k}}\frac{-\log p_{ $\omega$}}{\log\max_{z\in f_{i\mathrm{J}}^{-1}(J(G))}\Vert f_{ $\omega$}'(z)\Vert}\leq$\alpha$_{-}\leq$\alpha$_{+}\leq\max_{0)\in J^{k}}\frac{-\log p_{(i\mathrm{J}}}{\log\min_{z\in f_{\overline{ $\varpi$}}^{1}(J(G))}\Vert f_{ $\omega$}'(z)\Vert}<\infty.

In

_{particular, if}

_{p_{i}\displaystyle \min_{z\in f_{\mathrm{i}}^{-1}(J(G))}\Vert}

_fí’(z)

_||

>1

_for

everyi\in I,then

for

every non‐trivialC\in \mathscr{C}and

for

every

z\in J(G)

,wehave that

Höl(C,

z

)

\leq$\alpha$_{+}<1

and C.isnot

differentiable

atz.

Remark 1.12. Under

_{assumptions (1)(2)(3),}

_{suppose that the maps}

_{f_{i},i\in I}

,are

polynomials.

Then

J(G)\subset

\mathbb{C}. Since the

spherical

metric and the Euclidian metricare

equivalent

on

J(G)

,it follows thatwe can

replace

\Vert\cdot\Vert

in the definition of $\varphi$,Corollaries1.7, 1.9,1.11

by

the modulus

|\cdot|.

Remark 1.13. The function

_{C_{0}=T_{\mathrm{p}}}

is continuous

_(in

fact,it is Hölder

_continuous)

on

\hat{\mathbb{C}}

and varies

precisely

onthe Juliaset

_J(G)

.Note that

by assumptions (1)(2)

and

[Sum98],

wehave that

J(G)

isafractalsetwith

0<\dim_{H}\{J(G))<2

. The function

C_{0}

canbe

interpreted

as a

complex analogue

of the devil’s staircase

and

_{Lebesgue’s singular}

functions

_([Sumlla]).

Infact,the devil’s staircase is

_equal

tothe restrictionto

[0

,1

]

of the function of

probability

of

tending

to+\inftywhenweconsider random

dynamical

systemon\mathbb{R}

such thatat_everystepwechoose

_{f_{1}(x)=3x}

with

_probability

_1/2

and we choose

f_{2}(x)=3x-2

with

probability

1/2.

Similarly, Lebesgue’s

singular

function

_{L_{p}}

with_respecttotheparameter

p\in(0,1),p\neq

1/2

is

_equal

tothe restrictionto

_[0

,1

]

of the function of

probability

of

tending

to+\infty whenweconsider

random

_dynamical

_systemon\mathbb{R} such thatat_everystepwechoose

g_{1}(x)=2x

with

_probability

pandwe

choose

_{g_{2}(x)=2x-1}

with

_probability

_1-p

.Notethat theseare new

interpretations

ofthe devil’s staircase

and

_{LeUesgue’s singular}

functions obtained in

_{[Sumlla] by}

_{the second author of this paper.}

_Similarly,

itwas

_pointed

out

_by

him that the distributional functions of self‐similarmeasuresof FSs of orientation‐

preserving contracting diffeomorphisms h_{i}

on\mathbb{R}canbe

_interpreted

asthe functions of

_probability

of

_tending

to+\infty

regarding

the random

_dynamical

_systems

_generated

_by

_{(h_{i}^{-1})}

_([Sumlla]).

From the above

_point

of

view,when G isa

_{polynomial semigroup}

and

_L=\{\infty\}

,wecall

C_{0}=T_{\mathrm{p}}

adevil’s coliseum

([Sumlla]).

It

is well‐known

_([YHK97])

that the function

\displaystyle \frac{1}{2}\frac{\partial L_{p}(x)}{\partial p}|_{p=1/2}

on

[0

,1

]

is

equal

tothe

Takagi

function

$\Phi$(x)=

\displaystyle \sum_{n=0}^{\infty}\frac{1}{2^{n}}\min_{m\in \mathbb{Z}}|2^{n}x-m|

(also

referredtoas the

_{Blancmange function),}

which isafamous

_example

of

acontinuous but nowheredifferentiablefunctionon

[0

,1

]

. From this

point

ofview, the first derivatives

C\in \mathscr{C}canbe

_interpreted

as

_complex

_analogues

ofthe

_Takagi

function. The devil’sstaircase,

Lebesgue’s

singular

functions,the

_Takagi

fmction and the similar functions have been

_investigated

so

_long

in fractal geometryand the related fields. In_fact,the

_graphs

of these functions have certain kind of self‐similarities and these functions have many

interesting

and

_deep

_properties.

Thereare_many

_interesting

studies about the

_{original Takagi}

function and its related

_{topics ([AKII]).}

In [AK06], many

interesting

results

_(e.g.

continuity

and

_{non‐differentiauility,}

Hölder_order,the Hausdorff dimension of the

_graph,

thesetof

_points

where the functions takeontheir absolute maximum and minimum

_values)

ofthe

_higher

order

_partial

derivatives

\displaystyle \frac{\partial^{n}L_{p}(x)}{\partial p^{n}}|_{p=1/2}

of

_{L_{p}(x)}

with_respecttopareobtained. The first

_study

of the

_complex

_analogues

of the

_Takagi

functionwas

_{given by}

the second author in

[Sum13].

In

_particular,

some

_partial

resultson

the

_pointwise

Hölder_exponentsofthemwereobtained

([Sum13,

Theorem

3.40]).

However,it had beenan

open

problem

whether the

_{complex analogues}

ofthe

_Takagi

_{function vary}

_precisely

onthe Juliasetor_not,

until this paperwaswritten. The results of this paper

_{greatly improve}

the above results from[Sum13]. In

the

_prbofs

_{of the results of this paper,}we use

_completely

newideas and

_{systematic approaches}

whichare

explained

below. For the

figures

of the Juliaset

_J(G)

and the

_graphs

of

C_{0}

and

C_{1}

whichwedeal with in this paper whens=1,G isa

_polynomial

_semigroup

and

_L=\{\infty\}

,see

[Sumlla,

Sum13].

Remark 1.14. The resultsonthe classical

_Takagi

functionon

[0, 1|

_give

someevidence that the results stated

in Theorem 1.3are

_sharp.

_Indeed,

letusconsider the function

_{L_{1/2}}

and

$\psi$_{n}(x)=\displaystyle \frac{\partial^{n}L_{p}(x)}{\partial p^{n}}|_{p=1/2}

forn\geq 1.

Note that

_{\displaystyle \frac{1}{2}$\psi$_{1}}

is

_equal

tothe

_{original Takagi}

function. Sincewehave

L_{1/2}|_{[0,1]}(x)=x, L_{1/2}|_{(-\infty,0)}(x)=0

and

_{L_{1/2}|_{(1,\infty)}(x)=1}

, the function

L_{1/2}

is 1‐Hölder

(Lipschitz).

However,in

[AK06]

it is shown that the

functions

_{$\phi$_{n}}

on

[0

,1

]

are a‐Hölder for every a<1,butnot1‐Hölder continuous. It would be

interesting

to

further

_investigate

this

_phenomenon

for the

_complex

_analogues

of the

_Takagi

function.

Remark 1.15. We endow Rat with the

_topology

induced from the distance

\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}_{\mathrm{R}\mathrm{a}\mathrm{t}}

which is defined

_by

\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}_{\mathrm{R}\mathrm{a}\mathrm{t}}(f,g)

_{:=\displaystyle \sup_{z\in\hat{\mathbb{C}}}d(f(z),g(z))}

. Then

by

[Sum97,

Theorem

2.4.1],

the fact

J(G)=\displaystyle \bigcup_{i\in I}f_{i}^{-1}(J(G))

([Sum97,

Lemma

_{1.1.4]), [Sumlla,}

Remark

3.64],

and

[Sum13,

Theorem

_3.24]),

wehave that theset

{

(f_{i})_{i\in I}\in(\mathrm{R}\mathrm{a}\mathrm{t})^{I}

:

\deg(f_{\mathrm{i}})\geq 2(i\in I)

and the conditions

_(1)(2)(3)

hold for

_{(f_{i})_{i\in J}}

_}

is open in

(\mathrm{R}\mathrm{a}\mathrm{t})^{I}

.Also,wehave

plenty

of

examples

towhichwe can

apply

the main results of this paper.

SëeSection 2.

Remark 1.16. We remark that

_Uy

_using

_{the method in this paper,}we canshow similar resultstothose of this paper for random

dynamical

systemsof

_{diffeomorphisms}

on\mathbb{R}

(or

\mathbb{R}\cup\{\pm\infty\}

_).

Note that thecaseof the classical

_Takagi

function $\Phi$

_corresponds

tothe

_degenerated

case _{$\alpha$_{-}=$\alpha$_{+} in Theorem 1.2,}

_though

in thecaseof $\Phi$we_{have the open}setcondition but donothave the

_separating

condition. We

_emphasize

that in this paperwealso deal with the

_{non‐degenerated}

case,whichseems

_generic.

Remark 1.17. We remark that under

_{assumptions (1)(2)(3),}

the iteration of the transition_operator Mon

some

C^{a}(\hat{\mathbb{C}})

is well‐Uehaved

(e.g.,

there existsanM‐invanant finite‐dimensional

_subspace

U of

C^{a}(\hat{\mathbb{C}})

such that for every

h\in C^{a}(\hat{\mathbb{C}})

,

M^{n}(h)

tendsto Uas n\rightarrow\infty

exponentially

fast)

and Mhasa

spectral

gap

on

C^{a}(\hat{\mathbb{C}})

_([Sum97,

Lemma

_{1.1.4(2)], [Sumlla, Propositions}

3.63,

3.65],

[Sum13,Theorems_3.30,

_3.31]).

Note that this is.arandomness‐induced

phenomenon

_{(new phenomenon)}

in random

_dynamical

systems

for every

f\in

Rat with

_{\deg(f)\geq 2}

,the

dynamics,

of

f

on

J(f)

is chaotic.

Combining

the above

spectral

gappropertyof Mon

C^{a}(\hat{\mathbb{C}})

and the

_permrbation

_theory

forlinear_operators

_{([Kato80]) implies}

that the

map

\mathrm{x}=(x\mathrm{l}, \cdots,x_{s})\mapsto T_{(x_{1)}\ldots,\mathrm{x}_{s},1-$\Sigma$_{i=1}^{s}x_{j})}\in C^{a}(\hat{\mathbb{C}})

is

_{real‐analytic}

ina

_{neighbourhood}

_{of \mathrm{p} in the space}

W

_{:=\displaystyle \{(q_{i})_{i=1}^{s}\in(0,1)^{s}:\sum_{i=1}^{s}q_{i}<1\}}

_([Sum13,

Theorem

_3.32]).

_{Thus it is very natural and}

_important

for

the

_study

of the random

_dynamical

_systemtoconsider the

_higher

order

_partial

derivatives of

_{T_{\mathrm{p}}}

with_respect

tothe

_probability

vectors. Moreover,it is very

interesting

that

C_{\mathrm{n}}

isasolution of the functional

_equation

(Id‐M)

(C_{\mathrm{n}})=F

, whereFisafunction associated with lower order

partial

derivativesof

T_{\mathrm{p}}

. Infact,

by

using

the

_spectral

_gap

_properties

ofMon

C^{a}(\hat{\mathbb{C}})

and the_argumentsin the

_proof

of

[Sum13,

Theorem

3.32],

for any

\mathrm{n}\in \mathrm{N}_{0}^{s}\backslash \{0\}

,we canshow that

(I)

C_{\mathrm{n}}is the

unique

continuous solution of the above functional

equation

under the

_boundary

condition

_{C_{\mathrm{n}}|s_{G}=0}

and

(II)

_{C_{\mathrm{n}}=\displaystyle \sum_{j=0}^{\infty}M^{j}(F)}

in

C(\hat{\mathbb{C}})

and in

C^{ $\alpha$}(\hat{\mathbb{C}})

for small

$\alpha$>0.Thus,wehaveasystemof functional

equations

for elements

C_{\mathrm{n}}

.Note that this is the first paperto

investigate

the

_pointwise

Hölder_exponentsand other

_properties

of the

_higher

order

_partial

derivatives

C_{\mathrm{n}}

of the functions

_{T_{\mathrm{p}}}

of

_probability

of

_tending

tominimalsetswithrespecttothe

_probability

_parameters

regarding

random

_dynamical

_systemswhich have several variables of

_probability

_parameters. This isa

completely

new_concept.Infact,evenin the real_line,there has beenno

_{study regarding}

the

_objects

similar

tothe above. Evenmore,inthis paperwedeal with the

_complex

hnear combinations of

_partial

derivatives

C_{\mathrm{n}},whichareofcourse

completely

new

objects

in mathematics

coming naturally

from the

study

of random

dynamical

systemsand fractal_geometry. We also remark that the

_original

_Takagi

function is associated

with

_{Lebesgue’s singular}

_functions,but there has beenno

study

about the

higher

order

partial

derivatives of

the distribution functions of

_singular

measureswith_respecttothe_parameters.

The

_key

in the

_proof

of_{the main results of this paper is}toconsider the_systemof functional

_equations

satis‐ fied

_by

the elements of \mathscr{C}.The

composition

of these

equations along

oruits is best described intermsofan associated matrix

_cocycle

_{A( $\omega$,k)}

.

By using

combinatorialarguments,weshowaformula for the compon‐

entsof the matrix

_{A(\mathrm{o}\mathrm{J},k)}

,andwe

carefully

estimatethe

polynomial

growth

order of thesecomponents,as

k tendsto

_{infinity. Combining}

this withsomecalculations of the determinants of matrices whicharesimilar

tothe Vandermonde

detenninant,

wededuce the linear

_independence

ofthevectors

_{(C_{\mathrm{r}}(a)-C_{\mathrm{r}}(b))_{\mathrm{r}\leq \mathrm{n}}}

for cenain

_points

_{a,b\in J(G)}

whichareclosetoa

given point

x_{0}\in J(G)

. Here, \mathrm{r}\leq \mathrm{n}meansthatr_{i}\leq n_{i}for each i.From the‐linear

independence

of thesevectorswededuce thatacertain linear combination ofvec‐

tors

_{(C_{\mathrm{r}}(a)-C_{\mathrm{r}}(b))_{\mathrm{r}\leq \mathrm{n}}}

_{is bounded away from}zero. This

_gives

usthe upper bound of the

pointwise

Hölder

exponentsofC\in \mathscr{C}.Note that thisargumentis the

key

toprove Theorem 1.1 and it is the crucial

point

to derive that the elementsC\in \mathscr{C}arenot

_locally

constant_{in any}

_point

of the Juliaset

_(Corollary

_1.6).

We

emphasize

that those ideasare_verynewand

they

give

us_strongand

_systematic

toolsto

_analyze

random

dynamical

systems,

singular

functions,fractal functions and other related

_topics.

2. EXAMPLES

In thissection,we

_give

some

_examples

which illustrate the main results of this paper.

For

_{f\in \mathrm{R}\mathrm{a}\mathrm{t}}

,weset

F(f)

:=F

( \{f\rangle),J(f) :=J(\{f))

,and

P(f)=P(\langle f) ).

Wedenote

by

\mathscr{P} thesetof

poly‐

nomials of

_degree

twoor more. For

_{g\in \mathscr{P}}

,wedenote

by

K(g)

the filled‐in Juliaset. If G is arational

semigroup

and if K isa_{non‐empty compact}subset of

\hat{\mathbb{C}}

such that

g(K)\subset K

for each

_{g\in G}

,then Zorn’s

lemma

_implies

that there existsaminimalsetLof G with L\subset K

([Sumlla,

Remark

3.9]).

The

_following

_propositions

showusseveral methodsto

_produce

_many

_examples

of

_(fl,

\cdots,

f_{s+1}

)

\in(\mathrm{R}\mathrm{a}\mathrm{t})^{s+1}

(p_{i})_{i=1}^{s}\in(0,1)^{s}

with

_{\displaystyle \sum_{i=1}^{s}p_{i}<1}

, we can

apply

the results Theorems 1.1, 1.2, 1.3, 1.4, 1.5 and Corol‐

laries 1.6, 1.7,1.9 and 1.11 in Section 1.

Proposition

2.1. Laet

_(gl,

\cdots_, g_{s+1}

)

\in(\mathrm{R}\mathrm{a}\mathrm{t})^{s+1}

with

\deg(g_{i})\geq 2,i=1\ldots,s+1

.

Suppose

that

\langle g_{1},

g_{s+1}

}

is

_hyperbolic,

_{J(g_{i})\cap J(g_{j})=\emptyset}

_for

_every

_(i,j)

with

_{i\neq j}

,and that there existatleasttwodistinct minimal

sets

_of

_{\langle g_{1}}

,\cdots_, g_{s+1}

}.

Then thereexistsm\in \mathrm{N} such

thatfor

everyn\in \mathrm{N}withn\geq m

,

setting

f=g_{i}^{n},i=

1,

s+1,the element

(fl,

\cdots_,

f_{s+1}

)

satisfies assumptions (1)(2)(3) of

thispaper

Proposition

2.2. Let

_(gl,

\cdots_, g_{s+1}

)

\in(\mathrm{R}\mathrm{a}\mathrm{t})^{s+1}

with

\deg(g_{i})\geq 2,

i=1,

s+1.

Suppose

that

\displaystyle \bigcup_{i=1}^{s+1}P(g_{i})\subset

\displaystyle \bigcap_{i=1}^{s+1}F(g_{i})

,that

J(g_{i})\cap J(g_{j})=\emptyset

for

every

(i,j)

with

i\neq j

,and that thereexisttwocompact subsets

K_{1},K_{2}

of

\hat{\mathbb{C}}

with

K_{1}\cap K_{2}=\emptyset

such that

_{g_{i}(K_{j})\subset K_{j}}

_for

_{every i=1},\cdots_, s+1

andfor

\acute{j}=1,2

. Then there exists

m\in \mathbb{N} such that

for

everyn\in \mathbb{N} withn\geq m,

setting

f_{i}=g_{i}^{n},

i=1_,\cdots

, s+1, the element

(fl,

\cdots

,

f_{s+1}

)

satisfies assumptions (1)(2)(3) of

thispaper.

Combimng

[Sumlla,Remark

3.9]

with

_{[Sumlla, Proposition 6.1],}

wealsoobtain the

_following.

Proposition

2.3. Let

_{f_{1}\in \mathscr{P}}

be

_hyperbolic,

i.e.,

P(f_{1})\subset F(f_{1})

.

Suppose

that Int

(K(f_{1}))\neq\emptyset

, where Int

denotes theset

_of

interior

_points.

Let

_{b\in \mathrm{I}\mathrm{n}\mathrm{t}(K(f_{1}))}

be a

_point.

Letd\in \mathbb{N}withd\geq 2.

Suppose

that

(\deg(f_{1}), d)\neq(2,2)

. Then thereexistsanumber c>0 such

thatfor

each

$\lambda$\in\{ $\lambda$\in \mathbb{C}:0<| $\lambda$|<c\}

,

setting

f_{2, $\lambda$}(z) := $\lambda$(z-b)^{d}+b

,wehave

thefollowing.

(1)

(f_{1},f_{2, $\lambda$})

satisfies

assumpnons

(1)(2)(3) of

thispaperwith s=1.

(2) If

J(f_{1})

isconnected,then

_{P(\{f_{1},f_{2, $\lambda$}\})\backslash \{\infty\}}

isbounded in \mathbb{C}.

Thus

_combining

the above with Remark1.15,weobtain that for any

(f_{1},f_{2, $\lambda$})

inthe_above,there existsa

neighborhood

Vof

_{(f_{1},f_{2, $\lambda$})}

in

_(Rat)2

such thatforevery

(g_{1},g_{2})\in V

,

assumptions (1)(2)(3)

of this paper

aresatisfied and Theorems _{1.1, 1.2, 1.3, 1.4,} 1.5 and Corollaries

1.6,

1.7,1.9and 1.11 in Section 1 hold.

Also,

endowing

\mathscr{P} with the relative

_topology

fromRat,wehave that there existsan_open

_neighborhood

W of

_{(f_{1},f_{2, $\lambda$})}

in

\mathscr{P}^{2}

such that for every

(g_{1},g_{2})\in W

and for every

\mathrm{p}=p_{1}\in(0,1)

,

Corollary

1.8holds.

Example

2.4. Let

_{(f_{1},f_{2})\in \mathscr{P}^{2}}

beanelement such that

_{\{f_{1},f_{2}\}}

is

_hyperbolic,

_{P((f_{1},f_{2}))\backslash \{\infty\}}

isbounded in\mathbb{C} and

_{J(\{f_{1},h\rangle}

₎

is disconnected. Note that there are

_plenty

of

_examples

ofsuch elements

_{(f_{1},f_{2})}

(Proposition

2.3,

[Sumllb,

\mathrm{S}\mathrm{u}\mathrm{m}\mathrm{l}5\mathrm{b} Then

_Uy

[Sum09,

Theorems 1.5,

1.7],

wehave that

f_{1}^{-1}(J(G))\cap

f_{2}^{-1}(J(G))=\emptyset

where

_{G=(f_{1},f_{2}}

_}.

Thus

_{(f_{1},h)}

satisfies

_{assumptions (1)(2)(3)}

_{of this paper with s=1}

and \mathrm{a}\mathrm{U} results in Section 1 hold for

_{(f_{1},f_{2})}

_{and for every}

_{\mathrm{p}=p_{1}\in(0,1)}

.

Example

2.5. Let

_{g_{1}(z)=z^{2}-1,g_{2}(z)=z^{2}/4}

,and let

f_{i}=g_{i^{\mathrm{O}}}g_{i}, i=1,2

.Let

\mathrm{p}=p_{1}=1/2

. Let G=

\langle f_{1},h\}

.Then

(f_{1},h)

satisfies the

assumptions

(1)(2)(3)

of this paper with s=1 and

P(G)\backslash \{\infty\}

is bounded

in\mathbb{C}

_{([Sumlla, Example 6.2],[Suml3, Example 6.2]).}

Thus for this

_{(f_{1} ,h)}

,all results of Section 1 hold,

In

_particular,

_everynontrivialC\in \mathscr{C}is Hölder continuouson

\hat{\mathbb{C}}

and varies

_precisely

onthe Juliaset

_J(G)

(Corollary

1.6).

Moreover,

_{Uy Corollary}

1.8,there exists aBoreldense subset A of

_J(G)

with

_{$\mu$_{\mathrm{p}}(A)=}

1,

_{\displaystyle \dim_{H}(A)\geq\dim_{H}($\mu$_{\mathrm{p}})=\frac{3}{2}}

such that for everynontrivialC\in \mathscr{C}and for every z\in A,wehave $\alpha$_{-}\leq

Höl(C,

z

)

=\displaystyle \frac{1}{2}\leq$\alpha$_{+}

and C isnotdifferentiableatz.For the

figures

of

J(G)

and the

graphs

of

C_{0},C_{1}

with

L=\{\infty\}

,see

[Sum13,

Figures

2,3,4].

Notethat Theorem 1.2

implies

that$\alpha$_{-}<$\alpha$_{+}for every

probability

vector

_(parameter)

\mathrm{p}'\in(0,1)

.

Example

2.6. Let $\lambda$\in \mathbb{C}with

_{0<| $\lambda$|\leq 0.01}

and let

_{f_{1}(z)=z^{2}-1,f_{2}(z)= $\lambda$ z^{3}}

.Then

by

[\mathrm{S}\mathrm{u}\mathrm{m}\mathrm{i}\mathrm{l}5\mathrm{a}

,Exam‐

ple 5.4],

the element

_{(f_{1},f_{2})}

satisfies

_{assumptions (1)(2)(3)}

_{of this paper with s=1 and}

_{P((f_{1},f_{2}\rangle)\{\infty\}}

isbounded in \mathbb{C}.Thus all results in Section 1 hold for

(f_{1},f_{2})

and for every

probability

vector

(parameter)

\mathrm{p}=p_{1}\in(0,1)

.Thus, semng

p_{1}=\displaystyle \frac{1}{2}

,

G=\langle f_{1},h)

and

L=\{\infty\}

,every non‐tnvialC\in \mathscr{C}is Hölder contin‐

uous on

\hat{\mathbb{C}}

andvmes

_precisely

on

J(G)

,and

Corollary

1.8

implies

that there existsaBoreldense subset A

of

_J(G)

with

_{$\mu$_{\mathrm{p}}(A)=1}

and

\displaystyle \dim_{H}(A)'\geq 1+\frac{2\log 2}{\log 2+\log 3}

_{1.7737 such that for every non‐mvial}C\in \mathscr{C}and for everyz\in A,wehave $\alpha$‐\leq

Höl(C,

z

)

=\displaystyle \frac{2\log 2}{\log 2+\log 3}(=.0.7737)\leq$\alpha$_{+}

and C isnotdifferentiableatz.Also,

by

Theorem_1.2,wehave $\alpha$_{-}<$\alpha$_{+} for every

\mathrm{p}'.

\in(0,1)

.

Example

2.7. Let

_{g_{1},g_{2}\in \mathscr{P}}

be

_{hyperbolic. Suppose}

that

_{(J(g_{1})\cup J(g_{2}))\cap(P(g_{1})\cup P(g_{2}))=\emptyset, K(g_{1})\subset}

\mathrm{I}\mathrm{n}\mathrm{t}(K(g_{2}))

, and the union of

attracting

cycles

ofg_{2} in\mathbb{C} is included in Int

(K(g_{1}))

. Then

by [Sumlla,

Proposition 6.3],

there existsanm\in \mathbb{N} such that for each n\in \mathbb{N} withn\geq m,

setting

f_{1}=\cdot g_{1}^{n},f_{2}=g_{2}^{n}

,we

have that

_{(f_{1},h)}

satisfies

_{assumptions (1)(2)(3)}

_{of this paper with s=1}.Thus allstatementsofthe results in Section 1 holdfbr

_{(f_{1},h)}

and for every

\mathrm{p}=p_{1}\in(0,1)

.

The

_{following proposition provides}

us amethodto construct

_examples

of

_(fl,

\cdots_,

f_{s+1}

)

\in \mathscr{P}^{s+1}

for which

(1)(2)(3)

hold and

_{P((f_{1}, \ldots,f_{s+1}))\backslash \{\infty\}}

is bounded in\mathbb{C}. For such elements

(fl,

\cdots_,

f_{s+1}

)

andfor every

\mathrm{p}\in(0,1)^{S}

with

_{\displaystyle \sum_{i=1}^{s}p_{i}<1}

,we can

apply

all the results in Section 1.

Proposition

2.8. Let g_{1},

g_{s+1}\in \mathscr{P}

be

_hyperbolic

andsuppose that

J(f_{i})

isconnected

for

everyi= 1,\cdots_, s+1.

Suppose

that

J(f_{i})\subset Int(K(f_{i+1}))

for

everyi=1_,\cdots_, s.

Suppose

also that

\displaystyle \bigcup_{i=2}^{s+1}P(g_{i})\backslash \{\infty\}\subset

Int(K(f_{1}))

.Then there existsanm\in \mathrm{N}such

thatfor

everyn\in \mathbb{N} withn\geq m

,

setting

f=g_{\mathrm{i}}^{n},i=1

,\cdots, s+1,

the element

_(fl,

\cdots_,

f_{s+1}

)

satisfies assumptions (1)(2)(3)

and

P(\{f_{1}, \ldots,f_{s+1}\rangle)

\backslash \{\infty\}

isbounded in\mathbb{C}.

Example

2.9. Let

_{g_{1}(z)=z^{2}-1}

and let

g_{i}(z)=\displaystyle \frac{1}{10i}z^{2},

i=2,\cdots_, s+1. Then

(g_{1}, \cdots,g_{s+1})

satisfies the

assumptions

of

_Proposition

2.8. Note that

z^{2}-1

canbe

_{replaced by}

_any

_hyperbolic

element

_{f\in \mathscr{P}}

with connected Juliasetsuch that

_{J(f)\subset\{z\in \mathbb{C}:|z|<10\}}

and

_{0\in \mathrm{I}\mathrm{n}\mathrm{t}(K(f))}

.

Fromoneelement

(gl,

. g_{m}

) \in(\mathrm{R}\mathrm{a}\mathrm{t})^{m}

which satisfies

assumptions (1)(2)(3) (with

s+1=m

),

weobtain many elements which

satisfy assumptions (1)(2)(3)

ofour_paperasfollows.

Proposition

2.10. Let

_(gl,

. g_{m}

) \in(\mathrm{R}\mathrm{a}\mathrm{t})^{m}

with

\deg(g_{i})\geq 2,i=1

_,. m, andsupposethat

(gl,

\cdots_, g_{m}

)

satisfies

assumptions

(1)(2)(3) of

this_paper Letn\in \mathrm{N} withn\geq 2and let

_{f_{1}}

,\cdots_,

f_{s+1}

be

mutually

distinct

elements

_of

_{\{g_{0\}_{l}}\mathrm{o}\cdots \mathrm{o}g_{a11}| ($\omega$_{1}, \cdots, w)\in\{1, m\}^{n}\}}

wheres\geq 1. Thenwehave the

following.

(I)

(fl,

\cdots

,

f_{s+1}

)

satisfies assumptions (1)(2)(3) of

thispaper Thus allstatementsin Theorems1.1,

1.2, 1.3, 1.4, 1.5 and Corollaries1.6, 1.7, 1.9 and 1.1l in Section 1

_holdfor

_(fl,

\cdots_,

f_{s+1}

),

for

every minimalsetL

_of

_{\{f_{1}}

, ,

f_{s+1}\rangle

and

for

every

\mathrm{p}=(p_{1}, \ldots,p_{s})\in(0,1)^{s}

with

\displaystyle \sum_{i=1}^{s}p_{i}<1.

(II) If,

in additiontothe

_assumption,

_(fl,

\cdots_,

f_{s+1}

)

\in \mathscr{P}^{s+1}

,thenstatement

(1)

in

Corollary

1.8 holds

for

(fl,

\cdots_,

f_{s+1}

)

andfor

every\mathrm{p}, andstatement

(2)

in

Corollary

1.8 holds

for

(fl,

\cdots_,

f_{s+1}

)

and

for

every\mathrm{p}

provided

thatone

_{of (a)(b)(c)}

inthe

_{assumption of Corollary}

1.8

₍₂₎

holds.

(m) If

inadditiontothe

_{assumption ofourproposition,}

_(gl,

\cdots_, g_{m}

)

\in \mathscr{P}^{m}and

P((g_{1}, \ldots,g_{m}\rangle)\backslash \{\infty\}

isbounded in\mathbb{C}then

_{P(\langle f_{1},\ldots,f_{s+1}\})\backslash \{\infty\}}

isbounded in \mathbb{C}.Thus,

statement(2)

in

Corollary

1.8

holdsfor

(fl,

\cdots_,

f_{+1}

)

andfor

every\mathrm{p}.

Regarding

Remark1.15,wealso have the

_following.

Lemma 2.11. Lets\geq 1and

_{letl=\{1, s+1\}}

. Then theset

{

(f_{i})_{i\in I}\in \mathscr{P}^{I}

:

(f_{i})_{i\in I}

_{satisfies assumptions (1)(2)(3)}

and

_{P( (}

_{f_{1},}

\ldots,

f_{s+1}\})\backslash \{\infty\}

isboundedin\mathbb{C}

}

We remark that the above

_{examples, propositions}

and lemma in this section and Remark 1.15

_imply

thatwe

have

_plenty

of

_examples

towhichwe can

_apply

the results in Section 1. We

_{give examples}

towhichwe can

_{apply Corollary}

1.11.

Lemma 2.12. Let

_(gl,

\cdots_, g_{s+1}

)

beanelement which

satisfies assumptions (1)(2)(3).

Let

\mathrm{p}=(p_{i})_{i=1}^{s}\in

(0,1)^{S}

with

_{\displaystyle \sum_{i=1}^{s+1}p_{i}<1}

.Let

p_{s+1}=1-\displaystyle \sum_{i=\mathrm{i}}^{s}p_{i}

.Then there existsanm\in \mathrm{N} such that

for

every n\in \mathbb{N} with

n\geq m,

setting

f=g_{i}^{n},i=1\ldots,s+1

,and

setting

G

:=\{f_{1}

,\cdots_,

f_{s+1}

),

wehave that

(f_{1}, f_{s+1})

satisfies

assumptions

(1)(2)(3)

and

_{p_{i}\displaystyle \min_{z\in f_{i}^{-1}(J(G))}\Vert}

_fí

_(z)\Vert>1

_for

_every

_i=1,

s+1. Thus,

for

everyminimalset

L

_of

_{\langle f_{1}}

,\cdots_,

f_{s+1}\rangle

,

andfor

every

z\in J(G)

,wehave that every non‐trivialC\in \mathscr{C}

satisfies

Höl(C,

z

)

\leq$\alpha$_{+}<1

and C isnot

_{differentiable}

atz.

REFERENCES

[AK06] P. Allaartand K.Kawamura,Extreme valuesofsomecontinuous nowhere_{differentiablefunctions,}Math. Proc._Cambridge Philos. Soc. 140(2(\mathrm{K})6),no.2,269‐295.

[AKII] P. AJlaart and KKawamura,TheTakagifunction:a_{survey, Real Anal.Exchange}37(2011‐12),no.1,1‐54. MR 3016850

[FS91] J. E. Fomzess and N._Sibony,Random iterations_{of rationalfunctions, Ergodic Theory Dynam. Systems}11(1991),no._4,

687‐708. MR 1145616(93c:58l73)

[GR96] Z._Gongand F.Ren,Arandomdynamicalsystemformed by infinitely manyfunctions,J.Fudan Univ. Nat. Sci. 35(1996),

no.4,387‐392. MR 1435167_(98k:30032)

[HM96] A. Hinkkanen and G. J.Martin,The_{dynamics ofsemigroups ofrationalfunctions.}LProc. London Math. Soc.(3)73(1996),

no.2,358‐384. MR 1397693(97e:58l98)

[JKPS09]T.Jordan,M.Kesseb6hmer,M.Pollicott,and B. O.Stratmann,Sets_{ofnondifferentiabilityfor conjugacies}between_expand‐

inginterval maps, Fund. Math. 206(2009),161‐183. MR 2576266

[JS15a] J.Jaerisch and H.Sumi,_{Multifractalformalismfor expanding}rationalsemigroupsand randomcomplex dynamicalsystems, Nonlinearity28(2015),2913‐2938.

[JS15b] J. Jaerisch and H.Sumi,Dynamics ofinfinitely generated nicely.expandingrational_semigroupsand the_{inducing method,}to

appear in Trans. Amer. Math.Soc.,http://arxiv.oryabsi1501.06772.

[JS16] J. Jaerisch and H.Sumi,Pointwise HölderExponents ofthe_{Complex Analogues of}the_TakagiFunction in Random_Complex

Dynamics, preprint2016,https://arxiv.oryabs/1603.08744.

[KatoSO] T.Kato,Perturbation_TheoryforLinearOperators,Classics inMathematics,Springer,1980.

[KS08] M. Kesseböhmer and B. O.Stratmann,Fractal_analysisforsets_ofnon_{differentiability of}minkowski‘squestion markfunc‐ tion,Joumal of Number_Theory128(2008),2663‐2686.

[Roc70] R. T.Rockafellar,Convex_analysis,Princeton MathematicalSeries,No.28, Princeton_UniversityPress,Princeton,N.J.,

1970. MR MR0274683(43#445)

[SesOl] O._Sester,Combinatonal_{configurations offiberedpolynomials, Ergodic Theory Dynam. Systems}21(2001),915‐955.

[Sta12] R._{Stankewitz, Density of repellingfxed points}inthe Juliaset_ofarationalorentiresemigroup,Il,Discrete Contin._Dyn.

Syst.32(2012),no.7,2583‐2589. MR 2900562

[SSII] R.Stankewitz and H.Sumi,Dynamical propertiesandstructure_ofJuliasets_{ofpostcriticauy}boundedpolynomialsemi‐

groups, Trans. Amer. Math. Soc. 363(2011)5293‐5319.

[Sum97] H.Sumi,On_{dynamics ofhyperbolic}rationalsemigroups,J.Math.Kyoto.Univ. Vol.37,No.4,(1997),717‐733.

[Sum98] H.Sumi,On_{Hausdorffdimension of}Juliasets_ofhyperbolicrational_semigroups,Kodai Math. J. 21_(1998),no.1,1\mathrm{f}\succ 28.

MR 1625124(99h:30029)

[SumOO] H.Sumi,Skew_{productmops related}to_{finitely generated}rational_{semigroups, Nonlinearîty}13(2000),no.4,995−1019..

[SumOl] H.Sumi,Dynamics ofsub‐hyperbolicand_{semi‐hyperUolic}rationalsemigroupsand skewproducts,ErgodicTheory Dynam. Systems21_(2001),no._2,563‐603. MR 1827119_{(2002d:37073a)}

[Sum05]H.Sumi,DimensionsofJuliasets_ofexpandingrationalsemigroups,Kodai Math. J. 28_(2005),no.2,390‐422. MR 2153926

(2006c:37050)

[Sum09] H.Sumi,Interaction_{cohomology offorward}or_{backwardself‐similarsystems, Adv.Math.,}222(2009),no.3,729‐781.

[SumlO] H.Sumi,Dynamics ofpostciticallybounded_{polynomial semigroups}m.._{classification ofsemi‐kyperbolic semigroups}and

random JuliasetswhichareJordancurvesbutnot_{quasicircles, Ergodic Theory Dynam. Systems}30(6) (2010)1869‐1902.

[Sumlla] H.Sumi,Random_{complex dynamics}and_{semigroupsofholomorphicmaps, Proc. London Math. Soc.(1) (2011),}no.102,

[Sumllb]H.Sumi,Dynamics ofpostcnticallybounded_{polynomial semigroupsJ.. connected componentsof}the Julia sets, Discrete

and Continuous_{Dynamical Systems}SeriesA,Vol.29,No.3, 2011,1205‐1244.

[Sum13] H.Sumi,Cooperation principle, stabilityand_bifurcationinrandom_{complex dynamics,}Adv. Math.245(2013),137‐181.

[Sumi15a] H.Sumi,Random_{complex dynamics}and devil’scoliseums,Nonlinearity28(2015)1135‐1161.

[Sum15b] H.Sumi,The spaceof 2‐generator postcriticallyboundedpolynomial semigroupsand random_{complex dynamics,}Adv.

Math.,290(2016),809‐859.

[SU12] H. Sumi and M.Urbafiski,Bowen paroeoeter and Hausdo/ffdimensionfor expandingrational_semigroups,Discrete Contin.

Dyn. Syst.32(2012),no.7,2591‐2606. MR 2900563

[SU13] H. Sumi and M._{Urbański, Transversalityfamily ofexpanding}rational_semigroups,Adv. Math. 234

, (2013)697‐734. [Wa182] P.Walters,An introductionto_{ergodic theory,}Graduate Texts inMathematics,vol.79.Springer‐Verlag,NewYork,1982.

MR MR648108(84e:280l7)

[YHK97] M._Yamaguti,M.Hata,and J._Kigami,Mathematics_offractals,Translations of Mathematical_Monographs167(American