On the asymptotics of the α -Farey transfer operator
Johannes Kautzsch, Marc Kesseböhmer, Tony Samuel, Bernd O. Stratmann
OON THE ASYMPTOTICS OF THE α -FAREY TRANSFER OPERATOR J. KAUTZSCH, M. KESSEB ¨OHMER, T. SAMUEL, AND B. O. STRATMANNA bstract . We study the asymptotics of iterates of the transfer operator for non-uniformlyhyperbolic α -Farey maps. We provide a family of observables which are Riemann integrable,locally constant and of bounded variation, and for which the iterates of the transfer operator,when applied to one of these observables, is not asymptotic to a constant times the wanderingrate on the first element of the partition α . Subsequently, su ffi cient conditions on observablesare given under which this expected asymptotic holds. In particular, we obtain an extensiontheorem which establishes that, if the asymptotic behaviour of iterates of the transferoperator is known on the first element of the partition α , then the same asymptotic holds onany compact set bounded away from the indi ff erent fixed point.
1. I ntroduction
Expanding maps of the unit interval have been widely studied in the last decades andthe associated transfer operators have proven to be of vital importance in solving problemsconcerning the statistical behaviour of the underlying interval maps [3, 18].In recent years an increasing amount of interest has developed in maps which areexpanding everywhere except on an unstable fixed point (that is, an indi ff erent fixed point)at which trajectories are considerably slowed down. This leads to an interplay of chaoticand regular dynamics, a characteristic of intermittent systems [20, 22]. From an ergodictheory viewpoint, this phenomenon leads to an absolutely continuous invariant measurehaving infinite mass. Therefore, standard methods of ergodic theory cannot be applied inthis setting; indeed it is wellknown that Birkho ff ’s ergodic theorem does not hold underthese circumstances, see for instance [1].In this paper we will be concerned with α -Farey maps, see Figure 1. These maps are ofgreat interest since they are piecewise linear and expanding everywhere except for at theindi ff erent fixed point where they have (right) derivative one. This makes the α -Farey mapsa simple model for studying the physical phenomenon of intermittency [20]. Moreover, aninduced version of the α -Farey maps are given by the α -L¨uroth maps introduced in [17],which have significant meaning in number theory, see for instance [4, 15].Thaler [24] was the first to discern the asymptotics of the transfer operator of a classof interval maps preserving an infinite measure. This class of maps, to which the α -Fareymaps do not belong, have become to be known as Thaler maps. In an e ff ort to generalisethis work, by combining renewal theoretical arguments and functional analytic techniques,a new approach to estimate the decay of correlation of a dynamical system was achievedby Sarig [21]. Subsequently, Gou¨ezel [10, 11, 12] generalised these methods. Using theseideas and employing the methods of Garsia and Lamperti [9], Erickson [8] and Doney[7], recently Melbourne and Terhesiu [19, Theorem 2.1 to 2.3] proved a landmark resulton the asymptotic rate of convergence of iterates of the induced transfer operator andshowed that these result can be applied to Gibbs-Markov maps, Thaler maps, AFN maps,and Pomeau-Manneville maps. Thus, the question which naturally arises is, whether thisasymptotic rate can be related to the asymptotic rate of convergence of iterates of the actual The first two authors were supported by the German Research Foundation (DFG) grant
Renewal Theory andStatistics of Rare Events in Infinite Ergodic Theory (Gesch¨aftszeichen KE 1440 / a r X i v : . [ m a t h . D S ] O c t J. KAUTZSCH, M. KESSEB ¨OHMER, T. SAMUEL, AND B. O. STRATMANN transfer operator. The results of this paper give some positive answers to this question for δ -expansive α -Farey maps.As mentioned above, in this paper, we will consider the α -Farey map F α : [0 , → [0 , α (cid:66) { A n : n ∈ N } of (0 ,
1) by non-emptyintervals A n . It is assumed throughout that the atoms of α are ordered from right to left,starting with A , and that these atoms only accumulate at zero. Further, we assume that A n is right-open and left-closed, for all natural numbers n . We define the α - Farey mapF α : [0 , → [0 ,
1] by F α ( x ) (cid:66) (1 − x ) / a if x ∈ A (cid:66) A ∪ { } , a n − ( x − t n + ) / a n + t n if x ∈ A n , for n ≥ , x = , where a n is equal to the Lebesgue measure λ ( A n ) of the atom A n ∈ α and t n (cid:66) (cid:80) ∞ k = n a k denotes the Lebesgue measure of the n -th tail (cid:83) ∞ k = n A k of α , see Figure 1. Throughout,we will assume that the partition α satisfies the condition that the sequence ( t n ) n ∈ N is notsummable. For δ ∈ (0 , α -Farey map F α is said to be δ -expansive if the sequence( a n ) n ∈ N is regularly varying of order − (1 + δ ), that is, if there exists a slowly varyingfunction l : R → R such that a n = δ l ( n ) n − (1 + δ ) , for all n ∈ N . (Recall that l : [ a , ∞ ) → R is called a slowly varying function , if it is measurable, locally Riemann integrable andlim x →∞ l ( η x ) / l ( x ) =
1, for each η > a ∈ R , see [6, 23] for further details.) Inthis situation, [6, Theorem 1.5.10] implies thatlim n →∞ l ( n ) n − δ t n = lim n →∞ l ( n ) n − δ (cid:80) ∞ k = n a k = lim n →∞ l ( n ) n − δ (cid:80) ∞ k = n δ l ( n ) n − (1 + δ ) = . Therefore, the Lebesgue measure of the n -th tail of α is asymptotic to a regularly varyingfunction of order − δ . Thus, δ -expansive implies expansive of order δ in the sense of [15].However, an expansive α -Farey map of order δ is not necessarily δ -expansive. t ≔ t t t ... A A A ......... ( a ) δ = / t ≔ t t t ... A A A ......... ( b ) δ = F igure
1. The α -Farey map, where t n = n − δ , for all n ∈ N .Throughout, let µ α denote the F α -invariant measure which is determined by h α (cid:66) d µ α d λ = (cid:88) n ∈ N t n a n A n (1)and let B denote the Borel σ -algebra of [0 , B ∈ B , we let B denote the indicator function on B . It is verified in [15] that, since N THE ASYMPTOTICS OF THE α -FAREY TRANSFER OPERATOR 3 the sequence ( a n ) n ∈ N is regularly varying of order − (1 + δ ), the map F α is conservative,ergodic and measure preserving on the infinite and σ -finite measure space ([0 , , B , µ α ).The dynamical system ([0 , , B , µ α , F α ) will be referred to as a α -Farey system .Following the definitions and notations of [5], throughout, we let L µ α ([0 , L λ ([0 , f with domain [0 ,
1] for which | f | is µ α -integrable (respectively λ -integrable), and for f ∈ L µ α ([0 , L λ ([0 , (cid:107) f (cid:107) µ α (respectively (cid:107) f (cid:107) λ ) by (cid:107) f (cid:107) µ α (cid:66) (cid:90) | f | d µ α (cid:32) respectively (cid:107) f (cid:107) λ (cid:66) (cid:90) | f | d λ (cid:33) . Further, for a given measurable function w , we set (cid:107) w (cid:107) ∞ (cid:66) sup x ∈ [0 , | w ( x ) | . t ≔ t t t ... A A A ... t a t a t a t a ... ...... ( a ) δ = / t ≔ t t t ... A A A ... t a t a t a t a ... ...... ( b ) δ = F igure
2. Plot of the density function h α for the α -Farey map, where t n = n − δ for all n ∈ N .The α -Farey transfer operator (cid:98) F α : L µ α ([0 , → L µ α ([0 , (cid:98) F α ( v ) (cid:66) (cid:88) n ∈ N (cid:32) t n + t n v ◦ F α, + (cid:32) − t n + t n (cid:33) v ◦ F α, (cid:33) · A n , (2)where F α, (cid:66) ( F α | [0 , t ] ) − and F α, (cid:66) ( F α | [ t , ) − refer to the inverse branches of F α . Inparticular, for all v ∈ L µ α ([0 , w with (cid:107) w (cid:107) ∞ < ∞ , (cid:90) (cid:98) F α ( v ) · w d µ α = (cid:90) v · w ◦ F α d µ α . (3)(The above equality is a direct consequence of [15, Lemma 2.5].) Note that the equalitygiven in (3) is the usual defining relation for the transfer operator of F α . However, therelation in (3) only determines values of the transfer operator of F α applied to an observable µ α -almost everywhere. Thus the α -Farey transfer operator is a version of the transferoperator of F α .In order to state our main theorems, we will also require the following function spaces.We let φ : A → N ∪{ + ∞} denote the first return time , given by φ ( y ) (cid:66) inf { n ∈ N : F n α ( y ) ∈ A } ,and we write { φ = n } (cid:66) { y ∈ A : φ ( y ) = n } . Let β α denote the countable-infinite partition {{ φ = n } : n ∈ N } of A and let B α denote the set of functions with domain [0 ,
1] that aresupported on a subset of A and which have finite (cid:107)·(cid:107) B α -norm, where (cid:107)·(cid:107) B α (cid:66) (cid:107)·(cid:107) ∞ + D α ( · ) J. KAUTZSCH, M. KESSEB ¨OHMER, T. SAMUEL, AND B. O. STRATMANN and where D α ( f ) (cid:66) sup a ∈ β α sup x (cid:44) y ∈ a | f ( x ) − f ( y ) || x − y | . In particular, if f ∈ B α , then f is Lipschitz continuous on each atom of β α , zero outside of A and bounded (everywhere). We then define A α (cid:66) (cid:110) v ∈ L µ α ([0 , (cid:107) v (cid:107) ∞ < ∞ and (cid:98) F n − α ( v · A n ) ∈ B α for all n ∈ N (cid:111) . For examples of observables belonging to A α , we refer the reader to Example 4.4 and thediscussion succeeding our main results, Theorems 1.1 and 1.3. Let us also recall from [15]that the wandering rate of F α is given by w n = w n ( F α ) (cid:66) µ α n − (cid:91) k = F − k α ( A ) = µ α n (cid:91) k = A k = n (cid:88) k = t k . Further, as we will see in (12), if δ ∈ (0 ,
1) and if the given α -Farey system is δ -expansive,then the wandering rate is regularly varying of order 1 − δ . Also, in the case that δ =
1, if (cid:16) w n / w (cid:100) n · w n − (cid:101) (cid:17) n ∈ N is a bounded sequence, then we say that the wandering rate w n is moderately increasing .Here and in the sequel for r ∈ R we let (cid:100) r (cid:101) denote the smallest integer greater than or equalto r .With the above preparations, we are now in a position to state the main results, The-orems 1.1 and 1.3. Theorem 1.1 provides mild conditions under which the asymptoticbehavior of the iterates of an α -Farey transfer operator restricted to A can be extended toall of (0 ,
1] and is used in our proof of Theorem 1.3. (Note that, by (12), any δ -expansive α -Farey system satisfies the requirements of Theorem 1.1.) One of the facets of Theo-rem 1.3 is that it gives su ffi cient conditions on observables which guarantee that iterates ofan α -Farey transfer operator applied to such an observable is asymptotic to a constant timesthe wandering rate. These results complement [19, Theorem 10.5] and show that additionalassumptions are required in [19, Theorem 10.4]. Namely, in the case that δ =
1, we showthat the statement of [19, Theorem 10.4] holds true, with the additional assumption that thewandering rate is moderately increasing (Theorem 1.3(i)); for δ ∈ (1 / , ffi cient conditions (Theorem 1.3(ii)). Theorem 1.1.
For an α -Farey system ([0 , , B , µ α , F α ) for which the wandering ratesatisfies the condition lim n →∞ w n / w n + = , we have that, if v ∈ L µ α ([0 , satisfies lim n → + ∞ w n (cid:98) F n α ( v ) = Γ δ (cid:90) v d µ α uniformly on A , then the same holds on any compact subset of (0 , . The same statementholds when replacing uniform convergence by almost everywhere uniform convergence. Remark 1.2.
For δ ∈ (0 , δ -expansive α -Farey system has wandering rate satisfyinglim n →∞ w n / w n + = Theorem 1.3.
Let ([0 , , B , µ α , F α ) be a δ -expansive α -Farey system.(i) Let δ = and assume that the wandering rate is moderately increasing. If v ∈ A α and if ∞ (cid:88) k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:98) F k − α ( v · A k ) (cid:13)(cid:13)(cid:13)(cid:13) ∞ < + ∞ , (4) then uniformly on compact subsets of (0 , , lim n →∞ w n (cid:98) F n α ( v ) = (cid:90) v d µ α . (5) N THE ASYMPTOTICS OF THE α -FAREY TRANSFER OPERATOR 5 (ii) For δ ∈ (1 / , , if v ∈ L λ ([0 , with | v (1) | bounded and if(a) the sequence ( D α ( A · (cid:98) F n − α ( v · A n ))) n ∈ N is bounded and(b) there exist constants c > | v (1) | and η ∈ (0 , δ ) with (cid:107) v · A n (cid:107) ∞ ≤ cn η , for all n ∈ N .then uniformly on compact subsets of (0 , , lim n →∞ w n (cid:98) F n α ( v / h α ) = Γ δ (cid:90) v / h α d µ α . (6) Here, Γ δ (cid:66) ( Γ (1 + δ ) Γ (2 − δ )) − and Γ denotes the Gamma function.(iii) For δ ∈ (1 / , , there exists a positive, locally constant, Riemann integrable functionv ∈ A α of bounded variation satisfying the inequality in (4) , such that, for all x ∈ A , lim inf n →∞ w n (cid:98) F n α ( v )( x ) = Γ δ (cid:90) v d µ α and lim sup n →∞ w n (cid:98) F n α ( v )( x ) = + ∞ . (7) Remark 1.4.
It is immediate that if a n = n − ( n + − , then t n = n − and w n ∼ ln( n ), andthat these parameters give rise to an example of an α -Farey system which satisfies theconditions of Theorem 1.3(i). Indeed there exist many examples of α -Farey systems forwhich the conditions of Theorem 1.3(i) are satisfied, but where the wandering rate behavesvery di ff erently to the function n (cid:55)→ ln( n ). Letting δ =
1, as we will see in Lemma 2.6(iv),the sequence ( w n ) n ∈ N is slowly varying and lim n →∞ nt n / w n =
0. We also have that w n = n (cid:88) j = t j = n ∞ (cid:88) j = n + a j + n (cid:88) j = ja j = nt n + + n (cid:88) j = ja j . Using this we deduce the following.(1) If a n = n − (ln( n )) − / e (ln( n )) / , then t n ∼ n − (ln( n )) − / e (ln( n )) / and w n ∼ e (ln( n )) / .(2) If a n − = n − κ ( n )e ln( n ) / ln(ln( n )) , then t n ∼ n − κ ( n )e ln( n ) / ln(ln( n )) and w n ∼ e ln( n ) / ln(ln( n )) ,where κ ( n ) = (ln(ln( n )) − n ))) − .Indeed the above two sets of parameters give rise to examples of 1-expansive α -Fareysystems whose wandering rate is moderately increasing. Moreover,lim n →∞ ln( n )e (ln( n )) / = , lim n →∞ ln( n )e ln( n ) / ln(ln( n )) = n →∞ e (ln( n )) / e ln( n ) / ln(ln( n )) = , demonstrating that two moderately increasing wandering rates, although they are all slowlyvarying, do not have to be asymptotic to each other nor to the function n (cid:55)→ ln( n ). Remark 1.5.
In the case that F α is a 1-expansive α -Farey map, we have that the wanderingrate w n is a slowly varying function. We remark here that it is not the case that every slowlyvarying function is moderately increasing, namely, it is not the case that if l : [0 , ∞ ] → R isa slowly varying function, then the sequence (cid:16) l ( n ) / l ( (cid:100) n · l ( n ) − (cid:101) ) (cid:17) n ∈ N (8)is bounded. For instance consider the following. Let ( c k ) k ∈ N be a decreasing sequence ofpositive real numbers which converge to zero and, for k ∈ N , set x k + = k c k and b k + = ( c k − c k + ) x k + + b k , where x = b =
0. We define m : [0 , ∞ ) → R by m ( x ) (cid:66) c k x + b k , for x ∈ [ x k , x k + ]. The function l : [1 , ∞ ] → R defined by l ( x ) (cid:66) e m (ln( x )) is, by construction,slowly varying. However, the sequence given in (8) is unbounded. (We are grateful toFredrik Ekstr¨om for providing this example). J. KAUTZSCH, M. KESSEB ¨OHMER, T. SAMUEL, AND B. O. STRATMANN
Remark 1.6.
If in the definition of the norm (cid:107)·(cid:107) B α , one replaces the norm (cid:107)·(cid:107) ∞ by the essential supremum norm (cid:107)·(cid:107) ess sup , then by appropriately adapting the proofs given in thesequel, one can obtain a proof of Theorem 1.3 where the uniform convergence on compactsubsets of (0 ,
1] is replaced by uniform convergence almost everywhere on compact subsetsof (0 , Remark 1.7.
The first part of the proof of Theorem 1.3 (i) and (ii) are inspired by the firstparagraph in the proof of [19, Theorem 10.4].The structure of this paper is as follows. In Section 2 we collect basic properties of α -Farey maps and their corresponding transfer operators. In Section 3 we provide a proofof Theorem 1.1. This proof is inspired by arguments originally presented in [14]. Then inSection 4 we present the proof of Theorem 1.3, breaking the proof into three constituentparts. In Section 4.1 we obtain part (i) and give explicit examples of observables satisfyingthe given properties. In Section 4.2 we prove part (ii), for explicit examples of observableswhich satisfy the pre-requests of Theorem 1.3 (ii) we refer the reader to Remark 1.9. Finallywe conclude with Section 4.3 where part (iii) is proven using a constructive argument.Before we conclude this section with a series of remarks, Remarks 1.8 to 1.10, in whichwe comment on how Theorem 1.3, and hence Theorem 1.1, complement the results obtainedin [16, 24], we introduce the Perron-Frobenius operator P α : L λ ([0 , → L λ ([0 , P α ( f )( x ) (cid:66) (cid:88) y ∈ F − α ( x ) | F (cid:48) α ( y ) | − f ( y ) , where F (cid:48) α denotes the right derivative of F α and where F (cid:48) α (1) (cid:66) − a − . (Note, by construc-tion, if F α is δ -expansive, then the right derivative of F α at zero is equal to one.) A usefulrelation between the operators P α and (cid:98) F α is that (cid:98) F α ( f ) = P α ( h α · f ) / h α . (9)We refer the reader to [15, p. 1001] for a proof of the equality in (9). Remark 1.8.
For certain interval maps T : [0 , → [0 ,
1] with two monotonically increasing,di ff erentiable branches whose invariant measure has infinite mass and whose tail proba-bilities are regularly varying with exponent − δ ∈ [ − , P , namely,that for all Riemann integrable functions u with domain [0 , n → + ∞ w n ( T ) P n ( u ) = Γ δ (cid:32)(cid:90) u d λ (cid:33) h (10)uniformly almost everywhere on compact subsets of (0 , h denotes the associatedinvariant density and w n ( T ) denotes the wandering rate of T . However, α -Farey maps donot fall into this class of interval maps. Using the relationship between the transfer and thePerron-Frobenius operator, Theorem 1.3 (ii) together with the assumption that the Banachspace of functions of bounded variation with the norm (cid:107)·(cid:107) ess sup + Var( · ) satisfies certainfunctional analytic conditions (namely, conditions (H1) and (H2) given in Section 2), showthat Thaler’s result can be extended to δ -expansive α -Farey maps. Results of this form havealso been obtained in [27] for AFN maps. (Note, an α -Farey map is also not an AFN map.) Remark 1.9.
Kesseb¨ohmer and Slassi [16] showed that for the classical Farey map theconvergence given in (10) holds uniformly almost everywhere on [1 / ,
1] for convex C -observables. Likewise, for a δ -expansive α -Farey map, Theorems 1.3 (ii) impliesthat if u is a convex C -observable, then the convergence in (10) holds uniformly on com-pact subsets of (0 , C -observable satisfies the requirements ofTheorem 1.3 (ii), one employs arguments similar to those used in Example 4.4 together with(1) and (14). N THE ASYMPTOTICS OF THE α -FAREY TRANSFER OPERATOR 7 Remark 1.10.
The consequences of Theorem 1.3 go even further, in that for a δ -expansive α -Farey map, we are able to obtain that the convergence given in (10) holds uniformly oncompact subsets of (0 , v is an observable such that v · A n = n η A n , v (0) = v (1) =
1, for some η ∈ (0 , δ ), then, as we will see in Lemma 2.3, since 0 ≤ F n − ( A n ) ≤ t n A ,this observable fulfils the conditions of Theorem 1.3 (ii) and it is neither Riemann integrablenor is it bounded. Notation 1.11.
We use the symbol ∼ between the elements of two sequences of real numbers( b n ) n ∈ N and ( c n ) n ∈ N to mean that the sequences are asymptotically equivalent, namely thatlim n → + ∞ b n / c n =
1. We use the Landau notation b n = o ( c n ), if lim n → + ∞ b n / c n =
0. The samenotation is used between two real-valued function f and g , defined on the set of real numbers R , positive real numbers R + , natural numbers N or non-negative integers N . Specifically,if lim x → + ∞ f ( x ) / g ( x ) =
1, then we will write f ∼ g , and if lim x → + ∞ f ( x ) / g ( x ) =
0, then wewill write f ∈ o ( g ). 2. T he α -F arey system The map G α : A → A defined by G α ( x ) (cid:66) F φ ( x ) α ( x ) if x ∈ A , t if x = , is called the first return map and it is well known that G α is conservative, ergodic andmeasure preserving on ( A , B | A , µ α | A ), see for instance [1, Propositions 1.4.8 and 1.5.3].From this point on, we write µ α for both µ α and µ α | A and B for both B and B | A . Also,throughout, unless otherwise stated, we assume that F α is δ -expansive. ... t t = − a t t t ... ......... ( a ) δ = / − a t ... t t = − a t t t ... ......... ( b ) δ = F igure
3. Plot of G α , where t n = n − δ for all n ∈ N .We denote the open unit disk in C by D (cid:66) { z ∈ C : | z | < } , its closure by D and itsboundary by S . Given z ∈ D , define R n , R ( z ) : L µ α ([0 , → L µ α ([0 , R n ( v ) (cid:66) (cid:98) F α ( v · { φ = n } ) = A · (cid:98) F n α ( v · { φ = n } ) and R ( z ) (cid:66) (cid:88) n ∈ N z n R n . It is an easy exercise to show that R (1) is a version of the transfer operator of the map G α .Namely, for all v ∈ L µ α ([0 , w with (cid:107) w (cid:107) ∞ finite, we have that (cid:90) R (1)( v ) · w d µ α = (cid:90) v · w ◦ G α · A d µ α . (11) J. KAUTZSCH, M. KESSEB ¨OHMER, T. SAMUEL, AND B. O. STRATMANN
We will see in Proposition 2.4 that ( B α , (cid:107)·(cid:107) B α ) is a Banach space, that the operators R n and R (1) map B α into itself and that the following properties are fulfilled. (H1): There exists a constant c > R n : B α → B α is a bounded linearoperator with (cid:107) R n (cid:107) op ≤ c µ ( { φ = n } ), for all n ∈ N . (Here, the operator norm (cid:107)·(cid:107) op is takenwith respect to the Banach space ( B α , (cid:107)·(cid:107) B α ).) (H2): (i) The operator R (1) restricted to B α has a simple isolated eigenvalue at 1.(ii) For each z ∈ D \ { } , the value 1 is not in the spectrum of R ( z ) | B α .A result that will be crucial in the proof of Theorem 1.3 is [19, Theorem 2.1]. In orderto see how this result reads in our situation, note, for a δ -expansive α -Farey map, that µ ( { y ∈ A : φ ( y ) > n } ) = t n + ∼ l ( n ) n − δ , which is essential in the proof of [19, Theorem 2.1]given in [19]. Further, since t n ∼ l ( n ) n − δ and t n + < t n , for all n ∈ N , Karamata’s TauberianTheorem for power series [6, Corollary 1.7.3] implies that, for δ ∈ (0 , w n ∼ Γ δ n − δ l ( n ) . (12)Here, Γ δ (cid:66) Γ (1 − δ ) / Γ (2 − δ ). Theorem 2.1 ([19, Theorem 2.1]) . Assuming the above setting, in particular that conditions(H1) and (H2) are satisfied, we have that lim n → + ∞ sup v ∈B α : (cid:107) v (cid:107) B α = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) w n A · (cid:98) F n α ( v ) − Γ δ (cid:90) v d µ α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B α = . In the sequel, we will also use of the following auxiliary results, where we set Σ (cid:66) { , } and for each n ∈ N and for each word ω (cid:66) ( ω , ω , . . . , ω n ) ∈ Σ n we let F α,ω : [0 , → [0 , F α,ω ◦ F α,ω ◦ · · · ◦ F α,ω n . If ω is equal to the empty word, then we set F α,ω to be equal to the identity map. Lemma 2.2.
Let F α : [0 , → [0 , denote an arbitrary α -Farey map. For each k ∈ N , wehave that (cid:98) F k α ( u ) = (cid:88) n ∈ N (cid:88) ω ∈ Σ k c n ,ω u ◦ F α,ω · A n , where the constants c n ,ω are given recursively byc n , (0) (cid:66) t n + / t n , c n , ( ω ,...,ω k , (cid:66) c n , (0) c n + ,ω , c n , (1) (cid:66) − t n + / t n , c n , ( ω ,...,ω k , (cid:66) c n + , (1) c ,ω . (13) In particular, letting k (cid:66) (0 , , . . . , (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) k-times ) , we have that c , k = t k + , for each k ∈ N .Proof. We proceed by induction on k . The start of the induction is an immediate conse-quence of (2). Suppose that the statement is true for some k ∈ N . We then have that (cid:98) F k + α ( u ) = (cid:98) F α (cid:16)(cid:98) F k α ( u ) (cid:17) = (cid:98) F α (cid:88) n ∈ N (cid:88) ω ∈ Σ k c n ,ω u ◦ F α,ω · A n = ∞ (cid:88) m = (cid:88) n ∈ N (cid:88) ω ∈ Σ k t m + t m c n ,ω u ◦ F α,ω ◦ F α, · A n ◦ F α, + (cid:32) − t m + t m (cid:33) c n ,ω u ◦ F α,ω ◦ F α, · A n ◦ F α, (cid:33) · A m = ∞ (cid:88) m = (cid:88) ω ∈ Σ k t m + t m c m + ,ω u ◦ F α,ω ◦ F α, + (cid:32) − t m + t m (cid:33) c ,ω u ◦ F α,ω ◦ F α, · A m . This completes the proof of (13). The remaining assertion is proven by a straight forwardinductive argument, using the defining relations given in (13). (cid:3)
N THE ASYMPTOTICS OF THE α -FAREY TRANSFER OPERATOR 9 Lemma 2.3.
For each n ∈ N , we have that (cid:98) F n − α ( A n ) = t n A (14) and hence, by the definition of the norm, (cid:107) (cid:98) F n − α ( A n ) (cid:107) B α = (cid:107) (cid:98) F n − α ( A n ) (cid:107) ∞ = t n .Proof. For n = n (cid:44)
1, we have, by Lemma 2.2, that, on [0 , (cid:98) F n − α ( A n ) = ∞ (cid:88) k = c k , n − A n ◦ F α, n − · A k = ∞ (cid:88) k = c k , n − A · A k = t n A . To complete the proof, we need to evaluate the function (cid:98) F n − α ( A n ) at the point 1 for n ≥ (cid:98) F n − α ( A n )(1) = (cid:88) n ∈ N (cid:88) ω ∈ Σ c n ,ω A n ◦ F α,ω (1) · A n (1) = (cid:88) ω ∈ Σ c ,ω A n ◦ F α,ω (1) = . This completes the proof. (cid:3)
We will now show that conditions (H1) and (H2) are satisfied for every δ -expansive α -Farey system and for the Banach space ( B α , (cid:107)·(cid:107) B α ). Proposition 2.4.
The pair ( B α , (cid:107)·(cid:107) B α ) forms a Banach space and for a δ -expansive α -Fareysystem, the operators R n and R (1) map B α into itself. Moreover, (H1) and (H2) are satisfied. In the proof of the above proposition we will make use of the following lemma.
Lemma 2.5.
For any α -Farey map F α , we have that c , n − = µ α ( { φ = n } ) = a n = t n − t n + ,where n (cid:66) (1 , , , . . . , (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) n-times ) , for each n ∈ N .Proof. By construction of the α -Farey map F α , we have that { φ = } = [1 − a t , − a t ]and that { φ = n } = (1 − a t n , − a t n + ], for all integers n >
1. Thus, µ α ( { φ = n } ) = (cid:90) { φ = n } · d µ α d λ d λ = a − λ ( { φ = n } ) = t n − t n + . (15)We will now show by induction on n that, for each k ∈ N , c k , n − = t k + n − − t k + n t k . (16)From (2), we have that c k , (1) = − t k + / t k = ( t k − t k + ) / t k , for each k ∈ N . Suppose that thestatement in (16) is true for some n ∈ N . From (13), we have that c k , n (cid:66) ( t k + / t k ) c k + , n − , for each k ∈ N , which gives c k , n = t k + t k t k + n − t k + n + t k + = t k + n − t k + n + t k . This completes the proof of the statement in (16).Setting k = c , n − = t n − t n + , for all n ∈ N . Combining thiswith (15), completes the proof. (cid:3) Proof of Proposition 2.4.
It is shown in [2, Section 1] that the pair ( B α , (cid:107)·(cid:107) B α ) forms aBanach space.We now prove that condition (H1) holds and the invariance of B α . For this, let u ∈ B α and fix k ∈ N . Applying Lemmas 2.2 and 2.5 we have that R k ( u ) = A · (cid:98) F k α ( { φ = k } · u ) = A · µ α ( { φ = k } ) · u ◦ F α, k − . Hence, by definition of the partition β α , we have that (cid:107) R k ( u ) (cid:107) B α ≤ µ α ( { φ = k } ) (cid:107) u (cid:107) B α , and so,the operator R k maps B α into itself. Further, by definition of R (1), this gives that (cid:107) R (1) u (cid:107) B α ≤ (cid:88) n ∈ N (cid:107) R n ( u ) (cid:107) B α ≤ (cid:88) n ∈ N µ α ( { φ = n } ) (cid:107) u (cid:107) B α = (cid:107) u (cid:107) B α , and so, the operator R (1) maps B α into itself. Linearity of R k and R (1) follows from thelinearity of (cid:98) F α .For the proof of property (H2)(i), observe that G α is a piecewise linear expansive mapwith the following properties.(i) On the set { φ = n } , the absolute value of the derivative of G α is equal to 1 / ( t n − t n + ).Moreover, since ( t n ) n ∈ N is a positive monotonically decreasing sequence which isbounded above by 1, it follows that there exists a constant c > / ( t n − t n + ) > c ,for all n ∈ N .(ii) The partition β α is a countable-infinite partition of A and G α ( { φ = } ) = A and G α ( { φ = n } ) = A if n ≥
2, and hence, µ α ( G α ( { φ = n } )) = µ α ( A ) =
1, for all n ∈ N .Moreover, the σ -algebra generated by { G − n α ( { φ = m } ) : n , m ∈ N } is equal to the Borel σ -algebra on A .(iii) For each n ∈ N and ψ (cid:66) (1 , , , . . . , (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) ( n − − times , F α,ψ ([0 , = { φ = n } and d µ α ◦ F α,ψ d µ α = t n − t n + = a n . Given these properties, (H2)(i) is a consequence of [2, Theorem 1.6]: the proof of whichis based on the
Theorem on the di ff erence of two norms by Ionescu-Tulcea and Marinescu[13].For the proof of property (H2)(ii), we distinguish between the cases z ∈ D and z ∈ S \ { } . Case 1 . ( z ∈ D ): Sarig showed in [21, Section 3] that T ( z ) ◦ R ( z ) u = ( T ( z ) u ) − u , (17)where the operators T ( z ) , T n : L µ α ([0 , → L µ α ([0 , T ( z ) (cid:66) (cid:88) n ∈ N z n · T n and T n ( u ) (cid:66) A · (cid:98) F n α ( A · u ) . By way of contradiction, suppose that 1 is an eigenvalue of R ( z ) restricted to B α . Thenthere exists a non-zero measurable function w ∈ B α such that R ( z ) w = w . Substituting thisinto (17) shows that w is equal to zero µ α -almost everywhere, which gives a contradiction. Case 2 . ( z ∈ S \ { } ): We will now show that 1 is not an eigenvalue of R ( z ). (This partof the proof is based on the proof of [10, Lemma 6.7].) Since z ∈ S \ { } , there exists a t ∈ (0 , π ) such that z = e it . Suppose that R ( z ) f = f , for some non-zero f ∈ B α . Let L µ α ( A )denote the class of complex-valued measurable functions f with domain A for which | f | is µ α -integrable, and let it be equipped with the standard L µ α -inner product, (cid:104) u , u (cid:105) (cid:66) (cid:90) u · u d µ α . For each u ∈ L µ α ( A ) set (cid:107) u (cid:107) (cid:66) (cid:104) u , u (cid:105) / . Further, set L ∞ ( A ) (cid:66) { v : A → C : (cid:107) v (cid:107) ∞ < + ∞} and define V : L ∞ → L ∞ by V ( u ) (cid:66) e − it φ · u ◦ G α . Noting that R ( z ) v = R (1)(e it φ · v ) andusing (11), we have that, for all v ∈ B α and u ∈ L ∞ ( A ), (cid:104) u , R ( z ) v (cid:105) = (cid:90) u · R ( z ) v d µ α = (cid:90) u · R (1)(e it φ · v ) d µ α = (cid:90) u ◦ G α · e it φ · v d µ α = (cid:104) V ( u ) , v (cid:105) . Further, (cid:107) V ( f ) − f (cid:107) = (cid:107) V ( f ) (cid:107) − Re (cid:104) V ( f ) , f (cid:105) + (cid:107) f (cid:107) = (cid:107) V ( f ) (cid:107) − Re (cid:104) f , R ( z )( f ) (cid:105) + (cid:107) f (cid:107) = (cid:107) V ( f ) (cid:107) − Re (cid:104) f , f (cid:105) + (cid:107) f (cid:107) = (cid:107) V ( f ) (cid:107) − (cid:107) f (cid:107) (18) N THE ASYMPTOTICS OF THE α -FAREY TRANSFER OPERATOR 11 and, as G α preserves the measure µ α restricted to A , we have that (cid:107) V ( f ) (cid:107) = (cid:90) | f | ◦ G α d µ α = (cid:90) | f | d µ α = (cid:107) f (cid:107) . (19)Combining (18) and (19), it follows that V ( f ) − f vanishes µ α -almost everywhere on A .Thus, by taking the modulus, the ergodicity of G α implies that | f | is equal to a constant, µ α -almost everywhere on A . As f does not vanish µ α -almost everywhere, this constant isnon-zero, and so, we obtain that e − it φ = f / ( f ◦ G α ) almost everywhere on A . Now, foreach n ∈ N , let I n ⊆ { φ = n } be the interval of positive measure, such that G α ( I n ) = { φ = n } and let J n (cid:66) { x ∈ I n : G α ( x ) (cid:60) I n and e it φ ( x ) = f ( x ) / f ◦ G α ( x ) } . Since e it φ = f / f ◦ G α almost everywhere, and since the map G α is linear and expanding,we have that µ α ( J n ) >
0. In particular, the set J n is non-empty. We claim that there exists y n ∈ J n such that f ( y n ) = f ◦ G α ( y n ), for each n ∈ N . By way of contradiction, suppose that f ( x ) (cid:44) f ◦ G α ( x ), for all x ∈ J n . Since f is constant almost everywhere on A , we have that f is constant almost everywhere on J n ∪ G α ( J n ), which gives an immediate contradictionto the assumption that f ( x ) (cid:44) f ◦ G α ( x ), for all x ∈ J n . Therefore, we have that there exists y n ∈ A , such that φ ( y n ) = n , f ( y n ) = f ◦ G α ( y n ) and e it φ ( y n ) = f ( y n ) / f ◦ G α ( y n ), for each n ∈ N . Hence, we have that e itn =
1, for all n ∈ N , contradicting the initial choice of t . (cid:3) Finally, in preparation for the proof of Theorem 1.3, let us make note of the followingwell known properties of slowly varying functions.
Lemma 2.6.
Let L : [ a , + ∞ ] → R be a positive slowly varying function, for some a ∈ N .(i) [23, p. 2] For a compact interval I ⊂ R + we have that lim x → + ∞ L ( px ) / L ( x ) = holds uniformly with respect to p ∈ I, and hence, for a fixed b ∈ R + , lim x → + ∞ L ( x − b ) / L ( x ) = . (ii) [23, p. 18] For a fixed b ∈ R + we have that lim x → + ∞ L ( x ) x − b = and lim x → + ∞ L ( x ) x b = + ∞ . (iii) [23, p. 41] If L is continuous, strictly increasing and lim x → + ∞ L ( x ) = + ∞ , then, for a fixed c ∈ (0 , , lim x → + ∞ L − ( cx ) / L − ( x ) = . (iv) [23, p. 50] If M : [ a + , + ∞ ) → R is defined to be the linear interpolation of thefunction n (cid:55)→ n (cid:88) k = a + L ( k ) k − , then M is a slowly varying function and lim x →∞ L ( x ) M ( x ) = .
3. E xtending convergence : P roof of T heorem Proof of Theorem 1.1.
Let us first recall that, for x ∈ (0 ,
1] and n ∈ N , (cid:16) P n + α ( h α · v ) (cid:17) ( x ) = P α (cid:0)(cid:0) P n α ( h α · v ) (cid:1) ( x ) (cid:1) = (cid:0) P n α ( h α · v ) (cid:1) ( F α, ( x )) · (cid:12)(cid:12)(cid:12) F (cid:48) α, ( x ) (cid:12)(cid:12)(cid:12) + (cid:0) P n α ( h α · v ) (cid:1) ( F α, ( x )) · (cid:12)(cid:12)(cid:12) F (cid:48) α, ( x ) (cid:12)(cid:12)(cid:12) , which gives (cid:0) P n α ( h α · v ) (cid:1) ( F α, ( x )) = ( P n + α ( h α · v ))( x ) − ( P n α ( h α · v ))( F α, ( x )) · | F (cid:48) α, ( x ) || F (cid:48) α, ( x ) | . (20)We proceed by induction as follows. The start of the induction is given by the assumptionin the theorem. For the inductive step, assume that the statement holds for (cid:83) ki = A i , forsome k ∈ N . Then consider some arbitrary y ∈ A k + , and let x denote the unique elementin A k such that F α, ( x ) = y . Using (20), the fact that (cid:98) F α = h − α P α ( h α · v ) and the inductivehypothesis in tandem with the assumption that lim w n / w n + =
1, we obtain that w n (cid:16)(cid:98) F n α ( v ) (cid:17) ( y ) = w n (cid:16)(cid:98) F n α ( v ) (cid:17) ( F α, ( x )) = w n ( P n ( h α · v ))( F α, ( x )) h α ( F α, ( x )) = w n ( P n + α ( h α · v ))( x ) − | F (cid:48) α, ( x ) | · w n ( P n α ( h α · v ))( F α, ( x )) h α ( F α, ( x )) · | F (cid:48) α, ( x ) |∼ h α ( x ) − h α ( F α, ( x )) · | F (cid:48) α, ( x ) | h α ( F α, ( x )) · | F (cid:48) α, ( x ) | Γ δ (cid:90) v d µ α = Γ δ (cid:90) v d µ α , where the last equality is a consequence of the eigenequation P α h α = h α . (cid:3) Remark 3.1.
Using an analogous proof to that given above, one can obtain that the resultof Theorem 1.1 holds for other interval maps, such as Gibbs-Markov maps, Thaler mapsand Pomeau-Manneville maps. 4. P roof of T heorem Asymptotics of the α -Farey transfer operator for δ = . Throughout this section,we let ([0 , , B , µ α , F α ) be a 1-expansive α -Farey system. In order to prove Theorem 1.3 (i),we will use the following auxiliary results (Lemmas 4.1 and 4.2). Before which we requirethe following notation. Define the function (cid:96) : [0 , + ∞ ) → R by (cid:96) ( x ) (cid:66) x / + / x ∈ [0 , , t n + ( x − n ) + w n if x ∈ [ n , n + , for n ∈ N . (21)Note that (cid:96) is the linear interpolation of the function n (cid:55)→ w n defined on N , where w (cid:66) / σ ∈ R + , define j σ ( x ) (cid:66) x − (cid:96) − ((1 + σ ) − (cid:96) ( x )) , for all x ≥ (cid:96) − ((1 + σ ) / Lemma 4.1.
For a given σ ∈ R + , we have that j σ ( x ) ∼ x.Proof. For σ ∈ R + , we have thatlim x →∞ j σ ( x ) x = − lim x →∞ (cid:96) − ( (cid:96) ( x )(1 + σ ) − ) x = − lim x →∞ (cid:96) − ( (cid:96) ( x )(1 + σ ) − ) (cid:96) − ( (cid:96) ( x )) = , where the last equality follows from the fact that (cid:96) is a positive, strictly monotonicallyincreasing function and Lemma 2.6 (iii). (cid:3) Here and in the sequel we will use the following notation. For r ∈ R we let (cid:98) r (cid:99) denotethe largest integer not exceeding r . N THE ASYMPTOTICS OF THE α -FAREY TRANSFER OPERATOR 13 Lemma 4.2.
Let ( δ j ) j ∈ N denote a sequence of positive real numbers such that (cid:80) ∞ j = δ j t j < ∞ .If the wandering rate is moderately increasing, then lim n →∞ n (cid:88) j = w n w n − j δ j + t j + = ∞ (cid:88) j = δ j t j . Proof.
Without loss of generality, assume that sup { δ j : j ∈ N } = σ ∈ R + be fixed.By definition of (cid:96) , we have, for n ≥ (cid:96) − ((1 + σ ) / n (cid:88) j = w n w n − j δ j + t j + ≤ (cid:96) ( n ) (cid:96) ( n − j σ ( n )) (cid:98) j σ ( n ) (cid:99) (cid:88) j = δ j + t j + + (cid:96) ( n ) (cid:96) ( n ( (cid:96) ( n )) − ) (cid:98) n − n ( (cid:96) ( n )) − (cid:99) (cid:88) j = (cid:98) j σ ( n ) (cid:99) + δ j + t j + + (cid:96) ( n ) n (cid:88) j = (cid:98) n − n ( (cid:96) ( n )) − (cid:99) + t j + . By definition of j σ ( n ), we have that (cid:96) ( n ) /(cid:96) ( n − j σ ( n )) = (1 + σ ) − . Further, by Lemma 2.6(iv) and since ( t j ) j ∈ N is regularly varying sequence of order −
1, we have that,lim n →∞ (cid:96) ( n ) n (cid:88) j = (cid:98) n − n ( (cid:96) ( n )) − (cid:99) + t j + ≤ lim n →∞ t n + n (cid:96) ( n ) = . If (cid:98) j σ ( n ) (cid:99) + > (cid:98) n − n ( (cid:96) ( n )) − (cid:99) , then this completes the proof. Otherwise, note that, byLemma 2.6 (ii), we have that (cid:96) is a slowly varying function. Also, since (cid:96) is an unboundedmonotonically increasing function we have that n − n ( (cid:96) ( n )) − ∼ n and, by Lemma 4.1, wehave that j σ ( n ) ∼ n . The above three statements in tandem with the assumptions that (cid:80) ∞ j = δ j t j < ∞ and that the wandering rate is moderately increasing, yield the following:lim n →∞ (cid:96) ( n ) (cid:96) ( n ( (cid:96) ( n )) − ) (cid:98) n − n ( (cid:96) ( n )) − (cid:99) (cid:88) j = (cid:98) j σ ( n ) (cid:99) + δ j + t j + = . This completes the proof in the case (cid:98) j σ ( n ) (cid:99) + ≤ (cid:98) n − n ( (cid:96) ( n )) − (cid:99) . (cid:3) Proof of Theorem 1.3 (i).
By Theorem 2.1 and Proposition 2.4, we have for each n ∈ N thatthere exists θ n : (0 , → R such that sup {| θ n ( x ) | : x ∈ A } = o (1 / w n ) and A · (cid:98) F n α ( A · v ) = w n (cid:90) A · v d µ α · A + θ n · v · A . (22)Set τ j , n (cid:66) w n / w n − j −
1, for n ∈ N and j ∈ { , , , . . . , n } . By (22), we have on A that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w n (cid:98) F n α ( v ) − (cid:90) v d µ α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w n n (cid:88) j = A · (cid:98) F n − j α (cid:16) A · (cid:98) F j α ( v · A j + ) (cid:17) − (cid:90) v d µ α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w n n (cid:88) j = w n − j (cid:90) (cid:98) F j α ( v · A j + ) d µ α − (cid:90) v d µ α + w n n (cid:88) j = θ n − j · (cid:98) F j α ( v · A j + ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n (cid:88) j = τ n , j (cid:90) | v · A j + | d µ α + w n n (cid:88) j = (cid:107) θ n − j (cid:107) ∞ (cid:107) (cid:98) F j α ( v · A j + ) (cid:107) ∞ + ∞ (cid:88) j = n + (cid:90) | v · A j + | d µ α . (23)Since v ∈ A α ⊆ L µ α ([0 , (i) since v ∈ A α , we have that v ∈ L µ α ([0 , (cid:90) | v · A j | d µ α = t j a j (cid:90) v · A j d λ ;(ii) since v ∈ A α we have that (cid:107) v (cid:107) ∞ is finite, and so the sequence (cid:32) a j (cid:90) v · A j d λ (cid:33) j ∈ N is a bounded sequence;(iii) using Lemma 2.3 together with the fact that (cid:98) F α is positive and linear and the fact thatif v ∈ A α , then (cid:107) v (cid:107) ∞ is finite, we have that | (cid:98) F j − α ( v · A j )( x ) | ≤ (cid:107) v (cid:107) ∞ t j ;(iv) given (cid:15) >
0, there exists N (cid:15) ∈ N such that (cid:107) θ m (cid:107) ∞ ≤ (cid:15)/(cid:96) ( m ), for all m ≥ N (cid:15) .Combining these observations with Lemma 4.2 and (4), we have that the first and thesecond term in the final line of (23) converge to to zero. Since the arguments given aboveare independent of a given point in A , an application of Theorem 1.1 now finishes theproof. (cid:3) Remark 4.3.
In the proof of Theorem 1.3 (i) we have not used the specific structure of B α .We only used that B α is a Banach space which satisfies conditions (H1) and (H2). Thus,we may replace B α by an arbitrary Banach space which satisfies conditions (H1) and (H2).For such alternative Banach spaces see Remark 1.8. In doing such a substitution one maychange the uniform convergence to almost everywhere uniform convergence.To conclude, we give examples of 1-expansive α -Farey systems and of observables whichbelong to the set A α and which satisfy the summability condition given in (4). Example 4.4.
Let ([0 , , B , µ α , F α ) denote a 1-expansive α -Farey system with moderatelyincreasing wandering rate. Set u (cid:66) f / h α , where f ∈ D µ α , for D µ α (cid:66) { f : f ∈ L µ α ([0 , f ∈ C ((0 , f (cid:48) > f (cid:48)(cid:48) ≤ } . We claim that u ∈ A α and moreover, that u satisfies the summability condition given in (4).We first verify that u ∈ A α . For this, we are required to show that u ∈ L µ α ([0 , (cid:107) u (cid:107) ∞ < + ∞ and that (cid:98) F j − α ( u · A j ) ∈ B α , for all j ∈ N . By definition, any function belongingto D µ α is convex and continuous on (0 , ff erentiable and µ α -integrable. Thus, u ∈ L µ α ([0 , (cid:107) u (cid:107) ∞ < + ∞ . Combining this with the fact that 1 / h α is µ α integrable,non-negative and bounded, we have that u ∈ L µ α ([0 , (cid:107) u (cid:107) ∞ < + ∞ . Let us now turn tothe second assertion, namely that (cid:98) F n − α ( u · A n ) ∈ B α , for all n ∈ N . We immediately havethat (cid:98) F α ( u · A ) = u · A ∈ B α . For n ≥
2, note that, if g is a di ff erentiable Lipschitz functionon A , then D α ( g ) = sup {| g (cid:48) | : x ∈ A } . Thus, by Lemma 2.3 and the chain rule, we have that,for each integer n ≥ (cid:107) (cid:98) F n − α ( u · A n ) (cid:107) B α = (cid:107) c , n − (( f / h α ) ◦ F α, n − ) (cid:107) B α = (cid:13)(cid:13)(cid:13) a n f ◦ F α, n − (cid:13)(cid:13)(cid:13) B α = (cid:13)(cid:13)(cid:13) a n f ◦ F α, n − (cid:13)(cid:13)(cid:13) ∞ + D α ( a n f ◦ F α, n − ) ≤ a n ( (cid:107) f (cid:107) ∞ + (cid:107) f (cid:48) · A n (cid:107) ∞ ) . (24)Since f ∈ D µ α , we have that (cid:107) f (cid:107) ∞ < + ∞ and that 0 ≤ f (cid:48) ( x ) ≤ ( f ( t n + ) − f ( t n + )) / ( a n + ), forall x ∈ A n . Therefore, since a n = l ( n ) n − , it follows that there exists c >
0, such that (cid:88) n ∈ N , n (cid:44) a n (cid:107) f (cid:48) · A n (cid:107) ∞ ≤ (cid:88) n ∈ N , n (cid:44) a n f (cid:48) ( ξ n + ) = (cid:88) n ∈ N , n (cid:44) a n a n + ( f ( t n + ) − f ( t n + )) ≤ c f ( t ) . Combining this with (24) and using the facts that the sequence ( a n ) n ∈ N is summable and that (cid:107) f (cid:107) ∞ and (cid:107) u · A (cid:107) B α are finite, the summability condition in (4) follows. Hence it followsthat u ∈ A α . N THE ASYMPTOTICS OF THE α -FAREY TRANSFER OPERATOR 15 Asymptotics of the α -Farey transfer operator for δ ∈ (1 / , . Proof of Theorem 1.3 (ii).
Recall that ([0 , , B , µ α , F α ) is a δ -expansive α -Farey systemand that Γ δ = ( Γ (1 + δ ) Γ (2 − δ )) − . From (1) and (14), we have that, for all n ∈ N , (cid:107) A · (cid:98) F n − α ( v · A n / h α ) (cid:107) B α = (cid:107) A · (cid:98) F n − α ( v · A n / h α ) (cid:107) ∞ + D α ( A · (cid:98) F n − α ( v · A n / h α )) = a n (cid:107) v · A n (cid:107) ∞ + a n D α ( v · A n ◦ F α, n − ) . Combining this with the assumptions of the theorem, there exists a constant c (cid:48) > (cid:107) A · (cid:98) F n − α ( v · A n / h α ) (cid:107) B α < c (cid:48) , for all n ∈ N .As in the proof of Theorem 1.3 (i) we have, by Theorem 2.1 and Proposition 2.4, thatthere exists θ n : [0 , → R such that sup {| θ n ( x ) | : x ∈ A } = o (1 / w n ) and, for each n ∈ N , A · (cid:98) F n α ( A · v / h α ) = Γ δ w n (cid:90) A · v d λ A + θ n · v / h α · A . Set τ j , n (cid:66) w n / w n − j −
1, for each n ∈ N and j ∈ { , , , . . . , n } . By a calculation similar as in(23), we have on A that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w n (cid:98) F n α ( v / h α ) − Γ δ (cid:90) v d λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ w n n (cid:88) j = (cid:107) θ n − j (cid:107) ∞ (cid:107) (cid:98) F j α ( v · A j + / h α ) (cid:107) ∞ + Γ δ n (cid:88) j = τ n , j (cid:90) | v · A j + | d λ + Γ δ ∞ (cid:88) j = n + (cid:90) | v · A j + | d λ. (25)As v ∈ L λ ([0 , t n ) n ∈ N is positive, monotonically decreasing and bounded above by one and that by assumption a n = δ l ( n ) n − (1 + δ ) . Further, by (12), given ξ >
0, there exist constants c (cid:48)(cid:48) ∈ R and N = N ( ξ ) ∈ N such that, for all n , m ∈ N with n ≥ N , w m ≥ Γ δ (e − ξ l ( m ) m − δ + c (cid:48)(cid:48) ) and Γ δ e ξ l ( n ) n − δ ≥ w n ≥ Γ δ e − ξ l ( n ) n − δ . This implies that given (cid:15) ∈ (0 , n ∈ N such that (cid:100) (cid:15) n (cid:101) > N , n (cid:88) j = n −(cid:100) (cid:15) n (cid:101) + w n w n − j l ( j ) j + δ − η ≤ Γ δ e ξ ( l ( n )) w n δ − η + n − N (cid:88) j = n −(cid:100) (cid:15) n (cid:101) + e ξ n − δ l ( n )( n − j ) − δ l ( n − j ) l ( j ) j + δ − η + n − (cid:88) j = n − N + e ξ l ( n ) n − δ ( n − j ) − δ l ( n − j ) + c (cid:48)(cid:48) e ξ l ( j ) j + δ − η . (Recall that η is the value given in condition (b) of Theorem 1.3 (ii)). A simple calculationshows that the RHS of the latter inequality converges to zero as n tends to infinity. Further,since all of the summand are positive, we have that the limit as n tends to infinity exists andequals zero. This together with Lemma 2.6 (i) implies that, for each (cid:15), ξ ∈ (0 , n → + ∞ n (cid:88) j = w n w n − j (cid:90) | v · A j + | d λ ≤ lim sup n → + ∞ n −(cid:100) (cid:15) n (cid:101) (cid:88) j = e ξ l ( n ) n − δ l ( n − j )( n − j ) − δ (cid:90) | v · A j + | d λ = lim sup n → + ∞ n −(cid:100) (cid:15) n (cid:101) (cid:88) j = e ξ n − δ ( n − j ) − δ (cid:90) | v · A j + | d λ ≤ (cid:15) δ − e ξ (cid:90) | v | d λ. (26) Furthermore,lim inf n → + ∞ n (cid:88) j = w n w n − j (cid:90) | v · A j + | d λ ≥ lim inf n → + ∞ n −(cid:100) (cid:15) n (cid:101) (cid:88) j = e − ξ l ( n ) n − δ l ( n − j )( n − j ) − δ (cid:90) | v · A j + | d λ ≥ (cid:15) δ − e − ξ (cid:90) | v | d λ. Since (cid:15), ξ ∈ (0 ,
1) are arbitrary, it follows thatlim n → + ∞ n (cid:88) j = w n w n − j (cid:90) | v · A j | d λ = (cid:90) | v | d λ, and hence, the second term on the RHS of (25) converges to zero as n tends to infinity. Wenow show that the first term on the RHS of (25) converges to zero. By (12) and the factthat sup {| θ n ( x ) | : x ∈ A } = o (1 / w n ), given ξ > M = M ( ξ ) ∈ N such that, for all m ≥ M , Γ δ e − ξ l ( m ) m − δ ≤ w m ≤ Γ δ e ξ l ( m ) m − δ and (cid:107) θ m (cid:107) ∞ ≤ ξ/ w m . Moreover, there exists a constants c , c ∈ R such that, for all n ∈ N , w n ≥ Γ δ (e − ξ l ( n ) n − δ + c ) and (cid:107) θ n (cid:107) ∞ ≤ c / w n . Furthermore, since F α is δ -expansive, by condition (b) in Theorem 1.3 (ii), we have that thesequence ( a n (cid:107) v · A n (cid:107) ∞ ) n ∈ N is summable. These properties together with Lemma 2.3 and anargument similar to that presented in (26), imply the existence of a constant c > ≤ lim sup n → + ∞ w n n (cid:88) j = (cid:107) θ n − j (cid:107) ∞ (cid:107) (cid:98) F j α ( v · A j + / h α ) (cid:107) ∞ ≤ lim sup n → + ∞ ξ e ξ n − M (cid:88) j = l ( n ) n − δ l ( n − j )( n − j ) − δ (cid:107) v · A j + (cid:107) ∞ a j + + lim sup n → + ∞ c w n a n + w + lim sup n → + ∞ c e ξ n − (cid:88) j = n − M + l ( n ) n − δ ( n − j ) − δ l ( n − j ) + c e ξ a j + (cid:107) v · A j + (cid:107) ∞ = ξ e ξ ∞ (cid:88) j = a j + (cid:107) v · A j + (cid:107) ∞ . Since ξ > A , an application of Theorem 1.1 finishes the proof. (cid:3) Proof of Theorem 1.3 (iii) - Counterexamples for δ ∈ (1 / , . In this section weprovide a constructive proof of Theorem 1.3 (iii). The proof is divided into several parts.First, we define a class of observables V . Second, in Proposition 4.9 we will show thatif v ∈ V , then v is bounded, of bounded variation, Riemann integrable and belongs to L µ α ([0 , v ∈ V , then it belongs to thespace A α , and in Proposition 4.11 we will show that the summability condition given in (4)is satisfied for all v ∈ V . Finally, in Proposition 4.13 we will show that, if v ∈ V , thenlim inf n → + ∞ w n (cid:98) F n α ( v )( x ) = Γ δ (cid:90) v d µ α and lim sup n → + ∞ w n (cid:98) F n α ( v )( x ) = + ∞ . Combing these results will then yield a proof of Theorem 1.3 (iii)Let us now begin by defining the set V . We let V denote the class of observable v : [0 , → R which are of the following form: v (cid:66) ∞ (cid:88) k = s k N k (cid:88) j = N k − n k A j , (27) N THE ASYMPTOTICS OF THE α -FAREY TRANSFER OPERATOR 17 where N k (cid:66) (cid:100) g k (cid:101) , n k (cid:66) (cid:98) g k (cid:99) and s k (cid:66) − g k , and where g , g and g denote three positive constants, depending on δ , such that (C1): g > (1 − δ ) − , (C2): δ g > g , (C3): there exists (cid:15) ∈ (0 , δ − / g ( δ − (cid:15) ) > (2 δ + (cid:15) − g + g . Example 4.5.
For δ ∈ (1 / , g = (1 + (cid:15) ) / (1 − δ ) and g = δ/ (1 − δ ). Then it is clearthat g and g satisfy the conditions (C1) and (C2). With these choices one immediatelyverifies that (C3) is equivalent to g < (1 − δ ) − ρ(cid:15) , for ρ (cid:66) (3 δ + + (cid:15) ) / (1 − δ ). Hence, bychoosing (cid:15) > ffi ciently small, it follows that the conditions (C1), (C2) and (C3) can befulfilled simultaneously. Example 4.6.
For δ ∈ (1 / , g = (1 − δ ) − , g = ( δ + δ − δ ) − (1 − δ ) − , g = − and (cid:15) = δ (1 − δ ) (2 δ + δ − − . For each δ ∈ (1 / , s k ) k ∈ N is to ensure that v is of boundedvariation. Further, condition (C3) is only required in the proof of the second statement ofProposition 4.13, specifically when Lemma 4.12 is used.Before we begin with Proposition 4.9, we give two elementary lemmas which we willuse in its proof. Lemma 4.7.
If s ∈ (0 , and < b < a s , then ∞ (cid:88) k = a (1 − s ) k − ( a k − b k ) − s ≤ ∞ (cid:88) k = ( b / a s ) k < + ∞ . Proof.
By assumption, we have that a / b > − ( b / a ) k ≤ (1 − ( b / a ) k ) − s , whichimplies that ( a / b ) k − ( a / b ) k (1 − ( b / a ) k ) − s ≤
1, for each k ∈ N . Therefore, ∞ (cid:88) k = a (1 − s ) k − ( a k − b k ) − s = ∞ (cid:88) k = ( b / a s ) k (( a / b ) k − ( a / b ) k (1 − ( b / a ) k ) − s ) ≤ ∞ (cid:88) k = ( b / a s ) k < + ∞ . (cid:3) Lemma 4.8.
For each k ∈ N , we have that N k + − n k + > N k .Proof. By (C1) and (C2), we have that g > (1 − δ ) − and g − g <
0. This implies that, forall k ∈ N , N k + − n k + N k ≥ g ( k + − g ( k + − g k + = g (1 − ( g − g )( k + − − g ( k + )2 − g k + ≥ g (1 − g − g ) − − g )2 − g + . Using (C1) and (C2) once more immediately verifies that the latter term is strictly greaterthan one. (cid:3)
Proposition 4.9.
An observable v defined as in (27) is bounded, of bounded variation,Riemann integrable and belongs to the space L µ α ([0 , .Proof. Clearly the observable v is Riemann integrable. Moreover, v is measurable, as eachof the atoms of α is measurable and v is the sum of indicator functions of atoms of α .Further, the range of v is equal to { } ∪ { s k : k ∈ N } , and thus, (cid:107) v (cid:107) ∞ = s . By Lemma 4.8,we have that N k + − n k + > N k , and so the variation of v is equal to 2 (cid:80) ∞ k = s k =
1, whichis finite, as s k (cid:66) − g k and as g is positive. This shows that v is of bounded variation. It remains to show that v is µ α -integrable. For this recall that µ α ( A k ) = t k , for each k ∈ N .Choose a positive constant η < min { δ, g / g } and recall that t n ∼ l ( n ) n − δ . By Lemma 2.6 (ii),there exists a constant c > t n ≤ cl ( n ) n − δ ≤ cn η − δ , for each n ∈ N . Therefore, byLemma 4.7 and Lemma 4.8, we have that (cid:90) | v | d µ α = ∞ (cid:88) k = s k N k (cid:88) j = N k − n k t j ≤ ∞ (cid:88) k = − g k N k (cid:88) j = N k − n k cj δ − η ≤ (1 − δ + η ) − c ∞ (cid:88) k = (cid:18) − g k (cid:18) g k (1 − δ + η ) − (cid:16) g k − g k (cid:17) − δ + η (cid:19) + − g k N k η − δ (cid:19) ≤ (1 − δ + η ) − c ∞ (cid:88) k = (cid:16) ( g − δ g + η g − g ) k + − g k N k η − δ (cid:17) . The latter series converges, since η < min { δ, g / g } , g < δ g and N k >
1, for all k ∈ N . (cid:3) Our next aim is to show that v belongs to A α and satisfies the condition given in (4). Proposition 4.10.
An observable v defined as in (27) belongs to A α .Proof. By Proposition 4.9, we have that v ∈ L µ α ([0 , (cid:107) v (cid:107) ∞ =
1. Moreover, byLemma 2.3, we have on [0 , j ∈ N , (cid:98) F j − α ( v · A j )( x ) = t j if N k − n k ≤ j ≤ N k for some k ∈ N and if x ∈ A , (cid:98) F j − α ( v · A j ) ∈ B α , for all j ∈ N , and hence, it follows that v ∈ A α . (cid:3) Proposition 4.11.
An observable v defined as in (27) satisfies the summability conditiongiven in (4) .Proof.
Lemma 2.3 and (28) together imply that ∞ (cid:88) k = (cid:107) (cid:98) F k α ( v · A k + ) (cid:107) ∞ = ∞ (cid:88) k = s k N k (cid:88) j = N k − n k t j = ∞ (cid:88) k = s k N k (cid:88) j = N k − n k µ α ( A j ) = (cid:90) | v | d µ α . The latter term is finite, since v ∈ L µ α ([0 , (cid:3) In the proof of Proposition 4.13, we will require the following auxiliary result.
Lemma 4.12.
For each N ∈ N , the following sequence diverges to infinity: s k N k − N (cid:88) j = N k − n k N − δ k l ( N k )( N k − j ) − δ l ( N k − j ) l ( j ) j δ k ∈ N . Proof.
The result follows immediately from combining the following three observations.(i) Using the facts that δ ∈ (1 / ,
1) and (cid:15) >
0, that the sequence ( N k ) k ∈ N is not boundedabove and is strictly monotonically increasing, that s k (cid:66) − g k and that N is a fixednatural number, we have thatlim k → + ∞ s k (cid:32) N k N k − N (cid:33) δ N k − δ − (cid:15) N δ − (cid:15) = . N THE ASYMPTOTICS OF THE α -FAREY TRANSFER OPERATOR 19 (ii) For each k ∈ N , we have that s k N − δ − (cid:15) k n δ − (cid:15) k ≥ − g k g (1 − δ − (cid:15) ) k (2 g ( δ − (cid:15) ) − = ( g (1 − δ − (cid:15) ) + g ( δ − (cid:15) ) − g ) k − ( g (1 − δ − (cid:15) ) − g ) k . Using condition (C3) with the facts that δ ∈ (1 / , (cid:15) > g , g and g arepositive, it follows that lim k ∈ N s k N k − δ − (cid:15) n k δ − (cid:15) = + ∞ . (iii) There exist constants κ, ξ > k ∈ N su ffi ciently large, N k − N (cid:88) j = N k − n k N k − δ l ( N k )( N k − j ) − δ l ( N k − j ) l ( j ) j δ ≥ (cid:32) N k − N (cid:33) δ l ( N k − N ) l ( N k ) N − δ k e − ξ N k − N (cid:88) j = N k − n k N k − j ) − δ l ( N k − j ) ≥ κ e − ξ (cid:32) N k N k − N (cid:33) δ N − δ − (cid:15) k (cid:16) n δ − (cid:15) k − N δ − (cid:15) (cid:17) . Here, the first inequality follows from the facts that l is a slowly varying functionand that lim n →∞ ( N k − n k ) / N k = κ > κ − n (cid:15) ≥ l ( n ) ≥ κ n − (cid:15) , for all n ∈ N . (cid:3) Proposition 4.13.
For an observable v as defined in (27) , we have that, on A , lim inf n → + ∞ w n (cid:98) F n α ( v ) = Γ δ (cid:90) v d µ α and lim sup n → + ∞ w n (cid:98) F n α ( v ) = + ∞ . (29) Proof.
By Theorem 2.1 and Proposition 2.4, we have uniformly on A thatlim n → + ∞ l ( n ) n − δ A · (cid:98) F n α ( A ) = ( Γ δ / Γ δ ) µ α ( A ) A = ( Γ δ / Γ δ ) A . Thus, given ξ >
0, there exists N = N ( ξ ) ∈ N such that, for all n ≥ N on A ,e ξ Γ δ n δ − Γ δ l ( n ) ≥ (cid:98) F n α ( A ) ≥ e − ξ Γ δ n δ − Γ δ l ( n ) . (30)We will first show the second statement in (29). For this, observe that by (12) it is su ffi cientto show that, on A , lim sup k → + ∞ l ( N k ) N k − δ (cid:98) F N k α ( v )( x ) = + ∞ . In order to see this, let ξ > p = p ( ξ ) ∈ N denote the smallest integer forwhich n p > N . Since (cid:98) F α is a positive linear operator, we have, for all k > p , that l ( N k ) N k − δ (cid:98) F N k α ( v ) ≥ s k l ( N k ) N k − δ N k − N (cid:88) j = N k − n k (cid:98) F N k α ( A j ) . (31)Now, Lemma 2.6 (i) implies that lim n → + ∞ ( n − δ l ( n )) / (( n + − δ l ( n + =
1. As the sequence( a n ) n ∈ N is positive and since a n = δ n − − δ l ( n ) the value r (cid:66) inf { ( n − δ l ( n )) / (( n + − δ l ( n + } is finite and strictly greater than zero. Hence, by (14), (30) and (31) and the fact that t n ∼ l ( n ) n − δ , we have on A that, for each k ∈ N su ffi ciently large, l ( N k ) N k − δ (cid:98) F N k α ( v ) ≥ e − ξ Γ δ Γ δ s k N k − N (cid:88) j = N k − n k ( N k − j ) − δ l ( N k − j )( N k − j + − δ l ( N k − j + N − δ k l ( N k )( N k − j ) − δ l ( N k − j ) t j ≥ e − ξ Γ δ Γ δ rs k N k − N (cid:88) j = N k − n k N − δ k l ( N k )( N k − j ) − δ l ( N k − j ) l ( j ) j δ . By Lemma 4.12, the latter term diverges.All that remains to show is that the first statement of (29) holds. For this, observe that,by positivity and linearity of (cid:98) F , Theorem 2.1, Proposition 2.4 and (14), we have on A that,for each k ∈ N , Γ δ N k (cid:88) l = (cid:90) v · A l d µ α = Γ δ k (cid:88) m = s m N m (cid:88) j = N m − n m t j = k (cid:88) m = s m N m (cid:88) j = N m − n m lim inf n → + ∞ w n (cid:98) F n − j + ( (cid:98) F j + α ( A j )) ≤ lim inf n → + ∞ w n (cid:98) F n k (cid:88) m = s m N m (cid:88) j = N m − n m A j ≤ lim inf n → + ∞ w n (cid:98) F n ( v ) . Since k ∈ N was arbitrary, the above inequalities imply that on A ,lim inf n → + ∞ w n (cid:98) F n ( v ) ≥ Γ δ (cid:90) v d µ α . Suppose that the latter inequality is strict, namely, suppose that there exists a constant c > A , lim inf n → + ∞ w n (cid:98) F n ( v ) ≥ c > Γ δ (cid:90) v d µ α . This assumption together with (12) implies that, given ξ >
0, there exists M = M ( ξ ) ∈ N such that, for all n ≥ M and x ∈ A , (cid:98) F n ( v )( x ) ≥ e − ξ Γ δ − cn δ − / l ( n ) . Thus, by Karamata’s Tauberian Theorem for power series [6, Corollary 1.7.3], it followsthat, for all n ≥ M and x ∈ A , n (cid:88) k = (cid:98) F k ( v )( x ) ≥ M (cid:88) k = (cid:98) F k ( v )( x ) + n (cid:88) k = M + e − ξ Γ δ − ck δ − / l ( k ) ≥ M (cid:88) k = (cid:98) F k ( v )( x ) + e − ξ Γ − δ Γ δ − cn δ / l ( n ) . Hence,lim inf n ∈ + ∞ w n n n (cid:88) k = (cid:98) F n ( v )( x ) = lim inf n → + ∞ n − δ l ( n ) Γ δ n (cid:88) k = (cid:98) F n ( v )( x ) ≥ Γ − δ δ − c > Γ − δ Γ δ (cid:90) v d µ α . This is a contradiction, since by (12) and by combining Theorem 2.1 with Karamata’sTauberian Theorem for power series [6, Corollary 1.7.3], we have that the set A is aDarling-Kac set and therefore, by [1, Proposition 3.7.5], the α -Farey system is pointwisedual ergodic, meaning that, for µ α -almost every x ∈ [0 , n ∈ + ∞ w n n n (cid:88) k = (cid:98) F n ( v )( x ) = Γ − δ Γ δ (cid:90) v d µ α . (cid:3) Proof of Theorem 1.3 (iii).
This follows from Propositions 4.9, 4.10, 4.11 and 4.13. (cid:3)
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