A lower bound on the subriemannian distance for H"older distributions

A LO WER BOUND ON TH E SUBRIEMANNIAN DIST ANCE FOR H ¨ OLDER DISTRIBUTIONS SLOBODAN N. SIMI ´ C Abstra ct. Wherea s subriemannian geometry usually deals wi th smooth horizon tal distributions, partially hyp erb olic dynamical systems provide many examples of subriemannian geometries defined by non-smooth (namely , H¨ older conti nuous) distributions. These distributions are of great significance for the b eh a vior of the p aren t dynamical system. The study of H¨ older sub riemannian geometries could therefore offer new in- sigh ts into b oth dynamics and subriemannian geometry . In this pap er w e make a small step in that direction: w e pro ve a H¨ older-t yp e lo w er b ound on the sub riemannian distance for H ¨ older contin uous no where integ rable co dimension one d istributions. This b ound generalizes the w ell-know n square root bound vali d in the smo oth case. 1. Introduction The pur p ose of this note is to prov e a lo w er b ound in terms of the Rie- mannian d istance for subriemannian geometries defined by co dimension one distributions that are only H¨ older cont in uous, generalizing the we ll-kno wn square ro ot b ound v alid in th e smo oth c ase. A subriemannian or Ca rn ot-Carath ´ eo do ry geo metry is a pair ( M , H ), where M is a s mo oth manif old M and H is a nowher e inte gr able distribution (a notion we will define shortly) end ow ed with a R iemann ian metric g . Classi- cally , b oth H and g are assumed to b e smo oth, e.g., of class C ∞ ; H is usually called a ho rizontal di stribution (or a p olarizatio n, by Gromo v [Gro96]) and is tak en to b e brac k et- generating (see b elo w for a definition). Piecewise smo oth paths a.e. ta ngen t to H are called ho riz ontal pa ths . The subrieman- nian distance is defined b y d H ( p, q ) = in f {| γ | : γ is a h orizon tal path fr om p to q } , where th e length | γ | of a horizon tal path γ : I → M is defined in the usual w a y , | γ | = R I g ( ˙ γ ( t ) , ˙ γ ( t )) 1 / 2 dt . Recall th at a smo oth d istribution H is called brack et generating if an y lo cal smo oth f r ame { X 1 , . . . , X k } for H , together with all of its iterate d 2000 Mathematics Subje ct Classific ation. 51F99, 53B99. Key wor ds and phr ases. Distribution, H¨ older contin uit y , subriemannian d istance. 1 2 S. N. SIMI ´ C Lie brack ets spans the whole tangen t bun dle of M . (In PDEs, the b rac k et- generating co ndition is also ca lled the H¨ ormander cond ition.) By the w ell- kno wn theorem of Chow and Rashevskii [Mon02], if H is brack et generating, then d H ( p, q ) < ∞ , for all p, q ∈ M . Ho wev er, H d o es not ha ve to be brack et generating, or ev en smo oth, to define a reasonable subriemannian geometry . Imp ortant examples of suc h distributions co me from d ynamical systems. Let f : M → M b e a C r ( r ≥ 1) partially hyp erb olic diffeomo rphism of a smooth compact Riemannian manifold M . This means that f is lik e an Anoso v d iffeomorphism, but in ad - dition to h yp erb olic b eh avior (i.e ., exp onential con traction an d exp onent ial expansion), f also exhibits w eakly hyperb olic or n on -hyp erb olic b eha vior along certain tangen t directions. More precisely , f is partially hyp erb olic if the tangent bun dle of M splits con tinuously and inv arian tly into the s table, cen ter, and uns table bundles (i.e., distr ibution), T M = E s ⊕ E c ⊕ E u , suc h that the tangen t m ap T f of f exp onen tially con tracts E s , exp onen- tially expands E u and this h yp erb olic action on E s ⊕ E u dominates the action of T f on E c (see [Pes04, PS 04]). The stable and u n stable bun d le are alw a ys u niquely in tegrable, giving rise to the stable and u nstable foliati ons, W s , W u . A p artially hyp erb olic diffeomorphism is (rather unfortu nately) called a c- cessible if ev ery t w o p oint s of M can b e j oined by an su -path, i.e., a con tin- uous p iecewise smo oth path consisting of finitely man y arcs, eac h lying in a single leaf of W s or a single leaf of W u . If f is accessible, then the d istri- bution E s ⊕ E u is clearly non-in tegrable. Dolgo p y at and Wilkinson [D W03] pro v ed that in th e space of C r ( r ≥ 1) partially h yp erb olic diffeomorphisms there is a C 1 op en and dense set of acce ssible diffeomorphisms. Since in- v arian t distr ib utions of a p artially h yp erb olic diffeomorphism are in general only C θ , for some θ ∈ (0 , 1) (see [HPS77, P es04]), this result s ho ws that there is an abund an ce of subriemannian geometries defin ed b y non-smo oth, H¨ older con tin uous distributions. Since these distrib utions pla y a crucial role in the d ynamics of partially h yp erb olic systems and the acce ssibilit y p rop- ert y is frequently k ey to ergo dicit y (cf., e.g., [PS04]), a b etter un derstanding of subriemannian ge ometries defined b y H¨ older distrib utions co uld b e v ery useful in dyn amical sys tems. T his p ap er p ro vides a small initial step in that direction. When H is not smo oth, the d efinition of no where in teg rabilit y is more subtle than in the smooth case. On e should k eep in mind th e follo w ing example (we thank an anonymous referee for bringing it up): sup p ose H is smo oth and inte grable in a neigh b orho o d of a p oint p and non-inte grable elsewhere. Since ev ery tw o p oin ts can still b e conn ected b y a horizonta l path, d H is fi nite eve rywhere. Let L b e the leaf thr ou gh p of the lo cal foliation of U tangen t to H . Then unless q lies in L , d H ( p, q ) is b ounded a w ay fr om zero by a u n iform constan t, making the sub riemannian distance 3 P S f r a g r e p l a c e m e n t s γ U L p q Figure 1. Ev er y horizonta l path γ connecting p and q h as to lea v e U . discon tin uous with resp ect to the Riemannian distance for an y R iemann ian metric on M . See Figure 1. The follo wing d efinition prohibits th is t yp e of b eha vior. Definition. A distribution H on M is c al le d no where integrable if for every p ∈ M and eve ry ε > 0 , ther e exists a neighb orho o d U of p , such that any p oint in U c an b e c onne cte d to p by a horizontal p ath of length < ε . F rom n o w on, w e will assum e th at the Riema nnian metric on H is the r estriction of a Riemannia n metric fr om the ambient manifold M . If H is a smo oth brack et-generating distrib u tion, the relation b et w een the subrieman- nian d istance d H and th e Riemann ian distance, whic h w e denote b y dist, is w ell understo o d and is c haracterize d b y the Ball-B o x theorem [Mon02, Gro96]. When H is of co dimension one and brac k et-g enerating, this theo- rem states that in the v ertical direction, i.e., along an y short smo oth path γ transverse to H , d H is equiv alen t to √ dist. Th at is, there exist constan ts a, b > 0 such that for all p, q on γ , a p dist( p, q ) ≤ d H ( p, q ) ≤ b p dist( p, q ) . (1) In the horizon tal direction, i.e., along an y horizonta l p ath, d H and dist are clearly equiv alen t. This means that subr iemann ian geometry is n on - isotropic: it b eha v es d ifferen tly in differen t directions. S ubriemannian sp heres are far from b eing “round”. In the Heisen b erg group, for instance, su brie- mannian spheres lo ok lik e an apple [Mon02]. In [Gro96], M. Gromo v ga v e a short and eleg an t pr o of (without details) of the lo w er b ound for d H , i.e., the left hand side of (1). His pro of uses the assumption that H is C 1 only in the follo wing w a y: if α is a 1-form such that Ker( α ) = H , th en     Z ∂ D α     ≤ K | D | , (2) for ev ery C 1 immersed 2-disk with piecewise C 1 b ound ary , w here K is a constan t indep en d en t of D . Here, | D | denotes the area of D . If H (hence α ) is C 1 , this follo ws d irectly from the Stok es theo rem. 4 S. N. SIMI ´ C Gromo v also remarks that without the C 1 assumption, the squ are r o ot estimate probably fails. W e will sho w that this is indeed the case, in the sen se that if H is only C θ , for some 0 < θ < 1, and nowhere in tegrable, th en in the vertica l direction, d H ( p, q ) ≥ C dist( p, q ) 1 / (1+ θ ) . T o generali ze Gromo v’s approac h to C θ horizon tal distribu tions, one needs a generalizat ion of the estimate (2) to form s α that are only H¨ older. One suc h estimate w as recen tly pro v ed in [Sim09] and states that the integral of a C θ k -form α o v er the b ound ary of a sufficiently small ( k + 1)-disk D is b ound ed by a ce rtain m ultiplicativ e con vex com b ination of the ( k + 1)-v olume | D | of D and the k - dimensional area | ∂ D | of its b oundary; s ee Theorem 2.1 in the next s ection. Our main result is the follo w ing: Theorem. Supp ose tha t H is a nowher e inte gr able c o dimension one distri- bution of class C θ , for some 0 < θ < 1 , on a smo oth c omp act Riemannia n manifold M . Assume that the Riemannian metric on H is the r estriction of the ambient metric fr om M . Then ther e exists a c onsta nt  > 0 such that for any C 1 p ath γ tr ansverse to H and for every two p oints p, q on γ with R iemannian dista nc e less than  , we have d H ( p, q ) ≥ C dist( p, q ) 1 1+ θ , (3) wher e C > 0 is a c onstant th at dep ends only o n H and γ . Remark. (a) Observe that if dist( p, q ) = ε is sm all, then ε 1 / 2 ≫ ε 1 / (1+ θ ) , whic h means that the low er b ound on d H ( p, q ) is tighter for C 1 dis- tributions than f or C θ ones. (b) It is clear that in an y horizon tal direction, d H is equiv alen t to dist. 2. A uxiliar y resul t s The main to ol in the pro of will be the f ollo win g inequalit y . 2.1. The orem (Theorem A, [Sim 09]) . L et M b e a c omp act manifold and let α b e a C θ k -form on M , for some 0 < θ < 1 and 1 ≤ k ≤ n − 1 . Ther e exist c onstants σ, K > 0 , dep ending only on M , θ , and k , such that for every C 1 -immerse d ( k + 1) -disk D in M with pie c ewise C 1 b oundary satisfying max { diam( ∂ D ) , | ∂ D |} < σ , we have     Z ∂ D α     ≤ K k α k C θ | ∂ D | 1 − θ | D | θ . As b efore, | D | denotes the ( k + 1)-v olume of D and | ∂ D | , the k -v olume (or area) of its boun dary . Remark. If k = 1, then diam( ∂ D ) ≤ | ∂ D | , so the assumption diam( ∂ D ) < σ is sup erfluous. The H¨ older norm of α on M is defined in a natural w a y as follo w s. Let A = { ( U, ϕ ) } b e a finite C ∞ atlas of M . W e sa y that α is C θ on M if α is 5 C θ in eac h chart ( U, ϕ ); i.e., if ( ϕ − 1 ) ∗ α is C θ , for eac h ( U, ϕ ) ∈ A . W e set k α k C θ = max ( U,ϕ ) ∈ A   ( ϕ − 1 ) ∗ α   C θ ( ϕ ( U )) . F or a C θ form α = P a I dx I defined on an op en subset U ⊂ R n , we set k α k C θ ( U ) = max I k a I k C θ , and for a b ounded function f : U → R , k f k C θ ( U ) = k f k ∞ + su p x 6 = y | f ( x ) − f ( y ) | | x − y | θ . W e w ill also need the follo w ing version of the s olution to the isop erimetric problem “in the small”. 2.2. Lemma ([Gro83], Sublemma 3.4.B’ ) . F or every c omp act m anifold M , ther e exists a smal l p ositive c onstant δ M such that every k -dimensional cycle Z in M of volume less than δ M b ounds a chain Y in M , which is smal l in th e fol lowing sense: (i) | Y | ≤ c M | Z | ( k +1) /k , for so me c onstant c M dep ending only on M ; (ii) The chain Y i s c ontaine d in the  -neighb orh o o d of Z , wher e  ≤ c M | Z | 1 /k . The follo wing corollary is immediate. 2.3. Corollary . If Γ is a close d pie c ewise C 1 p ath in a c omp act manifold M with | Γ | < δ M , then th er e exists a 2 -disk D ⊂ M such that ∂ D = Γ , | D | ≤ c M | Γ | 2 , (4) and D is c ontaine d in the  - neighb orho o d of Γ , wher e  ≤ c M | Γ | . 3. Proof of the theo r em W e follo w Gromo v’s pro of in [Gro96], p. 116. Since the state men t is lo cal and concerns arbitrary d irections transv erse to H , we can assume without loss of generalit y that γ is a u nit sp eed Riemannian geodesic. Also without loss, we can assume that H is transversely orien table. If not, pass to a doub le cov er of M . Let X b e a unit vec tor field ev erywhere orthogonal to H and defi n e a 1-form α on M b y Ker( α ) = H , α ( X ) = 1 . Define another 1-form α γ on M b y requiring that Ker( α γ ) = H , α γ ( ˙ γ ) = 1 . Since H is C θ , so are α and α γ . ( It is reasonable to think of k α k C θ as the H¨ o lder norm of H .) Since α a nd α γ ha v e the same k ernel, there exists a f u nction λ su c h that α = λα γ . W riting ˙ γ = cX + w , for some w ∈ H and taking the Riemannian inner pr o duct with X , we obtain λ = c = cos ∢ ( ˙ γ , X ) = sin φ , where φ is the angle b et w een ˙ γ and H . Ev aluati ng 6 S. N. SIMI ´ C P S f r a g r e p l a c e m e n t s γ 0 γ 1 D p q Figure 2. V ertical path γ 0 and horizont al p ath γ 1 . b oth sides o f α = λα γ at ˙ γ , we see that λ = sin φ . Thus α = sin φ α γ . Set φ 0 = min φ . By assumption φ 0 > 0. Let τ = min( δ M , σ ), wher e δ M , σ are the constan ts fr om Theorem 2.1 and Lemma 2.2. Observe that n o where int egrabilit y of H imp lies that f or ev ery p ∈ M , th e function q 7→ d H ( p, q ) is con tin uous with resp ect to the Riemannian distance dist. By compactness of M , it follo ws that q 7→ d H ( p, q ) is in fact uniformly con tin u ous relativ e to dist. Th erefore, there exists η > 0 such that for all p, q ∈ M , d H ( p, q ) < τ / 2, w henev er dist( p, q ) < η . Set  = min η , τ 2 , 1 2 1+ θ θ c M K 1 /θ (sin φ 0 ) − 1 /θ k α k 1 /θ C θ ! . Observe that if dist( x, y ) <  , then dist( x, y ) + d H ( x, y ) < τ . T ak e any t w o p oin ts p, q on γ satisfying dist( p, q ) <  . Denote the s egment of γ starting at p and ending at q b y γ 0 . Let ε > 0 b e arbitrary but sufficien tly small so that dist( p, q ) + d H ( p, q ) + ε < δ M . Finally , let γ 1 b e a horizon tal path f rom p to q with | γ 1 | < d H ( p, q ) + ε . (Note th at a subriemann ian geo desic fr om p to q ma y not exist, since H is assumed to b e only H¨ older.) Define Γ = γ 0 − γ 1 ; Γ is a clo sed p iecewise C 1 path. Since | Γ | = | γ 0 | + | γ 1 | < dist( p, q ) + d H ( p, q ) + ε < τ < δ M , b y Corollary 2.3 there exists a 2-disk D suc h that ∂ D = Γ and | D | ≤ c M | Γ | 2 (see Fig. 2). On the other hand, since τ < σ , it follo w s that | ∂ D | = | Γ | < σ . Hence w e can apply Theorem 2.1 to α on D . Observe that along γ 0 , w e ha v e α γ = (sin φ ) − 1 α ≤ (sin φ 0 ) − 1 α . Along γ 1 , b oth α and α γ are zero, so 7 the inequalit y α γ ≤ (sin φ 0 ) − 1 α still h olds. Th erefore: | γ 0 | = Z ∂ D α γ ≤ (sin φ 0 ) − 1 Z ∂ D α ≤ K (sin φ 0 ) − 1 k α k C θ | ∂ D | 1 − θ | D | θ ≤ K (sin φ 0 ) − 1 k α k C θ | ∂ D | 1 − θ ( c M | ∂ D | 2 ) θ = c θ M K (sin φ 0 ) − 1 k α k C θ | ∂ D | 1+ θ = c θ M K (sin φ 0 ) − 1 k α k C θ ( | γ 0 | + | γ 1 | ) 1+ θ < c θ M K (sin φ 0 ) − 1 k α k C θ {| γ 0 | + d H ( p, q ) + ε } 1+ θ . T aking the (1 + θ )-th ro ot of eac h side and regrouping, we obtain | γ 0 | 1 1+ θ − c θ 1+ θ M K 1 1+ θ (sin φ 0 ) − 1 1+ θ k α k 1 1+ θ C θ | γ 0 | < c θ 1+ θ M K 1 1+ θ × × (sin φ 0 ) − 1 1+ θ k α k 1 1+ θ C θ { d H ( p, q ) + ε } . F actoring | γ 0 | 1 1+ θ out, the left-hand sid e b ecomes | γ 0 | 1 1+ θ  1 − c θ 1+ θ M K 1 1+ θ (sin φ 0 ) − 1 1+ θ k α k 1 1+ θ C θ | γ 0 | θ 1+ θ  . It is not hard to see that the assumption | γ 0 | = dist( p, q ) <  implies that the quan tit y in the cu r ly braces is ≥ 1 / 2. Therefore, 1 2 | γ 0 | 1 1+ θ < c θ 1+ θ M K 1 1+ θ (sin φ 0 ) − 1 1+ θ k α k 1 1+ θ C θ { d H ( p, q ) + ε } . Since ε > 0 can b e arbitrarily small and | γ 0 | = dist( p, q ), w e o btain dist( p, q ) 1 1+ θ ≤ 2 c θ 1+ θ M K 1 1+ θ (sin φ 0 ) − 1 1+ θ k α k 1 1+ θ C θ d H ( p, q ) . This completes the p ro of with C =  2 c θ 1+ θ M K 1 1+ θ (sin φ 0 ) − 1 1+ θ k α k 1 1+ θ C θ  − 1 .  (5) Remark. (a) It follo w s f rom [Sim09] that K s tays b oun ded as θ → 1 − . Therefore, by (5) the constant C = C ( θ ) do es n ot blo w up as θ → 1 − . This implies that if H is C 1 (hence C θ , for all 0 < θ < 1), w e reco ver the old quadratic estimate by letting θ → 1 − . (b) Note also that as the angle b etw een γ and H goes to zero (i.e., as γ tends to a horizon tal p ath), C → 0. This can b e inte rpreted as follo ws : alo ng horizon tal paths, d H and dist are equiv alen t, so (3) cannot hold for a n y p ositiv e C . The follo win g question app ears to be muc h harder than the one in v esti- gated in this pap er: 8 S. N. SIMI ´ C Question. Is ther e an analo gous upp er b ound for d H with r esp e ct to dist when H is C θ ? This and essenti ally all other questions of H¨ older su b riemannian geometry remain wide op en . Referen ces [DW0 3] D. D olgop y at and A. Wilkinson, Stable ac c essibility is C 1 dense , Ast´ erisque x vii (2003), n o. 287, 33–60. [Gro83] Mikhail Gromo v, Fil l ing Riemannian manifolds , J. Diff. Geometry 18 ( 1983), 1–147. [Gro96] , Carnot- Car ath´ eo dory sp ac es se en f r om within , Sub-Riemannian Geom- etry (A. Bellai che and J.-J. Risler, eds.), Progress in Mathematics, vol . 144, Birkh¨ auser, 1996, pp . 79–323. [HPS77] Morris W. Hirsc h, Charles C. Pugh, and Michael S hub, Invariant manifolds , Lecture Notes in Mathematics, v ol. 583, Springer-V erlag, Berlin-New Y ork, 1977. [Mon02] Richard Mon tgomery , A tou r of subriemannian ge ometries, their ge o desics and applic ations , Mathematical Surveys and Monographs, vol. 91, AMS, 2002. [P es04] Y ako v B. Pe sin, L e ctur es on p artial hyp erb oli city and stable er go di city , Zurich Lec- tures in Adv anced Mathematics, Europ ean Mathematical So ciety (EMS), Z ¨ urich, 2004. [PS04] Charles C. Pugh and Michael Shub, Stable er go dicity , Bull. A mer. Math. Soc. 41 (2004), 1–41. [Sim09] Slob o dan N . S imi ´ c, H¨ older forms and i nte gr abi l ity of i nvariant distributions , Dis- crete Contin. Dy n. Syst. 25 (2009), no. 2, 669–685. Dep ar tment of Ma thema tics, San Jos ´ e St a te Universi ty, San Jos ´ e, C A 95192- 0103 E-mail addr ess : simic@mat h.sjsu.edu