Regularity of multifractal spectra of conformal iterated function systems
aa r X i v : . [ m a t h . D S ] M a r REGULARITY OF MULTIFRACTAL SPECTRA OF CONFORMAL ITERATEDFUNCTION SYSTEMS
JOHANNES JAERISCH AND MARC KESSEBÖHMERA
BSTRACT . We investigate multifractal regularity for infinite conformal iterated functionsystems (cIFS). That is we determine to what extent the multifractal spectrum dependscontinuously on the cIFS and its thermodynamic potential. For this we introduce the no-tion of regular convergence for families of cIFS not necessarily sharing the same indexset, which guarantees the convergence of the multifractal spectra on the interior of theirdomain. In particular, we obtain an Exhausting Principle for infinite cIFS allowing us tocarry over results for finite to infinite systems, and in this way to establish a multifractalanalysis without the usual regularity conditions. Finally, we discuss the connections to the l -topology introduced by Roy and Urba´nski.
1. I
NTRODUCTION AND STATEMENT OF RESULTS
The theory of multifractals has its origin at the boarderline between statistical physics andmathematics - classical references are e. g. [FP85, Man74, Man88, HJK + F = ( j e : X → X ) e ∈ I , I ⊂ N , of conformal contractions on a compact connected subset X of the euclidean space (cid:0) R D , k · k (cid:1) , D ≥
1. The set of cIFS with fixed phase space X will be denoted by CIFS ( X ) (see Section 2 for definitions). For w ∈ I N we let w | k : = w · · · w k and j w | k : = j w ◦ · · ·◦ j w k .Then for each w ∈ I N the intersection T ¥ k = j w | k ( X ) is always a singleton given rise to acanonical coding map p F : I N → X . Its image L F : = p F (cid:0) I N (cid:1) will be called the limit set of F . Given a Hölder continuous function y : I N −→ R the multifractal analysis of thesystem F with respect to the potential y is in our context understood to be the analysis ofthe level sets F a : = p F ( w ∈ I N : lim k → ¥ S k y ( w ) log (cid:13)(cid:13) j ′ w | k (cid:13)(cid:13) X = a ) in terms of their Hausdorff dimension f ( a ) : = dim H ( F a ) . In here, S k y : = (cid:229) k − n = y ◦ s n denotes the Birkhoff sum of y with respect to the shift map s : I N → I N on the symbolicspace, and (cid:13)(cid:13) j ′ w | k (cid:13)(cid:13) X : = sup x ∈ X (cid:12)(cid:12) j ′ w | k ( x ) (cid:12)(cid:12) with (cid:12)(cid:12) j ′ w | k ( x ) (cid:12)(cid:12) denoting the operator norm of thederivative. A good reference for this kind of multifractal analysis is provided e. g. in[Pes97].Let us define the geometric potential function associated with F by z : I N → R − , z ( w ) : = log (cid:12)(cid:12) j ′ w ( p ( s ( w ))) (cid:12)(cid:12) . It is well known that in the case of finite cIFS, that is card ( I ) < ¥ , f can be related to the Legendre transform of the free energy function t : R → R , which isdefined implicitly by the pressure equation (cf. Definition 2.4) P ( t ( b ) z + by ) = , b ∈ R . (1.1) Date : November 7, 2018.2000
Mathematics Subject Classification.
Primary 37C45; Secondary 37D45, 37D35.
More precisely, there exists a closed finite interval J ⊂ R such that for all a ∈ J we have f ( a ) = − t ∗ ( − a ) : = − sup b {− ba − t ( b ) } = inf b { t ( b ) + ba } , (1.2)and for a / ∈ J we have F a = ∅ ([Pes97, Theorem 21.1], [Sch99]). If we consider infinitecIFS, i. e. card ( I ) = card ( N ) , we have to take into account that the pressure functionmight behave irregularly and hence it is not always possible to find a solution of (1.1).For the special case in which (1.1) has a unique solution the multifractal analysis has beendiscussed in [MU03, Section 4.9]. Further interesting results on the spectrum of localdimension for Gibbs states can be found [RU09].Our first task is to generalise this concept to the case when the free energy cannot be definedby the unique solution of (1.1). This leads to the following modified definition of the freeenergy function. Definition 1.1.
Let F ∈ CIFS ( X ) and y : I N −→ R be a potential function. Then the freeenergy function t : R −→ R ∪ { ¥ } for the pair ( F , y ) is given by t ( b ) : = inf { t ∈ R : P ( t z + by ) ≤ } . (1.3)Notice that our definition of the free energy function generalises the definition given for themultifractal analysis presented in [MU03, Section 4.9] or in [KU07], where the existenceof a zero of the pressure function t P ( t z + by ) is always required. Our definition israther in the spirit of [MU03, Theorem 4.2.13], which gives a version of Bowen’s formula,without assuming a zero of the pressure function to exist. More precisely, we havedim H ( L F ) = inf { t ∈ R : P ( t z ) ≤ } , which immediately implies that t ( ) = dim H ( L F ) . In fact, Lemma 3.1 shows that Defini-tion 1.1 gives rise to a proper convex function. This concept of the free energy function hasbeen investigated further in [JKL10] as a special case of the induced topological pressure for arbitrary countable Markov shifts. We would like to point out that this new formalismgives rise to further interesting exhausting principles similar to Example 1.6 and Corollary1.9 above.To state our first main result we set a − : = inf (cid:8) − t − ( x ) : x ∈ Int ( dom ( t )) (cid:9) and a + : = sup (cid:8) − t + ( x ) : x ∈ Int ( dom ( t )) (cid:9) , where t + , resp. t − , denotes the derivative of t from the right, resp. from the left, Int ( A ) denotes the interior of the set A , and dom ( t ) : = { x ∈ R : t ( x ) < + ¥ } refers to the effectivedomain of t . Theorem 1.2.
For a ∈ R we have f ( a ) ≤ max {− t ∗ ( − a ) , } and for a ∈ ( a − , a + ) wehave f ( a ) = − t ∗ ( − a ) . This first main result is essentially a consequence of the multifractal regularity propertyof sequences of tuples ( F n , y n ) n of iterated function systems and potentials, which is thesecond main concern of this paper.We adapt the definition of pointwise convergence in CIFS ( X ) as used by Roy and Urba´nskiin [RU05] to our setting, allowing us to investigate also families of cIFS with associatedpotentials not sharing the same index set N . To simplify notation let us write k h k W : = sup w ∈ W | h ( w ) | for the supremum norm of the map h : W −→ ( V , | · | ) from W to the normedspace ( V , | · | ) . For F , F ∈ CIFS ( X ) we define r (cid:0) F , F (cid:1) : = (cid:229) i ∈ I ∩ I − i (cid:0)(cid:13)(cid:13) j i − j i (cid:13)(cid:13) X + (cid:13)(cid:13) ( j i ) ′ − ( j i ) ′ (cid:13)(cid:13) X (cid:1) + (cid:229) i ∈ I △ I − i , (1.4) EGULARITY OF THE MULTIFRACTAL SPECTRUM OF CONFORMAL IFS 3 where A △ B denotes the usual symmetric difference of the sets A and B . It will turn out that r defines a metric on CIFS ( X ) . For w ∈ N k and k ∈ N we let [ w ] : = (cid:8) t ∈ N N : t | k = w (cid:9) denote the cylinder set of w .In order to set up a multifractal spectrum we restrict our analysis to families of Höldercontinuous functions y n : I N n −→ R and y : I N −→ R with I n ⊂ I ⊂ N , n ∈ N . Definition 1.3.
We say that ( F n , y n ) n −→ ( F , y ) converges pointwise if ,(A) F n −→ F in the r -metric and(B) for all k ∈ I we have lim n → ¥ (cid:13)(cid:13) y n − y (cid:13)(cid:13) [ k ] ∩ I N n = r -metric implies that (cid:13)(cid:13) y n − y (cid:13)(cid:13) [ k ] ∩ I N n in (B) is well definedfor all sufficiently large n . For a further discussion of the above defined property see alsothe remark succeeding Lemma 2.6.As discussed in [RU05] pointwise convergence topology leads to discontinuities of theHausdorff dimension of the limit sets. By introducing a weaker topology called the l -topology in [RU05] the Hausdorff dimension of the limit set depends continuously on thesystem (see also [RSU09]). Convergence in l -topology requires the additional condition(6.1) below. As a corollary we will also establish the continuity of the Hausdorff dimensionunder weaker assumptions.We are going to employ similar assumptions on the convergence of the pairs ( F n , y n ) n and ( F , y ) to obtain continuity of the multifractal spectra. This is the purpose of the followingdefinition. For this let z n denote the geometric potential associated with F n . Definition 1.4.
We say that ( F n , y n ) n converges regularly to ( F , y ) , if ( F n , y n ) n −→ ( F , y ) converges pointwise, and if for t , b ∈ R with P ( t z + by ) < ¥ there exists k ∈ N and a constant C > n ∈ N and all w ∈ ( I n ) k we haveexp sup t ∈ I N n ∩ [ w ] ( S k ( t z n + by n ) ( t )) ≤ C exp sup r ∈ I N ∩ [ w ] ( S k ( t z + by )( r )) . The assumption in Definition 1.4 is similar to the corresponding inequality in the definitionof the l -topology in [RU05] but depends additionally on the potentials y n and y . For par-ticular cases we will show that the convergence F n −→ F in the l -topology immediatelyimplies the conditions in Definition 1.4. This is demonstrated in the following exampleproviding an analysis of the (inverse) Lyapunov spectrum. This example is covered byProposition 6.4 (2) stated in Section 6. Example 1.5 ( lll -topology). Let F n = ( j ne ) e ∈ I n , F = ( j e ) e ∈ N be elements of CIFS ( X ) with F n → F converging in the l -topology and let y n = y =
1. Then ( F n , y n ) n −→ ( F , y ) converges regularly.The second example – eventhough straightforward to verify – is not only interesting foritself but will be of systematic importance for the proof of Theorem 1.2. See also Remark5.1 and Example 1.9 for further discussion of this example. Example 1.6 ( Exhausting Principle I).
Let F = ( j e ) e ∈ N be an element of CIFS ( X ) and y : I N −→ R be Hölder continuous. Define I n : = I ∩ { , . . . n } , n ∈ N and let F n = ( j e ) e ∈ I n and y n : = y (cid:12)(cid:12) I N n . Then ( F n , y n ) n −→ ( F , y ) converges regularly.If the multifractal regularity property is satisfied we are able to prove the regularity of thefree energy functions. Theorem 1.7. If ( F n , y n ) n −→ ( F , y ) converges regularly then t n converges pointwise tot on R . JOHANNES JAERISCH AND MARC KESSEBÖHMER
To state our second main result on the regularity of the multifractal spectra let F n a : = p F n ( w ∈ I N n : lim k → ¥ S k ( y n ) ( w ) log (cid:13)(cid:13) ( j n w | k ) ′ (cid:13)(cid:13) X = a ) , f n ( a ) : = dim H ( F n a ) , and with t n denoting the free energy function of ( F n , y n ) let a n − : = inf (cid:8) − t − n ( x ) : x ∈ Int ( dom ( t n )) (cid:9) and a n + : = sup (cid:8) − t + n ( x ) : x ∈ Int ( dom ( t n )) (cid:9) . Theorem 1.8.
Let F n = ( j ne ) e ∈ I n , F = ( j e ) e ∈ I be elements of CIFS ( X ) and y n , y beHölder potentials such that ( F n , y n ) n −→ ( F , y ) converges regularly. Then for each a ∈ ( a − , a + ) we have • lim n → ¥ − t ∗ n ( − a ) = f ( a ) = − t ∗ ( − a ) , • f n ( a ) = − t ∗ n ( − a ) , for all n sufficiently large.In particular, we have lim sup n a n − ≤ a − ≤ a + ≤ lim inf n a n + . If additionally sup dom ( t ) =+ ¥ then lim inf n a n − ≥ a − , whereas, if inf dom ( t ) = − ¥ then lim sup n a n + ≤ a + . Combining the above theorems with Example 1.6 we obtain the following application ofour analysis.
Corollary 1.9 ( Exhausting Principle II).
Let F = ( j e ) e ∈ N be an element of CIFS ( X ) , y : I N −→ R be Hölder continuous, and F n = ( j e ) e ∈ I n and y n : = y (cid:12)(cid:12) I N n with I n : = I ∩{ , . . . n } ,n ∈ N . Then for each a ∈ ( a − , a + ) we have lim n − t ∗ n ( − a ) = f ( a ) = − t ∗ ( − a ) and f n ( a ) = − t ∗ n ( − a ) , for all n sufficiently large. For the boundary points of the spectrum wehave the following. (1) If dom ( t ) = R then lim n → ¥ a n ± = a ± , (2) if sup dom ( t ) < + ¥ then lim sup n a n − = − ¥ and for all a < a − we have lim sup n → ¥ − t ∗ n ( − a ) ≤ f ( a ) , (3) if inf dom ( t ) > − ¥ then lim inf n a n + = + ¥ and for all a > a + we have lim sup n → ¥ − t ∗ n ( − a ) ≤ f ( a ) . In Example 1.13 below we demonstrate how the lower bound on f stated in Corollary 1.9(3) can be applied.Note that by virtue of Proposition 6.4 we have on the one hand that the convergence F n −→ F in the l -topology implies that ( F n , ) n −→ ( F , ) converges regularly. On the other handwe have t n ( ) = dim H ( L F n ) . Hence, the following corollary is straightforward and maybe viewed as a generalisation of the continuity results in [RSU09, RU05] for the Hausdorffdimension of the limit sets. Corollary 1.10 ( Continuity of Hausdorff dimension).
Let F n = ( j ne ) e ∈ I n , F = ( j e ) e ∈ I be elements of CIFS ( X ) such that ( F n , ) n −→ ( F , ) converges regularly. Then lim n → ¥ dim H ( L F n ) = dim H ( L F ) . Finite-to-infinite phase transition.
To complete the discussion of the Exhausting Princi-ple we would like to emphasise that the boundary values of the approximating spectra ingeneral do not converge to the corresponding value of the limiting system, i. e. we mayhave f n (cid:0) a n ± (cid:1) f ( a ± ) . (1.5)We refer to the property of an infinite system having a discontinuity of this kind in one ofthe boundary points as a finite-to-infinite phase transition in a + , resp. a − . Let us illustratethis property with the following concrete example. EGULARITY OF THE MULTIFRACTAL SPECTRUM OF CONFORMAL IFS 5
Example 1.11 ( Gauss system).
Let F : = { j e : x / ( x + e ) : e ∈ N } denote the Gausssystem and the potential y is given by y ( w ) : = − w , for w ∈ N N . In [JK10] we haveshown that the multifractal spectrum is unimodal, defined on [ , ] , and in the boundarypoints of the spectrum we have f ( ) = f ( ) = /
2. Nevertheless, for the exhaustingsystems F n : = ( j e ) ≤ e ≤ n and y n : = y (cid:12)(cid:12) { ,..., n } N we have for their corresponding multifrac-tal spectra f n (cid:0) a n + (cid:1) = n ∈ N giving rise to a finite-to-infinite phase transition (seeFig 1.1). A proof of this will be postponed to the end of Section 5.
11½ 00 α f ( ) α α αf ( ) α n f ( ) α m
1½ 0 βt ( ) β F IGURE f n : (cid:2) , a n + (cid:3) → R + and f m : (cid:2) , a m + (cid:3) → R + , m > n , offinite sub-systems to the multifractal spectrum f of the infinite system. Example 1.12 ( Lüroth system).
In the following example the effective domain of the freeenergy function is not equal to R , which leads to an interesting boundary behaviour. Forthis let us consider the Lüroth system F : = { j n : x x / ( n ( n + )) + / ( n + ) : n ∈ N } (essentially a linearised Gauss system) and the potential functions y given by y ( w ) : = − w , w ∈ N N . Then in virtue of our theorems the spectrum is given by the Legendre trans-form of t on ( / log ( ) , + ¥ ) via f ( a ) = − t ∗ ( − a ) . Similarly as for the Gauss system inthe example above, one can show that f ( / log ( )) = f n ( / log ( )) = f n (cid:0) a n + (cid:1) = n ∈ N .Since we have Lebesgue almost everywhere that lim k → ¥ (cid:229) ki = a i / (cid:229) ki = log ( a i ( a i + )) = ´ (cid:0) p − F ( x ) (cid:1) d l / ´ log (cid:0)(cid:0) p − F ( x ) (cid:1) (cid:0)(cid:0) p − F ( x ) (cid:1) + (cid:1)(cid:1) d l = + ¥ we find f (+ ¥ ) =
1. Henceas above, we have a finite-to-infinite phase transition – this time at infinity.
Example 1.13 ( Generalised Lüroth system).
In the following example the effectivedomain of the free energy function is again not equal to R and additionally we havea second order phase transition. Let us consider the generalised Lüroth system F : = { j n : x x / ( n ( n + )( n + ))+ / (( n + )( n + )) : n ∈ N } and the potential functions y given by y ( w ) : = − w , w ∈ N N . Then in virtue of our theorems the spectrum is givenby the Legendre transform of t on ( / log ( ) , a + ) via f ( a ) = − t ∗ ( − a ) , where a + : = (cid:16) (cid:229) n ≥ (( n ( n + )( n + )) / )( n ( n + )( n + )) (cid:17) − . Using the Corollary 1.9 (Exhausting Principle II) (3) wegather some extra information on the spectrum. Since we have lim n − t ∗ n ( − t ′ n ( )) = a n + = n / log ( n ( n + )( n + ) / ) → ¥ , we deduce that 1 is a lower bound for f ( a ) forall a ≥ a + . Similarly as for the Gauss system in the example above, one can show that f ( / log ( )) = f n ( / log ( )) = f n (cid:0) a n + (cid:1) = n ∈ N . Hence, f ( a ) = − t ∗ ( − a ) for all a ≥ / log ( ) . (cf. Fig. 1.2).Generalising further the latter two examples our analysis has successfully been applied in[KMS10] to determine the Lyapunov spectrum of a -Farey-Lüroth and a -Lüroth systems. JOHANNES JAERISCH AND MARC KESSEBÖHMER + αα α α ββt ( ) f ( ) nn n /log( ( +1)( +2)/4) n f ( ) n +∞ F IGURE t and the mul-tifractal spectrum f for the generalised Lüroth system. Thedashed graph is associated to the approximating spectra f n : [ / log ( ) , n / log ( n ( n + )( n + ) / )] → R + of finite sub-system to themultifractal spectrum f of the infinite system. Example 1.14 ( Irregular cIFS).
For this example we suppose that F is an irregular infi-nite cIFS, that is the range of the pressure function p : t P ( t z ) consists of the negativereals and infinity (see [MU03] for explicit examples), and let y be constantly equal to − p ( d ) = h <
0, where d is the critical value as well as the Hausdorff di-mension of the limit set. Then the free energy function t is given by t ( b ) = p − ( b ) for b < h and constantly equal to d for b ≥ h . The corresponding spectrum will have a linearpart in ( , a − ) if − p + ( d ) = / a − < ¥ and hence for a − >
0, we observe a second orderphase transition (see Fig. 1.3). f ( ) α α δ β t ( ) β δ α + α - η F IGURE t and the multifractalspectrum f for an irregular system with constant negative potential. Notethat in this situation we have a second order phase transition in a − andthe spectrum f is linear on ( , a − ) .The paper is organised as follows. In Section 2 we recall the basic notions relevant forcIFS. In Section 3 we show the regularity of the free energy function proving Theorem1.7. Section 4 provides us with the necessary prerequisites from convex analysis allowingus to deduce the multifractal regularity in Section 5. In particular, we prove Theorem 1.2and 1.8, and the finite-to-infinite phase transition for the Gauss system. The final sectionis devoted to the connection between our notion of regularity and the l -topology. EGULARITY OF THE MULTIFRACTAL SPECTRUM OF CONFORMAL IFS 7
2. P
RELIMINARIES
Let us recall the definition of a conformal iterated function system (see [MU03] for furtherdetails). Let X be a compact metric space. For an alphabet I ⊂ N with card ( I ) ≥ F = ( j e ) e ∈ I an iterated function system (IFS), where j e : X → X are injective contractions, e ∈ I , with Lipschitz constants globally bounded away from 1.Let I ∗ : = S n ∈ N I n denote the set of all finite subwords of I N . We will consider the left shiftmap s : I N → I N defined by s ( w i ) : = ( w i + ) i ≥ . For w ∈ I ∗ we let | w | denote the lengthof the word w , i. e. the unique n ∈ N such that w ∈ I n .The space I N is equipped with the metric d given by d ( w , t ) : = exp ( −| w ∧ t | ) , where w ∧ t ∈ I ∗ ∪ I N denotes the longest common initial block of the infinite words w and t .We now describe the limit set of the iterated function system F . For each w ∈ I ∗ , say w ∈ I n , we consider the map coded by w , j w : = j w ◦ · · · ◦ j w n : X → X . For w ∈ I N , the sets (cid:8) j w | n ( X ) (cid:9) n ≥ form a descending sequence of non-empty compact setsand therefore T n ≥ j w | n ( X ) = ∅ . Since for every n ∈ N , diam (cid:0) j w | n ( X ) (cid:1) ≤ s n F diam ( X ) ,we conclude that the intersection T j w | n ( X ) ∈ X is a singleton and we denote its onlyelement by p F ( w ) . In this way we have defined the coding map p = p F : I N → X . The set L = L F = p (cid:0) I N (cid:1) will be called the limit set of F . Definition 2.1.
We call an iterated function system conformal (cIFS) if the following con-ditions are satisfied.(a) The phase space X is a compact connected subset of a Euclidean space R D , D ≥ X is equal to the closure of its interior, i. e. X = Int ( X ) .(b) ( Open set condition (OSC)) For all a , b ∈ I , a = b , j a ( Int ( X )) ∩ j b ( Int ( X )) = ∅ . (c) There exists an open connected set W ⊃ X such that for every e ∈ I the map j e extends to a C conformal diffeomorphism of W into W .(d) ( Cone property ) There exist g , l > g < p /
2, such that for every x ∈ X ⊂ R D thereexists an open cone Con ( x , g , l ) ⊂ Int ( X ) with vertex x , central angle of measure g , and altitude l .(e) There are two constants L = L F ≥ a = a F > (cid:12)(cid:12) | j ′ e ( y ) | − | j ′ e ( x ) | (cid:12)(cid:12) ≤ L F k ( j ′ e ) − k X k y − x k a for every e ∈ I and every pair of points x , y ∈ X .For the following let s F : = sup e ∈ I (cid:13)(cid:13) ( j e ) ′ (cid:13)(cid:13) X < . (2.1)For a fixed phase space X satisfying (a) the set of conformal iterated function systems willbe denoted CIFS ( X ) : = (cid:8) F = ( j e : X −→ X ) e ∈ I cIFS, I ⊂ N (cid:9) . The following fact was proved in [MU03].
Proposition 2.2.
For D ≥ , any family F = ( j e ) e ∈ I satisfying condition (a) and (c) alsosatisfies condition (e) with a = . In [MU03] we also find the following straightforward consequence of (e).
JOHANNES JAERISCH AND MARC KESSEBÖHMER
Lemma 2.3. If F = ( j e ) e ∈ I is a cIFS, then for all w ∈ I ∗ and all x , y ∈ W , we have (cid:12)(cid:12) log | j ′ w ( y ) | − log | j ′ w ( x ) | (cid:12)(cid:12) ≤ L − s a k y − x k a . Another consequence of (e) is(f) (
Bounded distortion property ). There exists K F ≥ w ∈ I ∗ andall x , y ∈ X | j ′ w ( y ) | ≤ K F | j ′ w ( x ) | . In [MU03, Lemma 2.3.1] it has been shown that for a Hölder continuous function g : J N → R , J ⊂ I , we have for all w ∈ J ∗ and all x , y ∈ [ w ] thatexp S | w | g ( x ) ≤ K g exp S | w | g ( y ) . In here, the constant K g ≥ g , as well as the metric on J N . With Z n ( g ) : = (cid:229) w ∈ J n exp sup t ∈ [ w ] ∩ J N ( S n g ( t )) we will denote the n-th partition function of g . Definition 2.4.
The topological pressure P ( f ) of a continuous function f : I N → R isdefined by the following limit, which always exists (possibly equal to + ¥ ), P ( f ) : = lim n → ¥ n log Z n ( f ) = inf n n log Z n ( f ) . At the end of this section we would like to comment on the topology of pointwise conver-gence. r is well defined, since (cid:13)(cid:13) j i − j i (cid:13)(cid:13) X + (cid:13)(cid:13) ( j i ) ′ − ( j i ) ′ (cid:13)(cid:13) X is bounded by diam ( X )+ A △ C ⊂ A △ B ∪ B △ C for arbitrary sets A , B , C , we readily observethat r as given in (1.4) actually defines a metric on CIFS ( X ) . This metric induces the topology of pointwise convergence on CIFS ( X ) . Let F n = ( j ni : X −→ X ) i ∈ I n , n ∈ N , and F = ( j i : X −→ X ) i ∈ I be elements of CIFS ( X ) with F n → F pointwise. Then for every k ∈ N we find an integer N k such that and all n ≥ N k we have I n △ I ⊂ { k + , k + , . . . } . Similarly as in [RU05, Lemma 5.1] it follows that pointwise convergence in CIFS ( X ) isequivalent to the following condition. Condition 2.5.
We have ( F n , y n ) → ( F , y ) pointwise if and only if for every w ∈ I ∗ lim n → ¥ (cid:0)(cid:13)(cid:13) j n w − j w (cid:13)(cid:13) X + (cid:13)(cid:13) ( j n w ) ′ − j ′ w (cid:13)(cid:13) X (cid:1) = n → ¥ (cid:16)(cid:13)(cid:13) S | w | y n − S | w | y (cid:13)(cid:13) [ w ] ∩ I N n (cid:17) = Lemma 2.6.
Assume that ( F n , y n ) → ( F , y ) converges pointwise. Then there exists M > such that for every w ∈ I ∗ fixed and all sufficiently large n ∈ N (depending on w ) we havefor all h ∈ [ w ] ∩ I N n and t ∈ [ w ] ∩ I N e S | w | y n ( h ) e S | w | y ( t ) , e S | w | z n ( h ) e S | w | z ( t ) ∈ (cid:2) M − , M (cid:3) . Proof.
Using the above Condition 2.5 we find for w ∈ I ∗ and n sufficiently large thatmax n(cid:13)(cid:13) log (cid:12)(cid:12) ( j n w ) ′ (cid:12)(cid:12) − log (cid:12)(cid:12) ( j w ) ′ (cid:12)(cid:12)(cid:13)(cid:13) X , (cid:13)(cid:13) S | w | y n − S | w | y (cid:13)(cid:13) [ w ] ∩ I N n o ≤ . EGULARITY OF THE MULTIFRACTAL SPECTRUM OF CONFORMAL IFS 9
Then we have with K F and K y denoting the bounded distortion constants as defined above (cid:12)(cid:12) S | w | z n ( h ) − S | w | z ( t ) (cid:12)(cid:12) = (cid:12)(cid:12) log (cid:12)(cid:12) ( j n w ) ′ (cid:12)(cid:12) ( p F n ( s n ( h ))) − log (cid:12)(cid:12) ( j w ) ′ (cid:12)(cid:12) ( p F ( s n ( t ))) (cid:12)(cid:12) ≤ (cid:12)(cid:12) log (cid:12)(cid:12) ( j n w ) ′ (cid:12)(cid:12) ( p F n ( s n ( h ))) − log (cid:12)(cid:12) ( j w ) ′ (cid:12)(cid:12) ( p F n ( s n ( h ))) (cid:12)(cid:12) + (cid:12)(cid:12) log (cid:12)(cid:12) ( j w ) ′ (cid:12)(cid:12) ( p F n ( s n ( h ))) − log (cid:12)(cid:12) ( j w ) ′ (cid:12)(cid:12) ( p F ( s n ( t ))) (cid:12)(cid:12) ≤ + log K F as well as (cid:12)(cid:12) S | w | y n ( h ) − S | w | y ( t ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) S | w | y n ( h ) − S | w | y ( h ) (cid:12)(cid:12) + (cid:12)(cid:12) S | w | y ( h ) − S | w | y ( t ) (cid:12)(cid:12) ≤ + log K y . Letting M : = · max (cid:8) K F , K y (cid:9) the lemma follows. (cid:3) Remark . Note that we may replace condition (B) in Definition 1.3 by the slightlyweaker conditions on y n and y stated in the above Lemma combined with the conditionthat y n converges uniformly to y on compact s -invariant subsets of I N .3. R EGULARITY OF THE FREE ENERGY FUNCTION
In this section we give a proof of Theorem 1.7. Let z denote the geometric potential func-tion associated with F as defined in the introduction. For F ∈ CIFS ( X ) and a Hölder con-tinuous potential y : I N −→ R let t denote the free energy function of ( F , y ) as introducedin Definition 1.1, i. e. t ( b ) : = inf { t : P ( t z + by ) ≤ } . Clearly, if there exists a zero of t P ( t z + by ) then t ( b ) is the unique zero of this function (which in particular is thecase for a finite alphabet I ). Also, t ( b ) = + ¥ if and only if { t : P ( t z + by ) ≤ } = ∅ . Lemma 3.1.
The free energy t of ( F , y ) is a proper (not necessarily closed) convex func-tion on R .Proof. Fix b , b ∈ R , l ∈ ( , ) and e >
0. Using the convexity of the topological pressurewe have P (( l t ( b ) + ( − l ) t ( b ) + e ) z + ( lb + ( − l ) b ) y )= P ( l (( t ( b ) + e ) z + b y ) + ( − l )(( t ( b ) + e ) z + b y )) ≤ l P (( t ( b ) + e ) z + b y ) + ( − l ) P (( t ( b ) + e ) z + b y ) ≤ . Hence, by definition of t , this implies t ( lb + ( − l ) b ) ≤ l t ( b ) + ( − l ) t ( b ) + e .Since e > t is a proper convex function observe that − ¥ < P ( by ) and hence, for t < n log (cid:229) w ∈ I n exp sup t ∈ [ w ] ( S n t z ( t ) + by ) ≥ t log ( s F ) + P ( by ) → ¥ for t → − ¥ . Consequently, t ( b ) > − ¥ for all b ∈ R . Since also t ( ) = dim H ( L F ) < ¥ we have that t is proper. (cid:3) Lemma 3.2.
Let ( F n , y n ) n −→ ( F , y ) converge regularly. Then for all t , b ∈ R and ntending to infinity we have P ( t z n + by n ) −→ P ( t z + by ) . Proof.
Fix t , b ∈ R with P ( t z + by ) < ¥ and e >
0. With k ∈ N and C > m ∈ N large enough such that ( mk ) − log Z mk ( t z + by ) ≤ P ( t z + by ) + e / ( mk ) − log ( K + ) < e /
2, where K : = M | b | + | t | and M is the con-stant defined in the proof of Lemma 2.6. We prove that for all n ∈ N sufficiently large wehave ( mk ) − log Z mk ( t z n + by n ) ≤ ( mk ) − log Z mk ( t z + by ) + e / . (3.1)This would imply P ( t z n + by n ) ≤ ( mk ) − log Z mk ( t z n + by n ) ≤ ( mk ) − log Z mk ( t z + by ) + e / ≤ P ( t z + by ) + e for sufficiently large n ∈ N . To prove (3.1) we first choose a finite set F ⊂ I such that (cid:229) w ∈ I mk \ F mk exp sup r ∈ [ w ] ∩ I N ( S mk ( t z + by )( r )) < ( CK ) m Z mk ( t z + by ) . Then by Definition 1.4 and the choice of F we have (cid:229) w ∈ I mkn \ F mk exp sup r ∈ I N n ∩ [ w ] ( S mk ( t z n + by n ) ( r )) ≤ (cid:229) w ∈ I kn \ F k exp sup r ∈ I N n ∩ [ w ] ( S k ( t z n + by n ) ( r )) m ≤ C m (cid:229) w ∈ I kn \ F k exp sup r ∈ I N ∩ [ w ] ( S k ( t z + by )( r )) m ≤ ( CK ) m (cid:229) w ∈ I mkn \ F mk exp sup r ∈ I N ∩ [ w ] ( S mk ( t z + by )( r )) < Z mk ( t z + by ) Hence, on the one hand, we have Z mk ( t z n + by n ) = (cid:229) w ∈ I mkn ∩ F mk exp sup r ∈ I N n ∩ [ w ] ( S mk ( t z n + by n ) ( r ))+ (cid:229) w ∈ I mkn \ F mk exp sup r ∈ I N n ∩ [ w ] ( S mk ( t z n + by n ) ( r )) ≤ (cid:229) w ∈ I mkn ∩ F mk exp sup r ∈ I N n ∩ [ w ] ( S mk ( t z n + by n ) ( r )) + Z mk ( t z + by ) . To find an upper bound also for the finite sum in the latter inequality we note that byLemma 2.6 we have for every t ∈ R and w ∈ I mkn ∩ F mk and for sufficiently large n thatexp sup t ∈ I N n ∩ [ w ] ( S mk ( t z n + by n ) ( t )) ≤ K exp sup r ∈ I N ∩ [ w ] ( S mk ( t z + by )( r )) . Since I mkn ∩ F mk is finite we have on the other hand for n sufficiently large that (cid:229) w ∈ I mkn ∩ F mk e sup r ∈ I N n ∩ [ w ] ( S mk ( t z n + by n )( r )) ≤ K (cid:229) w ∈ I mkn ∩ F mk e sup r ∈ I N ∩ [ w ] ( S mk ( t z + by )( r )) . Combining both estimates we find for n sufficiently large Z mk ( t z n + by n ) ≤ ( K + ) Z mk ( t z + by ) . Taking logarithm and dividing by mk proves (3.1).To prove the reverse inequality lim inf n P ( t z n + by n ) ≥ P ( t z + by ) let us fix e > F ⊂ I such that P ( t z (cid:12)(cid:12) F N + by (cid:12)(cid:12) F N ) ≥ P ( t z + by ) − e . EGULARITY OF THE MULTIFRACTAL SPECTRUM OF CONFORMAL IFS 11
By [RU05, Lemma 4.2] and Definition 1.3 (B) we have (cid:13)(cid:13) z n (cid:12)(cid:12) F N − z (cid:12)(cid:12) F N (cid:13)(cid:13) F N → (cid:13)(cid:13) y n (cid:12)(cid:12) F N − y (cid:12)(cid:12) F N (cid:13)(cid:13) F N →
0. Since f P ( f ) is Lipschitz-continuous with respect to the k · k F N -norm (cf. [Wal82, Theorem 9.7]) we concludelim inf n P ( t z n + by n ) ≥ lim inf n P ( t z n (cid:12)(cid:12) F N + by n (cid:12)(cid:12) F N )= P ( t z (cid:12)(cid:12) F N + by (cid:12)(cid:12) F N ) ≥ P ( t z + by ) − e . (cid:3) Proof of Theorem 1.7.
Fix b ∈ R . To verify lim sup n t n ( b ) ≤ t ( b ) we may assume t ( b ) < ¥ . Since the map p b : t P ( t z + by ) is strictly decreasing on dom p b we have that P (( t ( b ) + d ) z + by ) < d >
0. As a consequence of Lemma 3.2 we have P (( t ( b ) + d ) z n + by n ) < n sufficiently large. This implies t n ( b ) ≤ t ( b ) + d for all d > n t n ( b ) < t ( b ) .To verify lim inf n t n ( b ) ≥ t ( b ) we first assume that t ( b ) < ¥ . By definition of t we have P (( t ( b ) − d ) z + by ) > d >
0. Then again by Lemma 3.2 we also have P (( t ( b ) − d ) z n + by n ) > n sufficiently large, which in turn implies t n ( b ) ≥ t ( b ) − d for all n large enough. Finally, let t ( b ) = ¥ , i. e. P ( t z + by ) = ¥ for all t .By Lemma 3.2 we have for any t ∈ R fixed P ( t z n + by n ) > n large enough andhence t n ( b ) ≥ t for n large enough. Since t ∈ R was arbitrary it follows that t n ( b ) tends toinfinity as n increases. (cid:3)
4. C
ONVERGENCE AND CONJUGACY OF CONVEX FUNCTIONS
In this section we collect the necessary basic facts from convex analysis needed for themultifractal analysis in Section 5. We closely follow [SW77], and all details can be foundeither therein or in [Roc70].The following proposition is a direct consequence of [SW77, Corollaries 2C and 3B] com-bined with the fact that Legendre conjugation is continuous with respect to the convergenceof epigraphs in the classical sense as defined e. g. by Kuratowski in [Kur66].
Proposition 4.1.
Let g n , g, n ∈ N , be closed convex functions on R such that Int ( dom ( g )) = ∅ and g n −→ g pointwise. Then pointwise on Int ( dom ( g ∗ )) , we have g ∗ n −→ g ∗ . The following corollary allows us to apply Proposition 4.1 also in the case when the func-tions g n , g are not closed. Corollary 4.2.
Let g n , g, n ∈ N , be convex functions on R and a ∈ R such that there existx , x ∈ Int ( dom ( g )) with x < x and g + ( x ) < a < g − ( x ) . Furthermore, assume thatthere exists an open neighbourhood U ⊂ dom ( g ) containing x , x such that g n (cid:12)(cid:12) U −→ g (cid:12)(cid:12) U pointwise. Then we have g ∗ n ( a ) −→ g ∗ ( a ) .Proof. Without loss of generality we have g n (cid:12)(cid:12) U < ¥ for all n ∈ N . Let A denote theindicator function on the set A and let e g n , e g denote the closed convex functions given by e g n : = g n U + ¥ R \ U and e g : = g U + ¥ R \ U . Notice that these closed convex functionsagree on U with the original functions. By Proposition 4.1 we conclude that e g ∗ n −→ e g ∗ pointwise on Int ( dom e g ∗ ) . Clearly by our assumptions, a ∈ Int ( dom e g ∗ ) and a belongsto the subdifferential ¶ g ( x ) : = { a ∈ R : ∀ x ′ ∈ R g ( x ′ ) − g ( x ) ≥ a ( x ′ − x ) } for some x ∈ U ,and hence e g ∗ ( a ) = a x − e g ( x ) = g ∗ ( a ) by [Roc70, Theorem 23.5]. It remains to showthat e g ∗ n ( a ) = g ∗ n ( a ) for n sufficiently large. Since by [Roc70, Theorem 24.5] the sub-differentials converge, the assumption g + ( x ) < a < g − ( x ) implies that a ∈ ¶ g n ( y n ) for some y n ∈ U and n large. Then again by [Roc70, Theorem 23.5] we have e g ∗ n ( a ) = a y n − g n ( y n ) = g ∗ n ( a ) . (cid:3)
5. R
EGULARITY OF THE MULTIFRACTAL SPECTRUM
We proceed by proving the Theorems 1.2 and 1.8. Recall that throughout we use thegeneralised version of the free energy function t as stated in Definition 1.1. Proof of Theorem 1.2.
Using the definition of topological pressure and a standard coveringargument (just cover F a with cylinder sets) we obtain f ( a ) ≤ max {− t ∗ ( − a ) , } forevery a ∈ R .We will use the Exhausting Principle to prove the reverse inequality. Let F n = ( j e ) e ∈ I n with I n : = I ∩ { , . . . n } and y n : = y (cid:12)(cid:12) I N n , n ∈ N . Clearly, ( F n , y n ) n −→ ( F , y ) convergesregularly (Example 1.6) and hence by Theorem 1.7, we conclude that t n −→ t pointwise on R . Note that for a ∈ ( a − , a + ) we find x , x ∈ Int ( dom ( t )) with x > x and − t − ( x ) < a < − t + ( x ) . Hence by Corollary 4.2, we concludelim n → ¥ − t ∗ n ( − a ) = − t ∗ ( − a ) . Since the functions t n are finite and differentiable on R we conclude by [Roc70, Theorem24.5] that a ∈ − t ′ n ( R ) for all n large enough. Recall that in the finite alphabet case it iswell-known that f n ( a ) = − t ∗ n ( − a ) . By construction we have F n a ⊂ F a and hence − t ∗ ( − a ) ≥ f ( a ) ≥ f n ( a ) = − t ∗ n ( − a ) −→ − t ∗ ( − a ) . (cid:3) Proof of Theorem 1.8.
By Theorem 1.7 the free energy functions t n converge pointwiseto t on R . For a ∈ ( a − , a + ) we have by Corollary 4.2 that lim n − t ∗ n ( − a ) = − t ∗ ( − a ) .Furthermore by Theorem 1.2, we have − t ∗ ( − a ) = f ( a ) . We also have − t ∗ n ( − a ) = f n ( a ) for large n by Theorem 1.2, since [Roc70, Theorem 24.5] implies a ∈ (cid:0) a n − , a n + (cid:1) for large n . This proves the first part of the theorem.We now consider the case sup dom ( t ) = + ¥ (the second case is proved along the samelines). Let us assume on the contrary that there exists e > a n − < a − − e for infinitelymany n . For fixed K > x ∈ dom ( t ) with x > ( K + ) / e and such that t ′ ( x ) exists. By Theorem 1.7 we find n ∈ N such that | t n ( x ) − t ( x ) | < a n − < a − − e .Furthermore, we can choose a ∈ (cid:0) a n − , a n + (cid:1) satisfying a < a − − e . By Theorem 1.2 wehave f n ( a ) = − t ∗ n ( − a ) . Since a + t ′ ( x ) ≤ a − a − < − e we have by [Roc70, Theorem23.5] f n ( a ) + t ∗ (cid:0) t ′ ( x ) (cid:1) = − t ∗ n ( − a ) + t ∗ (cid:0) t ′ ( x ) (cid:1) = inf b { t n ( b ) + ba } − t ( x ) + xt ′ ( x ) ≤ ( t n ( x ) + x a ) − t ( x ) + xt ′ ( x ) ≤ + x (cid:0) a + t ′ ( x ) (cid:1) < − e x < − K . Since t ∗ ( t ′ ( x )) ≥ − t ( ) = − dim H ( L ) and K can be chosen arbitrary large we get a con-tradiction to f n ( a ) ≥ (cid:3) Remark . Assume in the situation of Example 1.6 that ( a − , a + ) is a bounded interval.If t ∗ n ( a ± ) −→ t ∗ ( a ± ) then f n ( a ± ) −→ f ( a ± ) and we have − t ∗ ( − a ± ) = f ( a ± ) . To seethis notice that a
7→ − t ∗ ( − a ) is bounded on ( a − , a + ) since it coincides with the Hausdorffdimension of certain sets. Furthermore, by [Roc70, Theorem 12.2] we have that t ∗ is closed EGULARITY OF THE MULTIFRACTAL SPECTRUM OF CONFORMAL IFS 13 and hence − t ∗ ( − a ± ) < ¥ . By our assumption we have − t ∗ n ( − a ± ) < ¥ for n large, hence a ± ∈ − t ′ n ( R ) . Then the claim follows by observing that − t ∗ n ( − a ± ) = f n ( a ± ) ≤ f ( a ± ) ≤ − t ∗ ( − a ± ) . Finally, we will sketch the proof of (1.5) for the Gauss system as announced in the intro-duction. Let n : = ( n , n , n , . . . ) , n ≥
2, and suppose that w ∈ { , . . . , n } k , k ∈ N , differs from n (cid:12)(cid:12) k in at least ℓ positions. Let q k ( w ) denote the denominator of the k ’s approximant of thecontinued fraction expansion [ w , . . . , w k ] ∈ [ , ] . Then, by the recursive definition of q k ,we have q k ( w ) q k ( n ) > (cid:18) n − n (cid:19) ℓ . (5.1)From this it follows that a n + = y ( n ) z ( n ) = lim k → ¥ k log ( n ) log q k ( n ) = − log ( n ) log (cid:16) − n / + p n / + (cid:17) < f n (cid:0) a n + (cid:1) = (cid:26) m ∈ M (cid:0) { , . . . , n } N , s (cid:1) : ˆ y d m / ˆ z d m = a n + (cid:27) = (cid:8) d n (cid:9) , where M (cid:0) { , . . . , n } N , s (cid:1) denotes the set of shift invariant measures and d x the Diracmeasure centred on x . For the detailed argument see [KS07]. We prove this fact by wayof contradiction. Assume there exists m ∈ M (cid:16) { , . . . , n } N , s (cid:17) with m = d n . By convexityof the set of measures under consideration we may assume that m is ergodic. Then by ourassumption there exists ℓ < n with m ([ ℓ ]) = h >
0. Then for all m -typical points w wehave lim k S k y ( w ) / S k z ( w ) = lim k (cid:229) ki = log w i / log q k ( w ) = a n + and S k [ ℓ ] ( w ) ≥ h k / k ∈ N sufficiently large. Hence, using (5.1), we obtain k log ( n ) log q k ( n ) − (cid:229) k log w i log q k ( w ) == k log ( n ) log q k ( w ) − log q k ( n ) (cid:229) k log w i log q k ( w ) log q k ( n )= (cid:0) (cid:229) ki = log ( n ) − log ( w i ) (cid:1) log q k ( w ) + ( log q k ( w ) − log q k ( n )) (cid:229) k log w i log q k ( w ) log q k ( n ) ≥ k ( h / ) ( log ( n ) − log ( n − )) log q k ( w ) + ( log q k ( w ) − log q k ( n )) (cid:229) ki = log w i log q k ( w ) log q k ( n ) ≥ k h ( log ( n ) − log ( n − )) (cid:0) log q k ( w ) − (cid:229) ki = log w i (cid:1) q k ( w ) log q k ( n ) → ha n + n ( log ( n ) − log ( n − )) (cid:0) − a n + (cid:1) > . Since lim k → ¥ k log ( n ) / log q k ( n ) = a n + we obtain a contradiction.6. T HE EXTENDED l - TOPOLOGY
In this section we compare the notion of regular convergence with the l -topology intro-duced by Roy and Urba´nski. In particular, as a consequence of Proposition 6.4 we willverify Example 1.5. For ease of notation we will always assume I = N . Let us first recall the definition ofthe l -topology from [RU05] and then give a generalisation to adapt this concept to ourpurposes.For F n = ( j ne ) e ∈ I , F = ( j e ) e ∈ I elements of CIFS ( X ) sharing the same alphabet I we saythat F n converges to F in the l -topology, if F n → F in the r -metric and there exists R > n and all e ∈ I we have R − ≤ (cid:13)(cid:13) ( j ne ) ′ (cid:13)(cid:13) X (cid:13)(cid:13) j ′ e (cid:13)(cid:13) X ≤ R . (6.1)We shall generalise this to the case where we have I n ⊂ I = N . We say that F n = ( j ne ) e ∈ I n converges to F = ( j e ) e ∈ I in the extended l -topology of CIFS ( X ) , if they converge in the r -metric and there exists D > n sufficiently large and all e ∈ I n theassumption (6.1) holds.Let us begin with the following basic lemma. Lemma 6.1.
Let F n = ( j ne ) e ∈ I n , F = ( j e ) e ∈ N be elements of CIFS ( X ) with F n → F con-verging in the extended l -topology. Then with s F defined in (2.1), we havelim n s F n = s F < . Proof.
By the open set condition (OSC) there exists e ∈ I with (cid:13)(cid:13) j ′ e (cid:13)(cid:13) X = sup e (cid:13)(cid:13) j ′ e (cid:13)(cid:13) X = s F as well as for every n ∈ N there exists e n ∈ I n satisfying (cid:13)(cid:13) ( j ne n ) ′ (cid:13)(cid:13) X = sup e (cid:13)(cid:13) ( j ne ) ′ (cid:13)(cid:13) X = s F n .Since lim n (cid:13)(cid:13) ( j ne ) ′ (cid:13)(cid:13) X = (cid:13)(cid:13) j ′ e (cid:13)(cid:13) X we have lim inf n s F n ≥ s F .Next we conclude that { e n : n ∈ N } is contained in a finite set F ⊂ N . This follows by wayof contradiction. Assume the set is infinite. Then there exists a subsequence n k such thaton the one hand (cid:13)(cid:13) ( j ′ e nk ) (cid:13)(cid:13) X → (cid:13)(cid:13) ( j n k e nk ) ′ (cid:13)(cid:13) X → lim inf n s F n ≥ s F >
0. This would contradict property (6.1) defining the extended l -topology. Now by thedefinition of the r -metric we have for all ℓ ∈ F that lim n (cid:13)(cid:13) ( j n ℓ ) ′ (cid:13)(cid:13) X = (cid:13)(cid:13) ( j ℓ ) ′ (cid:13)(cid:13) X ≤ s F . Thisgives lim sup n s F n ≤ s F . (cid:3) For the following let y n : I N n −→ R and y : I N −→ R be Hölder continuous functions,satisfying condition (B) in Definition 1.3. Assumption 6.2.
Additionally, we assume that there exist k ∈ N and M ∈ N , such that forall n ∈ N and for all w ∈ I kn , t ∈ I N n ∩ [ w ] and h ∈ I N ∩ [ w ] we haveM − ≤ exp ( S k ( y n ) ( t )) exp ( S k ( y ) ( h )) ≤ M . Remark . Assumption 6.2 is for instance satisfied, if(1) sup n k y n k I N n < ¥ and k y k I N < ¥ , or(2) sup n k y n − y k I N n < ¥ .For the following proposition recall that K F denotes the bounded distortion constant for F as stated in condition (f) of the definition of a cIFS. Proposition 6.4.
Let F n = ( j ne ) e ∈ I n , F = ( j e ) e ∈ N be elements of CIFS ( X ) with F n → F converging in the extended l -topology. Let y n : I N n −→ R and y : I N −→ R be Hölder con-tinuous functions satisfying condition (B) in Definition 1.3 as well as Assumption 6.2. Then ( F n , y n ) n → ( F , y ) converges regularly, if one of the following conditions is satisfied: (1) sup n K F n < ¥ . (2) k y k I N < ¥ . (3) inf n a F n > and sup n L F n < ¥ . EGULARITY OF THE MULTIFRACTAL SPECTRUM OF CONFORMAL IFS 15 (4) D ≥ and the maps F ne extend to conformal diffeomorphisms on a common neigh-bourhood W ⊃ X into W for all e ∈ I n and n ∈ N .Proof. Clearly ( F n , y n ) n → ( F , y ) converges pointwise. Hence, we are left to verify thecondition in Definition 1.4 under the assumption (1) as well as under the assumption (2),and then show that both (3) and (4) imply (1). ad (1): For t ≥ k ∈ N , w ∈ I kn and n sufficiently large, such that(6.1) and Assumption 6.2 hold. Using this and the bounded distortion property of F fromLemma 2.3 with bounded distortion constant K = K F we obtainexp sup t ∈ I N n ∩ [ w ] ( S k ( z n ) ( t )) ≤ (cid:13)(cid:13) ( j n w ) ′ (cid:13)(cid:13) X ≤ k (cid:213) i = (cid:13)(cid:13) ( j n w i ) ′ (cid:13)(cid:13) X ≤ R k k (cid:213) i = (cid:13)(cid:13) ( j w i ) ′ (cid:13)(cid:13) X ≤ R k K k (cid:13)(cid:13) ( j w ) ′ (cid:13)(cid:13) X ≤ K k + R k exp sup r ∈ I N ∩ [ w ] ( S k ( z ) ( r )) . Combining this with Assumption 6.2 we have with C : = K t ( k + ) R tk M | b | and t ≥ t ∈ I N n ∩ [ w ] ( S k ( t z n + by n ) ( t )) ≤ exp sup t ∈ I N n ∩ [ w ] ( S k ( t z n ) ( t )) exp sup t ∈ I N n ∩ [ w ] ( S k ( by n ) ( t )) ≤ C exp sup r ∈ I N ∩ [ w ] ( S k ( t z ) ( r )) exp inf t ∈ I N ∩ [ w ] ( S k ( by ) ( t )) ≤ C exp sup r ∈ I N ∩ [ w ] ( S k ( t z + by )( r )) . For t < e K : = sup K F n < ¥ thatexp sup t ∈ I N n ∩ [ w ] ( S k ( z n ) ( t )) ≥ e K − (cid:13)(cid:13) ( j n w ) ′ (cid:13)(cid:13) X ≥ e K − k − k (cid:213) i = (cid:13)(cid:13) ( j n w i ) ′ (cid:13)(cid:13) X ≥ R − k e K − k − k (cid:213) i = (cid:13)(cid:13) ( j w i ) ′ (cid:13)(cid:13) X ≥ R − k e K − k − (cid:13)(cid:13) ( j w ) ′ (cid:13)(cid:13) X ≥ R − k e K − k − exp sup r ∈ I N ∩ [ w ] ( S k ( z ) ( r )) . As above we have with C : = R − tk e K − t ( k + ) M | b | and t < t ∈ I N n ∩ [ w ] ( S k ( t z n + by n ) ( t )) ≤ exp sup t ∈ I N n ∩ [ w ] ( S k ( t z n ) ( t )) exp sup t ∈ I N n ∩ [ w ] ( S k ( by n ) ( t )) ≤ C exp sup r ∈ I N ∩ [ w ] ( S k ( t z ) ( r )) exp inf t ∈ I N ∩ [ w ] ( S k ( by ) ( t )) ≤ C exp sup r ∈ I N ∩ [ w ] ( S k ( t z + by )( r )) . ad (2) : Since the potential y is bounded and the topological entropy infinite we have P ( by ) = ¥ . Hence, to verify the condition in Definition 1.4 we only have to concider thecase t >
0, since for t ≤ P ( t z + by ) ≥ P ( by ) = ¥ . But this case has beentreated in (1) without any additional assumption on F . (3) = ⇒ = ⇒ = ⇒ (1) : Since by Lemma 6.1 the contraction ratios s F n of F n defined in (2.1) convergeto s F < n K F n < ¥ . (4) = ⇒ = ⇒ = ⇒ (1) : This implication is an immediate consequence of [MU03, Theorems 4.1.2 and4.1.3]. See also Proof of Claim in the proof of Theorem 5.20 in [RSU09] for a similarargument. (cid:3)
Acknowledgement.
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