Tiểu Luận On the dimensions of conformal repellers. Randomness and parameter dependency

On the dimensions of conformal repellers.
Randomness and parameter dependency
By Hans Henrik Rugh
Abstract
Bowen’s formula relates the Hausdorff dimension of a conformal repeller to
the zero of a ‘pressure’ function. We present an elementary, self-contained proof
to show that Bowen’s formula holds for C
1
conformal repellers. We consider
time-dependent conformal repellers obtained as invariant subsets for sequences
of conformally expanding maps within a suitable class. We show that Bowen’s
formula generalizes to such a repeller and that if the sequence is picked at
random then the Hausdorff dimension of the repeller almost surely agrees with
its upper and lower box dimensions and is given by a natural generalization of
Bowen’s formula. For a random uniformly hyperbolic Julia set on the Riemann
sphere we show that if the family of maps and the probability law depend real-analytically on parameters then so does its almost sure Hausdorff dimension.
1. Random Julia sets and their dimensions
Let (U, dU
) be an open, connected subset of the Riemann sphere avoiding
at least three points and equipped with a hyperbolic metric. Let K ⊂ U be
a compact subset. We denote by E (K, U ) the space of unramified conformal
covering maps f : Df → U with the requirement that the covering domain
Df ⊂ K. Denote by Df : Df → R+ the conformal derivative of f , see equation
(2.4), and by kDf k = sup
f
ư1
K
Df the maximal value of this derivative over
the set f
ư1
K. Let F = (f
n
) ⊂ E (K, U ) be a sequence of such maps. The
intersection
(1.1) J (F ) =

n≥1
f
ư1
1
◦ · · · ◦ f
ư1
n
(U )
defines a uniformly hyperbolic Julia set for the sequence F . Let (Υ, ν ) be a
probability space and let ω ∈ Υ → f
ω
∈ E (K, U ) be a ν -measurable map.
Suppose that the elements in the sequence F are picked independently, each
according to the law ν . Then J (F ) becomes a random ‘variable’. Our main
objective is to establish the following
696 HANS HENRIK RUGH
Theorem 1.1. I. Suppose that E(log kDfω
k) < ∞. Then almost surely,
the Hausdorff dimension of J (F ) is constant and equals its upper and lower
box dimensions. The common value is given by a generalization of Bowen ’s
formula.
II. Suppose in addition that there is a real parameter t having a complex ex-tension so that: (a) The family of maps (f
t,ω
)
ω∈Υ
depends analytically upon t.
(b) The probability measure ν
t
depends real-analytically on t. (c) Given any
local inverse, f
ư1
t,ω
, the log-derivative log Dft,ω
◦f
ư1
t,ω
is (uniformly in ω ∈ Υ) Lip-schitz with respect to t. (d) For each t the condition number kDft,ω
k·k1/Df
t,ω
k
is uniformly bounded in ω ∈ Υ.
Then the almost sure Hausdorff dimension obtained in part I depends real-analytically on t. (For a precise definition of the parameter t we refer to Section
6.3, for conditions (a), (c) and (d) see Definition 6.8 and Assumption 6.13,
and for (b) see Definition 7.1 and Assumption 7.3. We prove Theorem 1.1 in
Section 7).
Example 1.2. Let a ∈ C and r ≥ 0 be such that |a| + r <
1
4
. Suppose
that c
n ∈ C, n ∈ N are i.i.d. random variables uniformly distributed in the
closed disk B(a, r) and that Nn
, n ∈ N are i.i.d. random variables distributed
according to a Poisson law of parameter λ ≥ 0. We consider the sequence of
maps F = (f
n
)
n∈N
given by
(1.2) f
n
(z) = z
Nn+2
+ c
n
.
An associated ‘random’ Julia set may be defined through
(1.3) J (F ) = ∂ {z ∈ C : f
n
◦ · · · ◦ f
1
(z) → ∞}.
We show in Section 6 that the family verifies all conditions of Theorem 1.1,
parts I and II with a 4-dimensional real parameter t = (re a, im a, r, λ) in the
domain determined by |a| + r < 1/4, r ≥ 0, λ ≥ 0. For a given parameter
the Hausdorff dimension of the random Julia set is almost surely constant and
equals the upper/lower box dimensions. The common value d(a, r, λ) depends
real-analytically upon re a, im a, r and λ. Note that the sequence of degrees
(Nn
)
n∈N
almost surely is unbounded when λ > 0.
Rufus Bowen, one of the founders of the Thermodynamic Formalism
(henceforth abbreviated TF), saw more than twenty years ago [Bow79] a natu-ral connection between the geometric properties of a conformal repeller and the
TF for the map(s) generating this repeller. The Hausdorff dimension dimH (Λ)
of a smooth and compact conformal repeller (Λ, f ) is precisely the unique zero
s
crit
of a ‘pressure’ function P (s, Λ, f ) having its origin in the TF. This relation-ship is now known as ‘Bowen’s formula’. The original proof by Bowen [Bow79]
was in the context of Kleinian groups and involved a finite Markov partition
and uniformly expanding conformal maps. Using TF he constructed a finite