It is shown by Denker and Yuri [Contemp. Math. 63, 93–108 (2015)] that “general” iterated function systems admit conformal families of measures. Here, we relate this property to estimating Hausdorff dimension in the expanding case, including overlaps. A correct formulation of Theorem 3.11 in the above paper is provided as well.

Topics
Fractals, Operator theory

Iterated function systems are a dynamical concept to study time series governed by several sources of behavior. Its set of initial values for which indefinite iteration is possible is one of the most important objectives. Here, the dimension of this collection of initial values is estimated using a novel approach based on thermodynamic formalism.

Denker and Yuri introduced an approach to treat general iterated function systems in 2015. In order to review the associated basic definitions, let us begin with an example.

Example
Consider the unit interval X = [ 0 , 1 ] and define a family of maps v J : J ~ → [ 0 , 1 ] defined on subsets J ~ of the form [ 0 , 1 3 ] or [ 2 3 , 1 ] by
v J ( x ) = 2 x mod 1 or v J ( x ) = 3 x mod 1.
Note that this defines four maps, e.g., labeled as J = ( J ~ , i ) with i ∈ { 0 , 1 }⁠, say. We are interested in studying the infinite forward orbits of points in X⁠, that is, the set of points for which, for all n ≥ 1⁠, its forward orbit never hits the interval ( 1 3 , 2 3 ) with all possible iterates at the same iteration level n⁠. The forward orbit Y ( x ) of a point x ∈ X is here understood to be the set of points
y ∈ ⋃ n ≥ 1 ⋃ v J 1 , v J 2 , … , v J n { v J n ° ⋯ ° v J 1 ( x ) } ∪ { x } .
Y ( x ) is said to be infinite if for each n ≥ 1 there are v J 1 , v J 2 , … , v J n such that v J n ° ⋯ ° v J 1 ( x ) is well defined. Of course, the union is only taken over those sequences where for each n v J n ° ⋯ ° v J 1 ( x ) is meaningful. It is easy to see that the set of infinite orbits is nonempty, even more: it contains the middle third Cantor set as a proper subset by just considering the map x → 3 x mod 1 alone.
In general, a “general” iterated function system is defined by a metric space X with metric d and a family V of homeomorphisms v : D ( v ) → v ( D ( v ) ) defined on some closed subset D ( v ) ⊂ X for each v ∈ V⁠. Define
Y = { x ∈ X | Y ( x ) is infinite } .
(1.1)
Example continued: The inverse branches of all four maps, restricted to their domains, are
f 1 : [ 0 , 1 3 ] → [ 0 , 1 6 ] , f 1 ( x ) = x 2 , f 2 ( x ) : [ 0 , 1 3 ] ∪ [ 2 3 , 1 ] → [ 0 , 1 3 ] , f 2 ( x ) = x 3 , f 3 ( x ) : [ 2 3 , 1 ] → [ 5 6 , 1 ] , f 3 ( x ) = 1 + x 2 , f 4 : [ 0 , 1 3 ] ∪ [ 2 3 , 1 ] → [ 2 3 , 1 ] , f 4 ( x ) = 2 + x 3 .
Restricting to connected components this can be written as the family
f 1 : [ 0 , 1 3 ] → [ 0 , 1 6 ] , f 1 ( x ) = x 2 , f 2 ( x ) : [ 0 , 1 3 ] → [ 0 , 1 9 ] , f 2 ( x ) = x 3 , f 2 ∗ ( x ) : [ 2 3 , 1 ] → [ 2 9 , 1 3 ] , f 2 ∗ ( x ) = x 3 , f 3 ( x ) : [ 2 3 , 1 ] → [ 5 6 , 1 ] , f 3 ( x ) = 1 + x 2 , f 4 : [ 0 , 1 3 ] → [ 2 3 , 7 9 ] , f 4 ( x ) = 2 + x 3 , f 4 ∗ : [ 2 3 , 1 ] → [ 8 9 , 1 ] , f 4 ∗ ( x ) = 2 + x 3 .
Clearly, these also define “general” iterated function systems.
The transfer operator for a “general” iterated function system is defined as
L f ( x ) = ∑ v ∈ V ; x ∈ D ( v ) f ( v ( x ) ) ϕ v ( x ) ,
(1.2)
where
Φ := { ϕ v = exp ⁡ [ φ v ] : D ( v ) → R + | v ∈ V }
is a family of continuous functions, called a potential. Let V n denote the collection of all well defined concatenations v n ° v n − 1 ° ⋯ ° v 1 where v i ∈ V for i = 1 , … , n⁠. Iteration yields for n ≥ 1⁠,
L n f ( x ) = ∑ w ∈ V n : x ∈ D ( w ) f ( w ( x ) ) ϕ w ( x ) = ∑ w ∈ V n : x ∈ D ( w ) f ( w ( x ) ) exp ⁡ [ φ w ( x ) ] ,
where D ( w ) denotes the domain of w = v n ° ⋯ ° v 1 ∈ V n⁠, v 0 ( x ) = x and
ϕ w ( x ) = ∏ i = 1 n ϕ v i ( v i − 1 ° ⋯ ° v 0 ( x ) ) = exp ⁡ [ ∑ i = 1 n φ v i ( v i − 1 ° ⋯ ° v 0 ( x ) ) ] .
Example continued: Let δ > 0⁠. We take the potential
ϕ v ( x ) = { 2 δ if the map is x → 2 x , 3 δ if the map is x → 3 x
to obtain the transfer operator
L 0 f ( x ) = f ( { 2 x } ) 2 δ + f ( { 3 x } ) 3 δ
for the first systems, where here { t } denotes the fractional part of t ∈ R⁠. Taking the potential ϕ f j = 2 − δ for j ∈ { 1 , 3 } and ϕ f = 3 − δ for the other four maps, we obtain the transfer operator,
L f ( x ) = { f ( x / 2 ) 2 δ + f ( x / 3 ) 3 δ + f ( ( 2 + x ) / 3 ) 3 δ , 0 ≤ x ≤ 1 / 3 , f ( x / 3 ) 3 δ + f ( ( 1 + x ) / 2 ) 2 δ + f ( ( 2 + x ) / 3 ) 3 δ , 2 / 3 ≤ x ≤ 1
for the second system.

The existence of invariant measures for transfer operators (or Perron Frobenius operator) in our situation has been established in Denker and Yuri.1 We recall the result in Sec. II. In Sec. III, we relate the eigenvalues of the transfer operator and some growth condition of overlaps of the domains of the “general” iterated function system to the Hausdorff dimension of the set Y of infinite orbits. More evolved results are certainly possible.

Recall1 that a conformal family of measures for a “general” iterated function system ( X , V ) and the potential Φ = { ϕ v = exp ⁡ [ φ v ] | v ∈ V } is a collection of measures { m , m v : v ∈ V } where m denotes a probability measure on X and m v a measure on D ( v ) for v ∈ V satisfying

  1. For some λ Φ > 0 and every f ∈ C ( X )⁠,
    λ Φ ∫ f d m = ∫ L Φ f d m + ∑ v ∈ V ∫ ∂ D ( v ) f ( v ( x ) ) ϕ v ( x ) ( m v − m ) ( d x ) ,
    (2.1)

  2. m v = m on the interior int D ( v ) of D ( v ) for all v ∈ V⁠.

In case m v = m | D ( v ) for all v ∈ V m is called a conformal measure. Here, ∂ D ( v ) denotes the boundary of D ( v )⁠.

Theorem II.1
(Ref. 1)
Let ( X , V ) be an iterated function system, and let Φ = { ϕ v = exp ⁡ [ φ v ] | v ∈ V } be a potential of uniformly continuous functions with
Z 1 ( Φ ) := ∑ v ∈ V sup x ∈ D ( v ) ϕ v ( x ) < ∞ .
Then, there exists a conformal family M = { m } ∪ { m v : v ∈ V } for ϕ⁠. It satisfies
∑ v ∈ V ∫ ∂ D ( v ) f ( v ( x ) ) ϕ v ( x ) ( m v − m ) ( d x ) ≤ 0
(2.2)
for every positive f ∈ C ( X )⁠.

The term on the left hand side in (2.2) is called the deficiency of the conformal family. So, m is a conformal measure if the deficiency vanishes.

Theorem 3.11 in Denker and Yuri1 is not correctly formulated as one crucial assumption is missing. Here is the correct formulation. (This is a corrigendum of Conformal Families Of Measures For General Iterated Function Systems authored and approved by Michiko Yuri, Department of Mathematics, Hokkaido University. We thank Rodrigo Bissacot, Thiago Costa Raszeja, and, in particular, Rodrigo S. Frausino for pointing out this omission.) For the definitions of P a l g ( Φ )⁠, P ( Φ , x ) and of a proper potential, we refer to Denker and Yuri.1 The notation in the correction below follows that of the present paper and differs slightly from the notation used in the former. It gives a condition to obtain conformal measures. Their existence is a basic assumption in Sec. III.

Theorem II.2
(Ref. 1)
Let ( X , V ) be an iterated function system such that each D ( v ) is compact. If Φ is a proper potential with P a l g ( Φ ) < ∞ and if there is x ∈ X with P ( Φ , x ) > 0 such that the forward orbit of x is contained in
X [ ∞ ] ° := ⋂ n = 1 ∞ ⋃ v ∈ V n v ( D ( v ) ) ,
then there exists a conformal measure with support in X [ ∞ ] °⁠.

The transfer operator in (2.1) can be connected to the estimation of the Hausdorff dimension of Y⁠. Here, we successively develop that relation.

Let ( X , V ) be a “general” iterated function system in some Euclidean space, Φ = { ϕ v = exp ⁡ [ φ v ] | v ∈ V } a potential and μ a conformal measure for the potential Φ⁠, which is positive on open sets contained in ⋃ v ∈ V D ( v )⁠. For n ≥ 1 and a measurable set B ⊂ X define
Λ ( B , n ) := ∫ B ∑ v ∈ V n exp ⁡ [ − φ v ] d μ .
(3.1)
Then, by (2.1),
λ Φ n Λ ( B , n ) = ∫ L Φ n ( I B ∑ v ∈ V n exp ⁡ [ − φ v ] ) ( y ) μ ( d y ) = ∫ ∑ w ∈ V n I B ( w ( y ) ) ∑ v ∈ V n exp ⁡ [ − φ v ( w ( y ) ) ] exp ⁡ [ φ w ( y ) ] μ ( d y ) .
We define the function
Γ ( B , n ) := ∫ ∑ w ∈ V n I B ( w ( y ) ) ∑ v ∈ V n e − φ v ( w ( y ) ) e φ w ( y ) μ ( d y ) ,
(3.2)
where B ⊂ X is measurable and n ≥ 1 .

Our next assumption is uniform distortion:

There is C 0 < ∞ such that for all n ≥ 1⁠, all v ∈ V n and all x , y ∈ D ( v )⁠,
| φ v ( x ) − φ v ( y ) | ≤ C 0 .
Proposition III.1

  1. If for a measurable set B ⊂ X⁠, n ≥ 1 and x ∈ B
    Γ ( B , n ) ≤ K γ n ,
    then
    μ ( B ) ∑ v ∈ V n exp ⁡ [ − φ v ( x ) ] ≤ e C 0 K λ Φ − n γ n .
    (3.3)
  2. If for a measurable set B ⊂ X and n ≥ 1
    Γ ( B , n ) ≥ K γ n ,
    then
    μ ( B ) sup x ∈ B ∑ v ∈ V n exp ⁡ [ − φ v ( x ) ] ≥ K λ Φ − n γ n .
    (3.4)
  3. Let B ⊂ X be a measurable set, x ∈ B and n ≥ 1⁠. Assume that
    Γ ( B , n ) ≥ K γ n
    and
    v ∈ V n & x ∉ D ( v ) ⟹ B ∩ D ( v ) = ∅ .
    Then,
    μ ( B ) ∑ v ∈ V n exp ⁡ [ φ v ( x ) ] ≥ e − C 0 K λ Φ − n γ n .
    (3.5)

Proof.

  1. By distortion and (2.1),
    μ ( B ) ∑ v ∈ V n exp ⁡ [ − φ v ( x ) ] ≤ e C 0 ∫ B ∑ v ∈ V n , x ∈ D ( v ) exp ⁡ [ − φ v ( y ) ] μ ( d y ) ≤ e C 0 ∫ B ∑ v ∈ V n exp ⁡ [ − φ v ( y ) ] μ ( d y ) = e C 0 Λ ( B , n ) = e C 0 λ Φ − n Γ ( B , n ) ≤ e C 0 K λ Φ − n γ n .

  2. Follows immediately by (2.1).

  3. Similarly to (2) observing that for y ∈ B⁠,
    ∑ v ∈ V n exp ⁡ [ − φ v ( x ) ] ≥ e − C 0 ∑ v ∈ V n ; x ∈ D ( v ) exp ⁡ [ − φ v ( y ) ]
    and that B ∩ D ( v ) ≠ ∅ implies x ∈ D ( v )⁠.

Consider a fixed Φ = { ϕ v = exp ⁡ [ φ v ] | v ∈ V } and the family of potentials
Φ t := { exp ⁡ [ t φ v ] | v ∈ V }
for t ≥ 0⁠. Assume that Φ t has a conformal measure μ t for every t⁠. Denote λ Φ t = λ ( t ) the eigenvalue for the transfer operator associated to Φ t⁠. We assume that Φ is strictly negative, i.e.,
sup { log ⁡ ϕ v ( y ) = φ v ( y ) : v ∈ V , y ∈ D ( v ) } < 0.

We first observe

Lemma III.2
λ ( t ) < λ ( s ) ∀ s < t
and
lim t → ∞ log ⁡ λ ( t ) = 0.
Proof.
The eigenvalue λ ( t ) satisfies
log ⁡ λ ( t ) = P ( Φ t , x 0 ) = lim sup s → ∞ 1 s log ⁡ L Φ t s 1 ( x 0 )
for some x 0 [see Denker and Yuri1]; hence,
log ⁡ λ ( t ) = lim sup s → ∞ 1 s log ⁡ ∑ v ∈ V s exp ⁡ [ t φ v ( x 0 ) ]
is decreasing and the assertions follow.
Let ε > 0 and x ∈ X have an infinite orbit. By assumption for Φ⁠, for each t > 0⁠,
lim n → ∞ ∑ v ∈ V n exp ⁡ [ − t φ v ( x ) ] = ∞
so that there exists a minimal n = n ( t , ε ) such that
ε t ∑ v ∈ V n exp ⁡ [ − t φ v ( x ) ] ≥ 1.
(3.6)
Also, for all ε sufficiently small, there exists a maximal m = m ( t , ε ) such that
ε t sup y ∈ B ( x , ε ) ∑ v ∈ V m exp ⁡ [ − t φ v ( y ) ] ≤ 1 ,
(3.7)
where B ( x , ε ) denotes the ball around x of radius ε⁠. Define
Γ t ( x , s ) = ∫ ∑ w ∈ V s I { B ( x , ε ) } ( w ( y ) ) ∑ v ∈ V s e − t φ v ( x ) e t φ w ( y ) μ t ( d y ) ,
(3.8)
which equals the Γ ( B , s ) defined in (3.2) for the potential Φ t⁠, the conformal measure μ t⁠, s ≥ 1 and B = B ( x , ε )⁠. Let
γ ( x , t ) := lim sup ε → 0 1 n ( t , ε ) log ⁡ Γ t ( x , n ( t , ε ) )
and
γ = lim sup ε → 0 1 n ( t , ε ) log ⁡ sup x ∈ Y Γ t ( x , n ( t , ε ) ) .
Lemma III.3
Let x ∈ X have an infinite orbit and for some t ≥ 0⁠,
log ⁡ λ ( t ) > γ ( x , t ) .
Then, there exists a constant C ( t , x ) < ∞ such that for all ε > 0⁠,
| B ( x , ε ) | t μ t ( B ( x , ε ) ) ≥ C ( t , x ) .
If
log ⁡ λ ( t ) > γ ( t ) ,
then inf x ∈ Y C ( t , x ) > 0⁠.
Proof.
For ε > 0 let n = n ( t , ε ) be as above. By assumption, there exists a constant K such that for all ε > 0⁠,
K λ ( t ) n ( t , ε ) ≥ Γ t ( x , n ( t , ε ) ) .
Then, by the choice of n⁠,
ε − t ≤ ∑ v ∈ V n exp ⁡ [ − t φ v ( x ) ] .
and, hence, by Proposition III.1,
μ t ( B ( x , ε ) ) ε − t ≤ e C 0 K λ ( t ) − n λ ( t ) n = e C 0 K < ∞ .
K can be chosen uniformly for x ∈ Y if log ⁡ λ ( t ) > γ ( t )⁠.
Let
γ 0 ( x , t ) := lim inf ε → 0 1 m ( t , ε ) log ⁡ Γ t ( x , m ( t , ε ) )
and
γ 0 ( t ) = lim inf ε → 0 1 m ( t , ε ) log ⁡ inf x ∈ Y Γ t ( x , m ( t , ε ) ) .
Lemma III.4
Let x ∈ X have an infinite orbit and for some t ≥ 0⁠,
log ⁡ λ ( t ) < γ 0 ( t , x ) .
Then, there exists a constant 0 < C ′ ( t , x ) < ∞ such that for all sufficiently small ε > 0⁠,
| B ( x , ε ) | t μ t ( B ( x , ε ) ) ≤ C ′ ( t , x ) .
If
log ⁡ λ ( t ) < γ 0 ( t ) ,
then sup x ∈ Y C ′ ( t , x ) < ∞⁠.
Proof.
By assumption, there exists a constant K > 0 such that for all ε > 0⁠,
K λ ( t ) m ( t , ε ) ≤ Γ t ( x , m ( t , ε ) ) .
By definition of m = m ( t , ε )⁠,
ε − t ≥ sup y ∈ B ( x , ε ) ∑ v ∈ V m exp ⁡ { − t φ v ( y ) } .
Then, by Proposition III.1,
μ t ( B ( x , ε ) ) ε − t ≥ K λ ( t ) − m λ ( t ) m ≥ K .

K can be chosen uniformly for x ∈ Y if log ⁡ λ ( t ) < γ 0 ( t )⁠.

Corollary III.5

Let ( X , V ) be a “general” iterated function system and Φ = { exp ⁡ [ φ v ] | v ∈ V } be a strictly negative potential satisfying uniform distortion and admitting the existence of conformal measures μ t for the potential { exp ⁡ [ t φ v ] | v ∈ V } with eigenvalue λ ( t )⁠, t ≥ 0⁠.

Then,
sup { t | log ⁡ λ ( t ) > γ ( t ) } ≤ Dim Y ≤ inf { t | log ⁡ λ ( t ) < γ 0 ( t ) } .
Proof.

Use Besicovic’s covering theorem2,3 and Lemmas III.3 and III.4.

If there is a constant C < ∞ with
lim inf r → 0 r t ν ( B ( x , r ) ) ≤ C ∀ x ∈ Y
except countably many x⁠, then
H t ( Y ) < ∞
and so Dim ( Y ) = sup { s : H s ( Y ) = ∞ } ≤ t⁠.
If there is a constant C > 0 with
lim sup r → 0 r t ν ( B ( x , r ) ) ≥ C ∀ x ∈ Y ,
then
H t ( Y ) > 0
and Dim ( Y ) = inf { s : H s ( Y ) = 0 } ≥ t⁠.

Note that the corollary reduces to the classical Bowen–Manning–McCluskey formula for dimension in the case of a Ruelle expanding map. It also applies to the example presented in the introduction, especially in the case where the potential is given by the derivative of the maps.

I would like to thank two referees for valuable comments and suggestions improving the clarity of notions, proofs, and formulas.

The authors have no conflicts to disclose.

Manfred Denker: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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