Genericity of pseudo-Anosov mapping classes, when seen as mapping classes
aa r X i v : . [ m a t h . G T ] J a n Genericity of pseudo-Anosov mapping classes,when seen as mapping classes
Viveka Erlandsson, Juan Souto, and Jing Tao
Abstract.
We prove that pseudo-Anosov mapping classes are generic with respect tocertain notions of genericity reflecting that we are dealing with mapping classes.
1. Introduction
Throughout this paper let Σ be a complete orientable hyperbolic surface of finite area,with genus g and r punctures, where ( g, r ) = (0 , ρ : Map(Σ) → R ≥ is aproper positive function, then we say that a set X ⊂ Map(Σ) is generic with respect to ρ ,or ρ -generic for short, if we have lim R →∞ | B ρ ( R ) ∩ X || B ρ ( R ) | = 1where B ρ ( R ) = { φ ∈ Map(Σ) with ρ ( φ ) ≤ R } . Here properness of ρ just means that B ρ ( R ) is a finite set for all R . A negligible set is one whose complement is generic.Maybe the first function that comes to mind is the word length with respect to a finitegenerating set G of Map(Σ), and Cumplido and Wiest [6] proved that indeed the set ofpseudo-Anosov elements is not negligible in this sense. It is not yet known if it is generic.However, one can make the case that the word length, while being related to the grouptheory of the mapping class group, has little to do with the fact that the mapping classgroup consists of mapping classes. To illustrate this point identify SL Z with the mappingclass group of the once punctured torus and note that the two matrices A = (cid:18) (cid:19) , B = (cid:18) (cid:19) have the same word length, namely 99, with respect to the standard generating set ofSL Z . Arguably, it would be more natural to say that A is farther from the identity than B . Not only because the coefficients of A are much larger than those of B but, moreimportantly, because the map induced by A on the torus distorts both the metric andconformal structure much more dramatically than the map induced by B . Genericity of pseudo-Anosovs
Our goal is to prove that pseudo-Anosov mapping classes are ρ -generic with respect toa number of functions on Map(Σ) measuring the complexity of mapping classes when seenas mapping classes: Theorem 1.1.
The set of pseudo-Anosov mapping classes is generic with respect to anyone of the functions ρ K ( φ ) = inf { K ( f ) where f ∈ Diff(Σ) represents φ } where K ( f ) is the quasi-conformal distortion of f , ρ Lip ( φ ) = inf { Lip( f ) where f ∈ Diff(Σ) represents φ } where Lip( f ) is the Lipschitz constant for f , and ρ σ,η ( φ ) = ι ( φ ( σ ) , η ) where σ and η are filling multicurves and ι ( · , · ) is the geometric intersection number.Remark. Note that, although amazingly it is not formally stated in the paper, the claimfor ρ K ( φ ) in Theorem 1.1 was obtained by Maher in [14]. Unfortunately, we were unawareof this fact until we finished writing our paper. Both the argument in [14] and ours havethe same starting point, namely an earlier, again not formally stated, result from [13].However, after that starting point, the arguments use different methods and techniques.We will return to this at the end of the introduction.We sketch now the proof of Theorem 1.1. We begin by addressing the reason why we areincluding ρ σ,η at all among the functions in Theorem 1.1. There are a few reasons. First,both quantities ρ K ( φ ) and ρ Lip ( φ ) can be estimated in terms of ρ σ,η . Second, there is themaybe not very important observation that, after identifying SL Z with the mapping classgroup of a punctured torus, the ℓ -norm on SL Z agrees with ρ σ,σ where σ is the unionof the two simple curves representing the standard generators of homology. However, themain reason to consider ρ σ,η is that it is the more natural quantity from the point of viewof proofs.In fact, if we denote by C (Σ) the space of geodesic currents on Σ endowed with theweak-* topology, and consider multicurves as currents, then what we will actually prove isthe following theorem: Theorem 1.2.
Let
R ⊂
Map(Σ) be the set of non-pseudo-Anosov mapping classes and let γ ⊂ Σ be a filling multicurve. Then we have lim L →∞ |{ φ ∈ R with F ( φ ( γ )) ≤ L }| L g − r = 0 for every continuous homogenous function F : C (Σ) → R ≥ which, for every compact K ⊂ Σ , is proper when restricted on the set C K (Σ) of currents supported by K . Recall that a function F : C (Σ) → R is homogenous if F ( t · λ ) = t · F ( λ ) for every t ≥ λ ∈ C (Σ). Note also that for Σ open, the properness condition we impose on F is muchweaker it being proper on C (Σ). For example, if η is a filling multicurve then F ( · ) = ι ( · , η )is not proper on C (Σ) but is proper on C K (Σ) for any K . Theorem 1.1 follows when weapply Theorem 1.2 to the corresponding functions combined with the fact, see [7, 26], thatlim inf L →∞ |{ φ ∈ Map(Σ) with F ( φ ( γ )) ≤ L }| L g − r > iveka Erlandsson, Juan Souto, and Jing Tao 3 for any F as in Theorem 1.2.The starting point of the proof of Theorem 1.2 is a result of Maher [13] asserting thatthe set R ⊂
Map( S ) of non-pseudo-Anosov mapping classes is the union, for each k , of k -isolated points (that is, points which are at distance at least k from any other elementof R ) together with the union of finitely many sets, each one of which consists of mappingclasses at relative distance L ( k ) around the centralizer of some mapping class. Here therelative distance is the semi-distance on Map( S ) arising, with the help of a base point, formits action on the curve complex. It follows that proving that R is negligible boils downto proving (1) that the set R k ⊂ R of k -isolated points has low density and (2) that setsof mapping classes with small relative distance of centralizers of elements are negligible.Rephrasing this in terms of measures (on the space of currents) it suffices to prove (1) thatlim k →∞ lim L →∞ L g − r X φ ∈I k δ L φ ( γ ) = 0 , (1.2)and (2) that lim L →∞ L g − r X φ ∈N rel ( C ( φ ) ,R ) δ L φ ( γ ) = 0 (1.3)for φ ∈ Map(Σ) non-central. Here δ x is the Dirac measure centred on x and the conver-gence takes place with respect to the weak-*-topology. We get (1.3) from the fact that anylimit is absolutely continuous to the Thurston measure — an immediate consequence of forexample Proposition 4.1 in [7] — and of the fact that the set of limits of sequences of theform ( φ i ( γ )) with φ ∈ N rel ( C ( φ ) , R ) has vanishing Thurston measure. To establish (1.2)we use again that any limit is absolutely continuous with respect to the Thurston measure,but this time we have to use Masur’s result [18] on the ergodicity of the Thurston measurewith respect to the action of the mapping class group. Remark.
Maher’s proof in [14] of Theorem 1.1 also relies on the decomposition of R as theunion of I k and finitely many sets consisting of mapping classes at bounded relative distancefrom the centralizer of some mapping class. At this point the two arguments diverge. Whilewe rely on the fact that every limit of (1.2) and (1.3) is absolutely continuous with respectto the Thurston measure, Maher makes use of a rather sophisticated lattice counting resultof Athreya-Bufetov-Eskin-Mirzakhani [2]. Similarly, while we rely on the ergodicity ofthe Thurston measure, that is the ergodicity of the Teichm¨uller flow, Maher relies on themixing property of that flow. We might be partial, but we believe that our argument isnot only different but also simpler than that of Maher. Remark.
As it is the case for Maher’s argument, all the results here hold with unchangedproofs if we replace the set R of non-pseudo Anosov elements by any set of elements forwhich there is a uniform upper bound for the translation length in the curve complex. Acknowledgments.
The last two authors started discussing the issues treated here whilevisiting the Fields Institute in the framework of the thematic program
Teichm¨uller Theoryand its Connections to Geometry, Topology and Dynamics . We are very thankful for thesupport of the Fields Institute and for many discussions with Kasra Rafi. The last authorwould also like to thank the second author and Anna Lenzhen for their hospitality lastsummer during which time this project was discussed. She also gratefully acknowledges
Genericity of pseudo-Anosovs
NSF DMS-1611758 and DMS-1651963. The first author was supported by EPSRC grantEP/T015926/1.
2. Maher’s theorem
As we already did in the introduction, we denote by R the set of all non-pseudo-Anosovmapping classes of Map(Σ). We also fix an arbitrary finite generating set G for Map(Σ)and let d G be the induced left-invariant distance: d G ( φ, ψ ) = word length with respect to G of ψ − φ. Given k > I k = { φ ∈ R with d G ( φ, φ ′ ) ≥ k for all φ ′ ∈ R \ { φ }} be the set of elements in R which do not have any other elements in R within distance lessthan k . We denote the complement of I k by D k = R \ I k . The notations are chosen to suggest that I k consists of k -isolated points and that D k consists of k -dense points .Recall that distances in the definition of I k (and thus in that of D k as well) are measuredwith respect to the distance d G . We stress that this is the case because we will also beworking with another distance, or rather a semi-distance, namely the relative distance d rel ( φ, ψ ) = d C (Σ) ( φ ( α ) , ψ ( α ))where d C (Σ) ( · , · ) denotes the distance in the curve complex C (Σ), and where α is a fixedbut otherwise arbitrary simple essential curve in Σ.Armed with this notation we can state Maher’s theorem: Theorem 2.1 (Maher) . For every k , there is a finite set of non-central mapping classes F ⊂
Map(Σ) \ C (Map(Σ)) and some L > such that D k ⊂ [ φ ∈F { ψ ∈ Map(Σ) with d rel ( ψ, C ( φ )) ≤ L } , where C ( φ ) is the centralizer of φ in Map( S ) and C (Map(Σ)) is the center of Map(Σ) . Although it is proved and used in [13] (see the discussion at the beginning of section 5 insaid paper), Theorem 2.1 is not explicitly stated therein. Hence we discuss how to deduceit from the stated results here:
Proof.
First, suppose that Map(Σ) is center free. Then, from the very definition of D k , weget that there is a finite subset F ⊂
Map(Σ) with D k ⊂ [ φ ∈F ( R ∩ R φ ) . (2.4)To see this, note that one can take F to be all non-trivial elements in the ball of radius k around the identity with respect to d G .Now, Theorem 4.1 in [13] implies that for each φ ∈ F there is some L such that R ∩ R φ ⊂ { ψ ∈ Map(Σ) with d rel ( ψ, C ( φ )) ≤ L } . iveka Erlandsson, Juan Souto, and Jing Tao 5 This theorem applies because the mapping class group is weakly relatively hyperbolic withrelative conjugacy bounds [13, Theorem 3.1] and because R consists of elements conjugatedto elements of bounded relative length [13, Lemma 5.5]. This concludes the discussion ofTheorem 2.1 if Map(Σ) is center free.In the presence of a non-trivial center the argument is almost the same: Note that R = R φ for every central element and hence the only change to the above argument isthat one has to take F to be the set of all non-central elements in the ball of radius k around the identity with respect to d G . (cid:3)
3. Currents
In this section we recall a few facts about the space of geodesic currents on Σ. Wethen describe the (projective) accumulation points of sequences of the form ( φ i ( γ )) where γ is an essential multicurve and where ( φ i ) is a sequence of mapping classes at boundedrelative distance of the centralizer of some φ ∈ Map(Σ). Recall that a multicurve is a finiteunion of (disjoint or not) of (simple or not) primitive essential curves in Σ. We say that amulticurve is filling if its geodesic representative cuts the surface into a collection of disksand once-punctured disks.
Properties of the space of currents.
Let Σ be a compact surface with interior Σ =Σ \ ∂ Σ, endowed with an arbitrary hyperbolic metric with totally geodesic boundary. Wesuggest the reader to think, in a first reading, that Σ = Σ; that is, Σ is closed.Geodesic currents on Σ are fundamental group invariant Radon measures on the spaceof geodesics on the universal cover of Σ. However, that they are such measures will notreally be relevant here—what is more important for our purposes are the properties thespace C (Σ) of currents have (when endowed with the weak-*-topology). We list the factsabout C (Σ) that we will use:(1) C (Σ) is a locally compact metrizable topological space.(2) C (Σ) is a cone as a topological vector space, meaning in particular that there arecontinuous maps C (Σ) × C (Σ) → C (Σ) , ( λ, µ ) λ + µ R ≥ × C (Σ) → C (Σ) , ( t, λ ) tλ satisfying the usual associativity, commutativity and distributivity properties as invector spaces.(3) The set { γ closed geodesic in Σ } is a subset of C (Σ) and in fact the set R + · { γ closed geodesic in Σ } of weighted closed geodesics is dense in C (Σ).(4) The inclusion of the set of weighted simple geodesics into C (Σ) extends to a con-tinuous embedding of the space ML (Σ) of measured laminations into C (Σ).(5) There is a continuous bilinear map ι : C (Σ) × C (Σ) → R ≥ such that ι ( γ, γ ′ ) is nothing other than the geometric intersection number for allclosed geodesics γ, γ ′ . Genericity of pseudo-Anosovs (6) The mapping class group acts continuously on C (Σ) by linear automorphisms.Moreover, the inclusion of the set of closed geodesics into C (Σ) is equivariant withrespect to this action.Moreover, for every compact K ⊂ Σ, let C K (Σ) ⊂ C (Σ) be the subcone consisting of thecurrents supported by K . Then the following holds:(7) The set { λ ∈ C K (Σ) with ι ( λ, η ) ≤ L } is compact for every L ≥ η . In particular, the image P C K (Σ) of C K (Σ) in the spaceP C (Σ) = ( C (Σ) \ { } ) / R > of projective currents is compact.(8) For every multicurve γ there is a compact K ⊂ Σ such thatMap(Σ) · γ ⊂ C K (Σ) . In particular, every sequence ( φ i ) in Map(Σ) contains a subsequence ( φ i j ) such thatthe limit lim j →∞ φ i j ( γ ) exists in P C (Σ).Currents were introduced by Bonahon in [3, 4] and all the facts here can be found ina more or less transparent way in these papers. In the case of closed surfaces, [1] is avery readable account of currents, measured laminations, and the relation between them.Finally, we hope that the presentation of currents, for both open and closed surfaces, inthe forthcoming book [10] will also be similarly readable. Accumulation points of thickened centralizers.
It will be important later on to knowthat projective accumulation points, in the space of currents, of sequences of the form( φ i ( γ )) where γ is a multicurve and with φ i ∈ N rel ( C ( φ ) , L ) = { ψ ∈ Map( S ) with d rel ( ψ, C ( φ )) ≤ L } are very particular: Proposition 3.1.
Let φ ∈ Map(Σ) \ C (Map(Σ)) be a non-central mapping class, let ( φ n ) bea sequence of pairwise distinct elements in N rel ( C ( φ ) , L ) , and let γ be a filling multicurve.If the sequence ( φ n ( γ )) converges projectively to a uniquely ergodic measured lamination λ , then φ ( λ ) is a multiple of λ . Recall that a measure lamination λ is uniquely ergodic if every measured lamination µ with ι ( λ, µ ) = 0 is a multiple of λ .We start with the following observation: Lemma 3.2.
Let γ ⊂ Σ be a filling multicurve and ( φ n ) and ( ψ n ) be sequences of mappingclasses with d rel ( φ n , ψ n ) ≤ L . Given any simple multicurve α , suppose that the sequences ( φ n ( γ )) and ( ψ n ( α )) converge projectively to λ, λ ′ ∈ P C (Σ) , respectively. If ( φ n ) consistsof pairwise distinct elements, then there is a chain λ = λ , λ , . . . , λ k = λ ′ of measured laminations with ι ( λ i , λ i +1 ) = 0 for all i = 0 , . . . , k − .Proof. We first prove the statement for α = α , where α is the base point in C (Σ) usedto define d rel . Assume that ( φ n ( γ )) and ( ψ n ( α )) converge projectively to λ, λ ′ ∈ P C (Σ).Abusing notation consider λ and λ ′ not only as projective currents but also as actualcurrents. The assumption that the sequences ( φ n ( γ )) and ( ψ n ( α )) converge projectively iveka Erlandsson, Juan Souto, and Jing Tao 7 to λ, λ ′ ∈ P C (Σ) implies that there are bounded sequences ( ǫ n ) and ( ǫ ′ n ) consisting ofpositive numbers and such that λ = lim n ǫ n φ n ( γ ) , λ ′ = lim n ǫ ′ n ψ n ( α ) . The assumptions that ( φ n ) consists of pairwise distinct elements and that γ is fillingimplies that the sequence ( φ n ( γ )) is not eventually constant, and thus that ǫ n → d rel ( φ n , ψ n ) ≤ L implies that for all n there is a chain ofsimple curves φ n ( α ) = β n , β n , . . . , β L +1 n = ψ n ( α )with ι ( β in , β i +1 n ) = 0 for all i = 1 , . . . , L and all n . Projective compactness of the spaceof currents (or rather of measured laminations) implies that passing to a subsequence wemight assume that there are bounded positive sequences ( ǫ n ) , . . . , ( ǫ L +1 n ) such thatlim n →∞ ǫ in β in = λ i = 0exists in the space ML (Σ) of measured lamination. We might also assume without loss ofgenerality that ǫ L +1 n = ǫ ′ n and thus that λ L +1 = λ ′ .The claim will follow when we show that ι ( λ, λ ) = ι ( λ , λ ) = ι ( λ , λ ) = · · · = ι ( λ L , λ L +1 ) = 0 . To do so, first note that ι ( λ, λ ) = lim n ǫ n · ǫ n · ι ( φ n ( γ ) , β n )= lim n ǫ n · ǫ n · ι ( φ n ( γ ) , φ n ( α ))= lim n ǫ n · ǫ n · ι ( γ , α ) = 0 , where the last equality follows from the fact that the sequence ( ǫ n ) is bounded while ( ǫ n )tends to 0. The proof of the other equalities is even simpler: since the curves β in and β i +1 n are disjoint for all n and i we have ι ( λ i , λ i +1 ) = lim n ǫ in · ǫ i +1 n · ι ( β in , β i +1 n ) = 0 . Now suppose α is an arbitrary simple multicurve with ( ψ n ( α )) converging to λ ′′ ∈ P C (Σ).There is a sequence α , α , . . . , α m = α of simple multicurves, with ι ( α i , α i +1 ) = 0 for all i = 0 , . . . , m −
1. By passing to subsequences of ( ψ n ( α i )) and taking limits as n → ∞ , weget a sequence of measured laminations λ ′ = λ ′ , . . . , λ ′ m = λ ′′ with ι ( λ ′ i , λ ′ i +1 ) = 0 for all i = 0 , . . . , m −
1. This chain extends the one from λ to λ ′ to achain from λ to λ ′′ . This finishes the proof of the lemma. (cid:3) We are ready to prove the proposition:
Proof of Proposition 3.1.
Take for all n some ψ n ∈ C ( φ ) with d rel ( φ n , ψ n ) ≤ L . Let α beany simple multicurve and let β = φ ( α ). Compactness of P ML (Σ) implies that, up topassing to a subsequence, we might assume that the limits λ ′ = lim n ψ n ( α ) and λ ′′ = lim n ψ n ( β )exist in P C (Σ). Genericity of pseudo-Anosovs
From Lemma 3.2, there is a chain of measure laminations λ = λ , λ , . . . , λ m = λ ′ with ι ( λ i , λ i +1 ) = 0 for i = 1 , . . . , m −
1. There is a similar chain from λ to λ ′′ .Recall now that λ = λ is uniquely ergodic. Since ι ( λ , λ ) = 0, we get that λ is amultiple of λ and thus uniquely ergodic. Then, since ι ( λ , λ ) = 0, we get that λ is amultiple of λ and thus of λ and uniquely ergodic and so on. Iteratively we get that λ ′ isa multiple of λ . Using the chain from λ to λ ′′ , we also get that λ ′′ is a multiple of λ .Finally, since β = φ ( α ) and ψ n ∈ C ( φ ), we have that, projectively, φ ( λ ′ ) = lim n →∞ φ ( ψ n ( α )) = lim n →∞ ψ n ( φ ( α )) = lim n →∞ ψ n ( β ) = λ ′′ . This implies that λ is projectively fixed by φ , so φ ( λ ) is a multiple of λ as claimed. (cid:3)
4. A technical result
The reason why we stressed earlier that C (Σ) is metrizable and locally compact is thatthese are the properties needed to work as customary with the weak-*-topology on thespace of measures on C (Σ). In fact, to establish Theorem 1.2 we will prove that themeasures m R γ ,L = 1 L g − r X φ ∈R δ L φ ( γ ) (4.5)converge when L → ∞ to the trivial measure. Here we consider the weighted multicurve L φ ( γ ) as a current and denote by δ L φ ( γ ) the Dirac measure on C (Σ) centered therein.In [7, 8, 11, 24] we considered a closely related family of measures and proved that thelimit C · m Thu = lim L →∞ L g − r X φ ∈ Map( S ) δ L φ ( γ ) . (4.6)exists (see also [10]). Here C = C ( γ ) is a positive real number and m Thu is the Thurstonmeasure on C (Σ). Recall that the Thurston measure is a Radon measure supported onthe space ML (Σ) of measured laminations. The Thurston measure can be constructedeither as a scaling limit [20, 10] or using the symplectic structure on ML (Σ). See [22] fora discussion of both points of view.The only facts about the Thurston measure we will need are that it is preserved by themapping class group, that the actionMap(Σ) y ( ML (Σ) , m Thu )is almost free in the sense that the fixed point set of every non-central element in Map(Σ)has vanishing Thurston measure—central elements act trivially on ML (Σ)—and that it isergodic with respect to Map(Σ) [18].In this section we prove: This is also the reason why we didn’t encourage the reader to think of currents as measures, becauseit is a well-established fact that thinking of ”the weak-*-topology on the space of measures on the spaceof measures endowed with the weak-*-topology” leads the unprepared reader to tremors, shaking and coldsweats. iveka Erlandsson, Juan Souto, and Jing Tao 9
Proposition 4.1.
Let γ ⊂ Σ be a filling multicurve. The family of measures (cid:0) m R γ ,L (cid:1) L ≥ is precompact with respect to the weak-*-topology on the space of Radon measures on C (Σ) .Moreover for any sequence L n → ∞ such that the limit m = lim n →∞ m R γ ,L n exists, one has that X φ ∈ Map(Σ) φ ∗ m ≤ C · m Thu where C is as in (4.6) . We start by proving that the family of measures in Proposition 4.1 is precompact andthat any limit must be uniformly continuous with respect to m Thu . Lemma 4.2.
The family of measures (cid:0) m R γ ,L (cid:1) L ≥ is precompact with respect to the weak-*-topology on the space Radon measures on C (Σ) . Moreover, any accumulation point isabsolutely continuous with respect to the Thurston measure.Proof. The measure m R γ ,L is bounded from above for all L by the measure m γ ,L = 1 L g − r X φ ∈ Map( S ) δ L φ ( γ ) . (4.7)From the existence of the limit (4.6) we getlim sup Z f dm R γ ,L ≤ lim sup Z f dm γ ,L = C · Z f d m Thu < ∞ (4.8)for every continuous function f : C (Σ) → R with compact support. This implies thatthe family ( m R γ ,L ) L ≥ is bounded and thus precompact in the weak-*-topology. Moreover,(4.8) implies that any accumulation point of m R γ ,L is bounded from above by C · m Thu andhence is absolutely continuous to the Thurston measure, as we had claimed. (cid:3)
Note that the same argument also proves that both families m I k γ ,L = 1 L g − r X φ ∈I k δ L φ ( γ ) and m D k γ ,L = 1 L g − r X φ ∈D k δ L φ ( γ ) are precompact and that any limit when L → ∞ is absolutely continuous with respectto the Thurston measure. Here I k and D k are, as before, the subsets of R consisting of k -isolated points and k -dense points, respectively.We can from now on fix a sequence ( L n ) with L n → ∞ such that the following limits allexist: m γ = lim n →∞ m R γ ,L n , m I k γ = lim n →∞ m I k γ ,L n , and m D k γ = lim n →∞ m D k γ ,L n . (4.9)Since R is the disjoint union of I k and D k they automatically satisfy that m γ = m I k γ + m D k γ . Our next goal is to prove that the second of these limits is 0:
Lemma 4.3.
We have m D k γ = 0 .Proof. By Maher’s Theorem 2.1 it is enough to prove that, for any non-central φ ∈ Map( S )and any R ≥
0, the trivial measure is the only accumulation point when L → ∞ of thefamily of measures m N γ ,L = 1 L g − r X φ ∈N rel ( C ( φ ) ,R ) δ L φ ( γ ) . Well, each m N γ ,L is bounded by the measure m γ ,L given by (4.7) and hence any suchaccumulation point m ′ = lim n →∞ m N γ ,L n is bounded by C · m Thu by (4.6). The claim willthen follow when we say that the support of m ′ is contained in a set of vanishing Thurstonmeasure.First, the support of the limiting measure m ′ is contained in the set of accumulationpoints of sequences ( x n ) where x n is in the support of m N γ ,L n , that is, a multiple of φ n ( γ ) forsome φ n ∈ N rel ( C ( φ ) , R ). On the other hand, since the set of uniquely ergodic laminationhas full m Thu -measure [17], we also get that m ′ is supported by uniquely ergodic laminations.It thus follows from Proposition 3.1 that m ′ is supported by the set of measured laminationsprojectively fixed by φ . Since this set has vanishing m Thu -measure we get that m ′ is trivial,as we needed to prove. (cid:3) As a final step towards the proof of Proposition 4.1 we establish an equivariance propertyfor the limits of the measures m R γ ,L : Lemma 4.4.
We have φ ∗ ( m γ ) = lim n →∞ m R φ ( γ ) ,L n for all φ ∈ Map(Σ) .Proof.
Noting that the set R is closed under conjugation we get that R φ = φ R . Thismeans that m R φ ( γ ) ,L n = 1 L g − r X ψ ∈R δ L ψφ ( γ ) = 1 L g − r X ψ ∈R φ δ L ψ ( γ ) = 1 L g − r X ψ ∈ φ R δ L ψ ( γ ) = 1 L g − r X ψ ∈R δ L φψ ( γ ) = 1 L g − r X ψ ∈R φ ∗ (cid:16) δ L ψ ( γ ) (cid:17) = φ ∗ L g − r X ψ ∈R δ L ψ ( γ ) = φ ∗ (cid:0) m R γ ,L n (cid:1) The claim follows now from (4.9) and the continuity of the action of Map(Σ) on the spaceof currents. (cid:3)
We are ready to prove the proposition:
Proof of Proposition 4.1.
Recall that Lemma 4.2 asserts that the given family of measuresis precompact and hence we can assume that we are given a sequence ( L n ) with L n → ∞ such that the limit m γ = lim n →∞ m R γ ,L n iveka Erlandsson, Juan Souto, and Jing Tao 11 exists. To prove Proposition 4.1 it will suffice to show, with C as in (4.6), that for everyfinite set Z ⊂
Map(Σ) we have X φ ∈Z φ ∗ ( m γ ) ≤ C · m Thu . Fixing such a finite set Z choose k > · max { d G (id , φ ) where φ ∈ Z} . Lemma 4.4 and Lemma 4.3 imply, respectively, the first and last of the following equalities: X φ ∈Z φ ∗ ( m γ ) = X φ ∈Z lim n →∞ m R φ ( γ ) ,L n = lim n →∞ X φ ∈Z m R φ ( γ ) ,L n = lim n →∞ X φ ∈Z m I k φ ( γ ) ,L n . Moreover, from the choice of k we get that I k φ ∩ I k φ ′ = ∅ for any two distinct φ, φ ′ ∈ Z and we can thus rewrite X φ ∈Z m I k φ ( γ ) ,L n = 1 L g − rn X φ ∈Z X ψ ∈I k δ Ln ψφ ( γ ) = 1 L g − rn X φ ∈Z X ψ ∈I k φ δ Ln ψ ( γ ) = 1 L n g − r X ψ ∈ S φ ∈Z I k φ δ Ln ψ ( γ ) . It thus follows that X φ ∈Z m I k φ ( γ ) ,L n ≤ L g − rn X ψ ∈ Map( S ) δ Ln ψ ( γ ) and hence that X φ ∈Z φ ∗ ( m γ ) ≤ lim n →∞ L g − rn X φ ∈ Map( S ) δ Ln φ ( γ ) = C · m Thu . We are done. (cid:3)
5. Proofs of the theorems
We are now ready to prove the main results.
Theorem 1.2.
Let
R ⊂
Map(Σ) be the set of non pseudo-Anosov mapping classes and let γ ⊂ Σ be a filling multicurve. Then we have lim L →∞ |{ φ ∈ R with F ( φ ( γ )) ≤ L }| L g − r = 0 for every continuous homogenous function F : C (Σ) → R ≥ which, for every K ⊂ Σ compact, is proper when restricted on the set C K (Σ) of currents supported by K .Proof. The claim will follow easily once we prove thatlim L →∞ m R γ ,L = 0 (5.10) with m R γ ,L as in (4.5). Since this family of measures is precompact by Proposition 4.1, itsuffices to prove that 0 is the only accumulation point when L → ∞ . So let ( L n ) be asequence tending to ∞ and such that the limit m γ = lim n →∞ m R γ ,L n exists. By Lemma 4.2 m γ is absolutely continuous with respect to m Thu . This means thatthere is a function (the Radon-Nikodym derivative) κ : C (Σ) → R ≥ with the propertythat Z C (Σ) f ( ζ ) d m γ ( ζ ) = Z C (Σ) f ( ζ ) · κ ( ζ ) d m Thu ( ζ )for any continuous compactly supported function f on the space of currents.Proposition 4.1 asserts that the measure P φ ∈ Map( S ) φ ∗ m γ is not only finite, but actuallybounded by a multiple C · m Thu of the Thurston measure. In terms of the function κ , thisimplies that X φ ∈ Map( S ) κ ( φ ( ζ )) ≤ C for m Thu -almost every ζ ∈ C (Σ) . (5.11)We claim that this implies that κ ( ζ ) = 0 almost surely: Claim. κ ( ζ ) = 0 almost surely with respect to the Thurston measure. In a nutshell, the claim follows from the fact that ergodic actions of discrete groups onnon-atomic measure spaces are recurrent (the condition on the measure being non-atomicis just there to rule out actions with only one orbit). In any case, we give a direct argumentto prove the claim:
Proof of the Claim.
If the claim fails to be true, then there is a positive m Thu -measure set U ⊂ C (Σ) with κ ( ζ ) ≥ ǫ > ζ ∈ U . Noting that the action Map(Σ) y C (Σ) isalmost free we get from (5.11) that, for almost every ζ ∈ U , { φ ∈ Map(Σ) with φ ( ζ ) ∈ U } ≤ Cǫ .
It follows that there is a set V ⊂ U of positive Thurston measure such that the set Z = { φ ∈ Map(Σ) with φ ( V ) ∩ U = ∅} is finite. Now, since the action is essentially free we can in fact find W ⊂ V of positiveThurston measure with W ∩ φ ( W ) = ∅ for all φ / ∈ C (Map(Σ)). This contradicts theergodicity of the action of the mapping class group on ( ML (Σ) , m Thu ). (cid:3) The claim implies that the limiting measure vanishes, that is m γ = 0, establishing(5.10). We can now conclude the proof: let F : C (Σ) → R ≥ be as in the statement andnote that |{ γ ∈ R · γ with F ( γ ) ≤ L }| L g − r ≤ |{ φ ∈ R with F (cid:0) L φ ( γ ) (cid:1) ≤ }| L g − r = m R γ ,L ( { F ( · ) ≤ } )and by (5.10) together with the fact that { F ( · ) ≤ } is compact we have thatlim L →∞ m R γ ,L ( { F ( · ) ≤ } ) = 0 . (cid:3) iveka Erlandsson, Juan Souto, and Jing Tao 13 Finally, we prove Theorem 1.1:
Theorem 1.1.
The set of pseudo-Anosov mapping classes is generic with respect to eitherone of the functions ρ K ( φ ) = inf { K ( f ) where f ∈ Diff(Σ) represents φ } where K ( f ) is the quasi-conformal distortion, ρ Lip ( φ ) = inf { Lip( f ) where f ∈ Diff(Σ) represents φ } where Lip( f ) is the lipschitz constant, and ρ σ,η ( φ ) = ι ( φ ( σ ) , η ) where σ and η are filling multicurves and ι ( · , · ) is the geometric intersection number.Proof. We start by proving that the set of pseudo-Anosov mapping classes is ρ σ,η -genericfor filling multicurves σ and η . Well, the function C (Σ) → R ≥ , λ ι ( λ, η )is continuous and proper on the set C K (Σ) of currents supported by compact sets K ⊂ Σ.We thus get from Theorem 1.2 thatlim L →∞ |{ φ ∈ R with ι ( φ ( σ ) , η ) ≤ L }| L g − r = 0 (5.12)On the other hand we get from [26] or [21] (see also [9, 10]) thatlim inf L →∞ |{ φ ∈ Map(Σ) with ι ( φ ( σ ) , η ) ≤ L }| L g − r = const( σ, η ) > . (5.13)Since ρ σ,η ( φ ) = ι ( φ ( σ ) , η ) we get from (5.12) and (5.13) thatlim L →∞ |{ φ ∈ R with ρ σ,η ( φ ) ≤ L }||{ φ ∈ Map(Σ) with ρ σ,η ( φ ) ≤ L }| = 0 . This shows the set of pseudo-Anosov mapping classes is generic with respect to ρ σ,η .We consider now genericity with respect to ρ Lip . Fix once and for all a filling multicurve σ . Although it does not really matter, we could for example assume that σ is a markingin the sense of [19]. We need the following fact: Fact 1.
There is C = C (Σ , σ ) ≥ with C · ρ Lip ( φ ) ≤ ρ σ,σ ( φ ) ≤ C · ρ Lip ( φ ) for all φ ∈ Map(Σ) . Fact 1 is well known but, for the convenience of the reader, we will comment on its proofonce we are done with Theorem 1.1. From Fact 1 we get that |{ φ ∈ R with ρ Lip ( φ ) ≤ L }| ≤ |{ φ ∈ R with ρ σ,σ ( φ ) ≤ CL }||{ φ ∈ Map(Σ) with ρ Lip ( φ ) ≤ L }| ≥ (cid:12)(cid:12)(cid:12)(cid:12)n φ ∈ Map(Σ) with ρ σ,σ ( φ ) ≤ LC o(cid:12)(cid:12)(cid:12)(cid:12) . We thus get from (5.12) and (5.13) thatlim L →∞ |{ φ ∈ R with ρ Lip ( φ ) ≤ L }||{ φ ∈ Map(Σ) with ρ Lip ( φ ) ≤ L }| = 0 , as we had claimed.The genericity with respect to ρ K follows by the same argument when we replace Fact1 by the following also well-known fact: Fact 2.
There is C = C (Σ) ≥ with C · ρ Lip ( φ ) ≤ ρ K ( φ ) ≤ C · ρ Lip ( φ ) for all φ ∈ Map(Σ) . We have proved Theorem 1.1. (cid:3)
We comment now on the proofs of the two facts used in the proof above. By properties(7) and (8) of the space of currents, we have that for any other filling multicurve σ ′ thereis a constant C = C (Σ , σ, σ ′ ) with1 C ι ( σ, φ ( σ )) ≤ ℓ Σ ( φ ( σ ′ )) ≤ C ι ( σ, φ ( σ )) , (5.14)where ℓ Σ ( · ) is the hyperbolic length function. Choosing σ ′ to be a short marking in thesense of [12], we get from Theorem 4.1 in that paper that there is a constant C = C (Σ , σ ′ )such that 1 C ℓ Σ ( σ ′ ) ≤ ρ Lip ( φ ) ≤ C ℓ Σ ( σ ′ ) (5.15)Fact 1 follows, with C = C · C , from these two inequalities.A similar argument, replacing results from [12] by results from [23], yields Fact 2. Al-ternatively one can directly refer to Theorem B in [5].For the reader who feels cheated by a proof which only consists of a sequence of references,we sketch a more direct proof of Fact 1 and Fact 2. Suppose Σ is closed. By the Arzel´a-Ascoli theorem, there is a Lipschitz map f on Σ representing φ with L f = ρ Lip ( φ ). ByTeichm¨uller’s theorem, there is a unit-area quadratic differential q on Σ and a map g representing φ , such that ρ K ( φ ) = L g , where L g is the Lipschitz constant of g with respectto the singular Euclidean metric induced by q . Moreover, L g is the minimal Lipschitzconstant of all maps on q representing φ . By compactness of Σ, the q –metric and thehyperbolic metric on Σ are bilipschitz equivalent. By compactness of the space of unit-area quadratic differentials, this bilipschitz equivalence is uniform. Therefore, there is aconstant B depending only on Σ such that1 B L f ≤ L g ≤ BL f . This obtains Fact 2 with C = B .Let σ be a filling multicurve which we realize by a q –geodesic. Because σ is filling, itcannot be entirely q –vertical. Compactness of the space of unit-area quadratic differentialsimplies that in fact the horizontal length of σ is a definite proportion of its total length.Under the map g , the q –horizontal direction gets stretched by the factor L g , so the q –length of φ ( σ ) grows proportionally to L g . By comparing to the hyperbolic metric andusing compactness of Σ again, we get Equation (5.15) with σ = σ ′ . We still have (5.14)(with σ = σ ′ ). This shows Fact 1.For the general case, losing compactness of Σ means losing bilipschitz equivalence be-tween the q –metric and the hyperbolic metric. However, the argument we just sketched iveka Erlandsson, Juan Souto, and Jing Tao 15 can be modified to take care of this issue and we refer to the above listed references for thedetails. References [1] J. Aramayona and C.J. Leininger,
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Viveka Erlandsson
School of Mathematics, University of BristolUiT The Arctic University of Norway [email protected]
Juan Souto
IRMAR, Universit´e de Rennes 1 [email protected]
Jing Tao