Dubins' problem on surfaces. II. Nonpositive curvature
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Dubins’ problem on surfaces. II. Nonpositive curvature Mario Sigalotti — Yacine Chitour. N° 5378 November 2004. ISSN 0249-6399. ISRN INRIA/RR--5378--FR+ENG. Thème NUM. apport de recherche.
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(13) . . Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France) Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65. .
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(39) ¹Ä ¢ o&jqbcwNc\µ]+mRpCÑ=f`cP,^_NPRI`PR?pC^_f}R>`"R?]]jC K fh]pC`!jqP]_^pqn?hRIjqe^_NPR½njq`=^_e+jqhRr!^_bce_`cf`c pqn?^_fhjq`¤Á¹QSj=rcR?hRr=\ g Ä ^+jj~}yR?e+njqQSR/^_NPRa]_mce+Rpqr:f`cUjC-^+NPRyRj=rcR]_fhn] ¢
(40) £ NPR?` |u| ≤ ε ≤ 1 ^_NPRU R #Rn?^jC u g fh]`Pjq^]+^_e+jq`cSR?`PjqbcwN»pC`Pr{njqQmchR?^+R njq`=^_e+jqEpCcff^\ÍpCfh]a^+jNPjqhr ¢ `Ôpmce+R?}3fhjqbP]Um"pCmR?e½Á u¹Ä> R'pCe+Rpqr:\¥mce+jqmjw]+Rr ]+jqQSRSyRjqQSR?^_e_fhn½pC`Pr¥^+jqmjqhjqwfhnpC Ð njq`Pr:f^_fhjq`P]Sjq` aNcfhnNÆR?`P]_bce+R½^_N"pC^^_NPRVsÂaÂmce+jqmR?e_^\ÔNPjqhrc]^_e_bPR `!m"pCe_^_fhn?bcEpCe f^apq]mce+j~}Rr¿^_MN"pC^pC`=\¿njqQm"pqn?^ M ]pC^_fh])¦"R]^_NPRSVsÂa mce+jqmR?e_^\ ¢SÐ `¥¢µ^_NPÐ Rbc`=jqbc`PrPR?r npq]+Rwc^_NPRU]pCQSRnjq`Pn?bP]+fhjq`{/pq]jqc^pCf`PRr{aNPR?` M fh] ¶C«>ºC²&³¬Í±y¬®¶q´6´¸
(41) º a¶w¬¯:f ¢ R ¢ K ^+R?`Prc] ^+j¡ÏR?e+j8pC^Uf`:¦P`cf^\ ¢ M/NPRSnjq`=^_e+jq.]+^_epC^+R?w\¿yj=R?]pq]jqhj~] ¢IÐ ^Ufh]s¦"e+]_^ ]_NPj~a`µ^_N"pC^ ^_NPR njq`=^_e+jqEpCcff^Ì\!jC (D ) fh]Rg=bcf}qpChR?`=^^+j»^_NPRI¯pqn?^^_N"pC^R?}R?e_\ q = (p, v) ∈ N npC` R ]_^+RR?e+Rr ^+j q = (p, −v) =\ pC`!pqr:Qfh]]_fchR'^_epk_Rn?^+jqe_\µjC (D ) ¢ M/NPR'EpC^_^+R?emce+jqmR?e_^\y npChRr .ª_¶ ®+±q¨-¬©_±q´6´¶·>¹´ ¬º/jC (D ) wfh] mce+j~}yRr=\^_epqnÑ3f`c pK^+RpCe+r:e+jqmShj=jqmf`'psnj~}yR?e_f`c rcjqQ'pCf`¥j}yR?ep'mcfhRnRjC M pC^Kf`:¦"`cf^\ ¢ l]_bcf^pCchRnj~}yR?e_f`cQ'pC`cf¯jqhr¿npC`¿Rwhjq"pC\ rcR]+n?e_fRr{=\¡p]_f`cwhR pCmcmce+jqmce_fEpC^+RUyRj=rcR]+fhnUnN"pCe_^ ¢ jq`PRUÑRR?mP]f`¤Qf`Pr¤^_NPRU`Pjq`:njq`=^_e+jq6pCcff^Ì\¤jC ^_NPR¾Kbccf`P]ÀPmce+jqchR?Q jq` Pf^&fh] e+Rpq]+Ð jq`"pCchR^+j»njq`P]+fhrcR?e^_NPR½]+f^_b"pC^_fhjq` aNPR?e+R K ≥ 0 ¢¥Ð `PrcRRr# f` ^_N"pC^npq]+Rw H`Pj8hj=npC ]_mce+Rpqr:f`cµ R #Rn?^r:bPR½^+j»^_NPR¡r:e_f¹^^+R?e_Q N"pq]^+j»RnjqQmR?`P]pC^+Rr ¢ W»jqe+Rj~}R?e f M fh] pC`¡jqmR?`¡Q'pC`cf¯jqhraf^_N¡`Pjq`c`PR?pC^_f}yRUn?bce_}wpC^_bce+RwPpe+R]_bc^a=\ÂjqNc`: Îjw]+]+R?` Á ¹Ä.fQmPfhR] ^_N"pC^ Z Á¯yÄ KdA < ∞ , aNPR?e+R dA fh],^_NPRa]_bce)ÍpqnR/R?hR?QSR?`=^f` M ¢ MaNPRf`=^+R?wepC"rcRnp\jC K ^+jKÏR?e+jUpC^ f`:¦P`cf^\npC` Rf`=^+R?e_mce+R?^+Rr!pq] p¡]+jqe_^UjCpq]_\3Qmc^+jq^_fhn pC^_`PR]+] njq`Pr:f^_fhjq` pC`Pr¥f^U]+bcwyR]_^+]^_N"pC^ (D ) ]_NPjqbchrRnjqQmchR?^+R?\njq`=^_e+jqEpCchRjqe,R>}R?e_\ ` u qRÎ/R>eR/pCchR.^+jKnjq`:¦Pe_Q ^+N"pC^ f`=^_bcf^_fhjq`»bc`PrcR?e&^_NPRpqrcr:f^_fhjq`"pC pq]+]_bcQmc^_fhjq`8ε^_N->pC^ 0K¢ Ð fh] jqbc`PrcRr8j~}R?e M "f ¢ R ¢ sup K ε. ε. ε. 2. ε. ε. −. ε. ε. 2. M. ε. M. . ù.
(42) U§c·?6¨c«³-©_±·?´ª?² ±q¨8«§3©?¶y®ª«. u. fh]¦P`cf^+RwGpC`Pr¿pCh]+jS^+jIR>Ç:^+R?`Pr¤^_N"pC^&e+R]_bc^&^+j'^_NPRnpq]+R aNPR?e+R K ≥ 0 jqbc^+]_fhrcRp'njqQm"pqn?^ ]+R?^ ¢ M/NPR]+^_epC^+R?w\½jC
(43) ^_NPRsmce+j=jC,]_^_fõnjq`P]_fh]_^+]/f`¡^_epqnÑ3f`c'p^+RpCe+r:e+jqm{hjjqm{f`{pnj~}R>e+f`P rcjqQ'pCf` D cbc^K^_NPRnjq`P]_^_e_bPn?^_fhjq`¥jC D RnjqQSR]KQbPnN¿QSjqe+RrcR?fhnpC^+R^_N"pC`»^_N"pC^sjCÎ^_NPR pq]_\3Qmc^+jq^_fhn pC^npq]+R ¢ o&RnR?`=^_\yyi ¢ i,pC`P]+bmjqf`3^+Rrjqbc^,NPj~!^_NPR/R>Ç:fh]_^+R?`PnRjCGp&n?hjw]+RryRj=rcR]_fhn/jq`pK]_bce)¯pqnR a f^_N¿njq`P]_^pC`=^K`PR?pC^_f}Rn?bce_}wpC^_bce+RfQmcfhR]K^_N"pC^-¯jqeKR?}yR?e_\{mjqf`=^ p ∈ M pC`Pr8R?}R?e_\ M q^_NPR?e+RaR>Ç:fh]_^+] p&`Pjq`:njq`=^_epqn?^+fchRpqr:Qfh]]_fchR/^_epk)Rn?^+jqe_\jC aNcfhnNS]_^+RR?e+] ]+jqQSR ε>0 mjqf`=^ q ∈ U M ^+j{^_NPR'mjqf`=^ q ¢ Z=bPnN p¯RpC^_bce+R'fh] m"pCe_^_fhn?bcEpC(De_\ )f`=^+R?e+R]_^_f`cP,^pCÑ3f`c f`=^+jpqnnjqbc`=^/^_N"pC^/VsÂa¼rcj=R]`Pjq^.NPjqhr'¯jqe.^_NPRKi
(44) jqf`PnpCe+dKN"pC¹ÍmcEpC`PR H ¢ l`½fQQSRr:fEpC^+R g=bPR]_^_fhjq`Éfh]IaNPR?^_NPR?e¡VsÂa npC`ÉR8e+Rnj~}yR?e+RrÓjqe¡p g=bPjq^_fhR?`=^½Q'pC`cf¯jqhr mce+j~}3fhrcRr^_N"pC^ÎRpqrcrSR?`PjqbcwN'^+jqmjqhjqw\^+j H Á¹f ¢ R ¢ mce+j~}=fhrcRrS^_N"pC^^+NPRawe+jqMbcm =Γ fhH]hpC/Γe+yR R?`PjqbcwN-Ä>
(45) jqeUf_ jq`µ^_NPRSnjq`=^_epCe_\ ^_NPRS¾Kbccf`P]À#mce+jqchR?Q jq`ÔpC`=\¿gbPjq^_fhR?`=^jC H fh]U`Pjq^ N"pq]s^_NPRVsÂa VÂa ^s^_bce_`P]jqbc^^_N"pC^^_NPRpC`P]_/R?enpC`¥RRpq]+f\»]_^pC^+Rr#Å mce+jqmR?¢Ge_^Ð \¤f.pC`Pr»jq`c\{fÎf^&fh]KjCÎ^_NPR ¦Pe+]_^&Ñ3f`Pr#-f ¢ R ¢ f^+]&fQf^sM]+R?^=Á¯]+HRR¾s/ΓR>¦P`cf^_fhjq`µv ¢
(46) Äfh] ^_NPRUR?`=^_fe+Rjqbc`PrPpCe_\{pC^f`:¦P`cf^\¡jC H ¢ `S^_NPRmce+R]R?`=^Îjqe_ÑRf`=}R]_^_fpC^+R^_NPRaQSjqeRyR?`PR?epCnpq]+RjC#ofhR?Q'pC`c`cfEpC`½]_bce)ÍpqnR?] Ð af^_N¿`Pjq`cmjw]_f^_f}Rn?bce_}wpC^_bce+R ¢UÐ ^Kfh]KR?ÍÑ=`Pj~`¿^_N"pC^UpI]_bce)ÍpqnR M jC.]_bPnN¿Ñ=f`Pr»npC`¿R fhrcR?`=^_f¦"Rraf^_N^_NPR.g=bPjq^_fhR?`=^,]_m"pqnR X/Γ CaNPR?e+R X fh],p KpqrPpCQ'pCe+r]+bce)¯pqnRKÁ¹f ¢ R ¢ yp]+fQmc\ njq`c`PRn?^+Rr#njqQmchR?^+R/ofhR>Q'pC`P`cfEpC`Q'pC`cfjqhrjC"`Pjq`cmjw]+f^_f}yR/n?bce_}wpC^_bce+R~Ä
(47) pC`Pr Γ fh] pwe+jqbcm jC jqe_fhR?`=^pC^_fhjq`:ÍmceR]+R?e_}3f`cIfh]+jqQSR?^_e_fhR]aNcfhnN»pqn?^+]a¹e+RR?\¡pC`Pr¤r:fh]+njq`=^_f`=bPjqbP]_\{jq` X Á6jqe pC,^_NPRrPR¦"`cf^_fhjq`P]#]RRZ3Rn?^_fhjq`µvyÄ Rmce+j~}=fhrcRnjq`Pr:f^_fhjq`P]Ujq` G]+b 'n?fhR?`=^K¯jqesVsÂa ^+j'NPjqhr ¢ M/NPRU¦Pe+]+^&jq`PR yR?`PR?epCfÏR]s¢^_£ NPR N=\3mR?e_jqfhnnpq]+R ¢Ð ^&]p\:]M^_N"pC^ (i) M = X/Γ fh] jC.^_NPR¦"e+]_^sÑ=f`Pr#Gf ¢ R ¢ #f^+]sfQf^U]+R?^ L(Γ) fh]Rgb"pC,^+j½^_NPRfhrcRpC,jqbP`PrPpCe_\ X(∞) jC X ¢ M/NPR]+Rnjq`Pr8n?jq`"r:f^_fhjq` (ii) fh]e+R?Qf`cfh]nR?`=^KjC ^_NPR `Pjq`c`PR?pC^_f}Rn?bce_}wpC^_bce+Rnpq]+Rw"`"pCQSR?\y f^]_^pC^+R]^_N"pC^¯jqeR?}R?e_\ r > 0 pC`Pr!R?}R?e_\ ]+Rn?^+jqe S f` X sup R KdA = 0 aNPR?e+R B (p, r) rcR?`Pjq^+R]Î^_NPRa"pC"jC#nR?`=^+R?e p pC`PrSepqr:fbP] r ¢ M/NPR?`õRamce+R]+R?`=^`PRnR]+]pCe_\ njq`Pr:f^_fhjq`P]Îjq` M jqe,mce+jqmR?e_^\VÂa ^+jKNPjqhr ¢ l¼g=bcf^+R/]_bce_mPe_fh]_f`cse+R]+bc^,fh],^_NPRjqhj~af`c ¢ V&`PrcR?eõ^_NPRÎpq]+]_bcQmc^_fhjq` ^_N"pC^ R?f^_NPR?e K fh]õjqbc`PrcRr jqe sup K < 0 ~f M }R?e_f¦"R]#mPe+jqmR?e_^\ VÂaÂK ^_NPR?` R?f^_NPR?enjq`Pr:f^_fhjq` (i) jqenjq`Pr:f^_fhjq` (ii) QbP]_^NPjqhrÔ^+e_bPR ¢¿Ð `Ô^_N"pC^/p\y R R>ÇPpqn?^_\»nN"pCepqn?^+R?e_fÏR'^_NPR]_bce)ÍpqnR] õaf^_N¥`Pjq`cmjw]_f^_f}RSn?bce_}wpC^_bce+R R?f^_NPR?eUjqbc`PrcRr jqe]_bPnN¡^_N"pC^ sup K < 0 :}yR?e_f\=f`cMmce+jqmR?e_^\VÂa ¢cÐ `¤pqrcr:f^_fhjq`õP]+jqKQSRnjq`=^_e+jqEpCcffh^\ pC`PrI`Pjq`:njq`=^_e+jqEpqcff^\Ie+R]_bP^].f`½^_NPRKnpq]+RaNPR?e+R K fh]`Pjq`cmjw]_f^_f}RKjqbc^+]_fhrcRspnjqQm"pqn?^ ]_bcP]R?^ jC pCe+Rwf}R?` R'pCh]+j{mce+j~}3fhrcR']_b 'n?fhR?`=^njq`Prcf^+fjq`"]R?`P]_bce_f`c¤^_N"pC^ N"pq] ^_NPRVÂa mcMe+jqmR?e_^\y=af^_N'¢I`P£ j ]_fw`½pq]+]+bcQmc^_fhjq`Ijq` K `"pCQSR?\ (a) M N"pq]¦P`cf^+R&pCe+MRp: (b) ^_NPR yRj=rcR]_fhn j~ jq` M f]^+jqmjqhjqwfhnpC\¤^_epC`P]_f^_f}R ¢/£ RUnjq`Pn?bPrcRaf^_N»]+jqQSRUe+R?Q'pCe_Ñ:] jq`Ô^_NPR']+^_e_bPn?^_bce+RIjCa^_fQSRIjqmc^_fQ'pC^_epk_Rn?^+jqe_fhR] jqe K ≤ −ε RI]_NPj~ ^_N"pC^/^_NPR?\S¯jqhj~ p¾Kbccf`P]À3m"pC^_^+R?e_`õc`"pCQSR?\'^_N"pC^/^_NP(DR?\½pC) e¢¡RKÐ njqsup `PnpC^+R?`"pC^_fhjw`¤jC
(48) p"pC`cPcp ε. −. p. 2. 2. 2. 2. 2. 2. p∈S BX (p,r). X. M. M. ε. . ð !#". M.
(49) ¬Í±q§3©-y¶w´±w¬¯¬. . ]_f`cwbcEpCe pC`Prµp½"pC`c¤pCe+n{Á¹aN"R>eR]+jqQSRSpCe+nnpC`¥mjw]]_fc\8N"p}yRÏR?e+jhR?`cw^_N-Ä> af^_N¥^_NPR Á¹mjw]+]+fchR~Ä/]_f`cwbcEpCe&pCe+nR?f`cIpyR?j=rcR]+fhnUjC,^_NPR]+bce)¯pqnR ¢ M/N"R/m"pCmR?efh]jqe_pC`cfÏRr'pq],¯jqhj~] `IZ3Rn?^_fhjq`I:wR/jqe_QbcEpC^+Ra^_NPRnjq`=^_e+jqPmce+jqchR?Q pC^N"pC`PrµpC`Pr¥rcR]+n?e_fRf^+]"pq]_fhnRpC^_bc¢
(50) e+RÐ ] ¢Ð ` Z3Rn?^_fhjq`Ôv:GyR?`PR?epC Ípqn?^+]jq`µofhR?Q'pC`c`cfEpC` ]_bce)ÍpqnR]jCK`Pjq`cmjw]_f^_f}R¡n?bce_}wpC^_bce+R¤pCe+Re+RnpChRr ¢ M/NPRQ'pCf`Æe+R]_bc^+]SjC&^_NPRm"pCmR?eIpCe+R ]_^pC^+Rr!f`ÆZ3Rn?^_fhjq`Ë¥pC`Pr!^_NPR?femce+j=jC¯]pCe+RImcej~}=fhrcRr!f`¼Z3Rn?^_fhjq`Æu ¢ f`"pC\.Z3Rn?^_fhjq` njq`=^pCf`P]&]+jqQSRse+R?Q'pCe_Ñ:]jq`{^_fQSRUjqmc^_fQ'pCõ^_epk_Rn?^+jqe_fhR] ¢ Û3Ú × Ü × ^&fh]sp'mchRpq]_bce+R ¯jqe&bP]&^+jI^_N"pC`cѤi ¢ i,pC`P]+b8jqe&NPf]s]+R?Qf`"pC,]_bcq yR]_^_fhjq`8pC`Pr¡Ncfh]`=bcQSR?e+jqÐ bP]&pqr:}3fhnR ¢ ( G (G+ ¤ R?^ RSp½^jCr:fQSR?`P]_fhjq`"pq.ofhR?Q'pC`c`cfEpC` Q'pC`cfjqhr# jqe pq]RafRg=bcf}wpChR?`=^_\¿npC f^K p M
(51) ¯ª?²¶w¨P¨P¶w¨É«§3© ¶y®ª ¢ l&]]_bcQSR8^_N"pC^ M fh]jqe_fhR?`=^+Rr pC`PrÉ^_N"pC^f^+]½QSR?^_e_fhn m fh] njqQmchR?^+R ¢ ¾sR?`Pjq^+R'=\ N ^_NPR'bc`cf^^pC`cyR?`=^cbc`Pr:hR U M = {q ∈ T M | m(q, q) = 1} pC`Pr =\ π : N → M ^_NPRInpC`Pjq`cfhnpCcbc`Pr:hRImce+jCk)Rn?^_fhjq` jC N jq`=^+j M ¢ R?^ K R'^_NPR pCbP]]_fEpC`¿n?bce_}qpC^_bPe+Rjq` M ¢s£ Raf,bP]+R^_NPR]_\3Qjq K pCh]+jIjqeK^_NPR^_e_f}=fEpC R>Ç:^+R?`P]_fhjq` Ánjq`P]_^pC`=^jq` ¦PR?e+]ÄUjC K jq` N ¢ MaNPR'r:fh]+^pC`PnRIjq` M f`"r:bPnRr =\ m fh]rcR?`Pjq^+RrÔ=\ Ájqe d(·, ·) aNPR?`¿`Pj¡njq`:¹bP]+fjq`¥fh]smjw]+]_fchR~Ä> pC`Pr#GjqeR?}R?e_\ p ∈ M pC`Pr r > 0 d (·, ·) ]_^pC`Prc]ajqe^_NPRs"pC jC nR?`=^+R?e pC`Pr¡epqr:fbP] B (p, r) R?^ f RU^_NPR ª+±~°ª«¯®«Í³-©_¶qº jq` T Mp "aNPjw]+R e+R]_^_e_rfh¢n?^_fhjq`8^j N Á]+^_f rcR?`Pjq^+Rr8=\ f Ä fh] p½R?rcR>¦P`PRrµ}yRn?^+jqe¦"R?hr¥jq` N ¢ oRnpC ^_N-pC^ f fh]UnN"pCepqn?^+R?e_fÏRrµ=\8^_NPRjqhj~af`c mce+jqmR?e_^\GÅ p(·) fh]&pyRj=rcR]+fnjq` M fÎpC`Pr¤jq`c\¡f (p(·), p(·)) fh]KpC`¤f`=^+R?wepC
(52) n?bce_}R jC f ¢ ˙ `Ð ¡m"pCe_^_fhn?bcEpCe f ]pC^_fh])¦"R]a^_NPRe+R?EpC^_fhjq` Á¯vyÄ π (f (q)) = q , ¯jqeR?}yR?e_\ q ∈ N ¢ M/NPRIª ³c±q¨ª?¨"¬¶q´ ²S¶³ ±q¨ M f]rcR>¦P`PRr¡=\ Á¹=Ä exp(t, q) = π(e (q)) , aNPR?e+R e : N → N rcR?`Pjq^+R]^_NPR j~ÈjC,^_NPR}yRn?^+jqea¦"R?hr f pC^^_fQSR t ¢ R?^ g R&^_NPRK]_QSj=jq^_N}Rn?^+jqe.¦"R?hr½jq` N =aNPjw]RKnjqe_e+R]_mjq`Pr:f`c j~ÒpC^/^_fQSR t fh]^_NPR ¦PR?e_afh]+Rae+jq^pC^_fhjq`SjC#pC`cwhR t ¢ jqe R?}R?e_\ q ∈ N wR/]+R?^ q = e (q) pC`Pr Rq = e (q) ¢ jqe chR?^ Rs^_NPRUnjq`=^_e+jq ]_\:]_^+R?Q . M. M. ?. tf. tf. −. . ε>0. πg. − π2 g. (DεM ). (DεM ) :. l&` ¶°q²E««¯·?´ª»®±q¨-¬©_±q´ fh]p¥QSRpq]_bcepCPR8¹bc`Pn?^_fhjq` arcR>¦P`PRrÉjq`É]+jqQSR¤f`=^+R?e_}qpC jC R af^_NÓ}wpCbPR]'f` [−ε, ε] ¢ M/NPR8]+jqbc^_fhjq`P]IjC (D u(·) n jqe_e+R]_mjq`Pr:f`c!^+jÔpqr:Qfh]+]+fchR ) q˙ = f (q) + ug(q) ,. q ∈ N,. u ∈ [−ε, ε] .. M ε. . ù.
(53) U§c·?6¨c«³-©_±·?´ª?² ±q¨8«§3©?¶y®ª«. x. n jq`=^_e+jqh]pCe+RKnpChRr¥¶y°q²6««Í·>´hª¬©_¶ ª+®¬Í±w©Íª« ¢ ÁÍZ3jqQSR?^_fQSR]:^+jmce+R?}R?`=^apC`=\'njq`:bP]_fhjq`õ3R af ]_mRpCÑ¡pCh]+jSjC ε ¶°q²E««¯·?´ª®±q¨"¬©_±q´ «apC`Pr ε ¶y°q²6««Í·>´hª ¬©_¶ ª+®¬Í±q©>¯ª>«_Ä ¢ jqeR?}R?e_\ q ∈ N pC`Pr T > 0 ^_NPR8¶w¬¯¬Í¶w6¨-¶=·>´ª'«ª¬ ©_±w² q §³Ó¬Í±8¬¹²'ª T fh] ^_NPR']+R?^ njq`P]+fh]_^_f`c¡jC.^_NPRR?`Pr:mjqf`=^+]UjC/pCpqr:Qfh]]_fchR^_epk_Rn?^+jqe_fhR]sjqe (D ) A = A (M, ε) ]_^pCe_^_f`cS¹e+jqQ q PjC
(54) hR?`cw^_N8]_Q'pChR?e^_N"pC`¤jqeRgb"pCõ^+j T ¢ £ RUpCh]jae_f^+R . . T q. . . T q. M ε. Aq = Aq (M, ε) = ∪T >0 ATq (M, ε) .. /M NPRnjq`=^_e+jq-]_\:]_^+R?Q (D ) f]ÎnpChRr¤®±q²&³-´hª?¬ª>´¸º®±q¨"¬©_±w´6´¶=·>´ªf A = N jqeÎR?}yR?e_\ q ∈ N ¢ ¡× 6 Û ª«¶qºs¬ :¶w¬#¬ :ª U§c·?6¨c«Ì³-©+±·>´hª>²±q¨ M 3¶C«/¬ :ª/§=¨"©+ª«>¬©>¯®¬ª_°®±q²&³-´hª?¬ª ®±q¨"¬©_±w´6´¶·?6´¸¹¬º
(55) K³-©_±³Pª>©?¬º ̱q© /ª ?§3 ¶q´hª>¨"¬´¸º /¬ 3¶y¬.¬Î6« S ±q©ª Cª>©>º ε > 0 E« ®±q²&³-´ª¬ª>´ º®±q¨"¬©+±q´6´¶·>´hª (D ) R8]_^_e+R]+]I^_N"pC^½^_NPR8`Pjq^_fhjq`ÊjCVÂa mce+jqmR?e_^\Ónjqe_e+R]_mjq`Prc]^+jÔ^_NPR8jqe_fwf`"pCÍ&njq`=^_e+jq £ e+RRUjqe_QbcEpC^_fhjq`¥jC^_NPR¾Kbccf`P]~À"mce+jqchR?Q jq` M ÅaVsÂa NPjqhrc]&f.pC`Pr»jq`c\¡f_P¯jqe&R?}R?e_\ pC`PrÉR?}R?e_\ (p , v ) (p , v ) f` T M ^_NPR?e+R»R>Ç:fh]_^+]p mcfhRnR?afh]+R¥]_QSj=jq^_NÊn?bce_}R ε > 0 af^_NÒyRj=rcR]_fhn¥n?bce_}qpC^_bce+Rµ]_Q'pChR?e^_N"pC` ε ]_bPnNÊ^_N"pC^ γ(T ) = p γ : [T , T ] → M Vmµ^+j8pe+R?m"pCepCQSR?^+R?e_fÏpC^_fhjq`!=\µpCe+n>ÍhR?`cw^_Nõ
(56) ^_NPRf^ jC γ jq` N γ(T ˙ ) = v i = 1, 2 ¢ fh]pC` εpqr:Qfh]+]_fchRI^_epk)Rn?^+jqe_\
(57) aNPjw]+RInjqe_e+R]_mjq`Pr:f`c¥njq`=^_e+jq u(t) fh]Rg=b"pCÍ pC^pCQSjw]+^ R?}R?e_\ t ∈ [0, T ] c^+j^_NPRyRj=rcR]_fhn n?bce_}qpC^_bce+R jC γ pC^^_NPRsmjqf`=^ γ(t) ¢ ^,fh] Rpq]_\^+jnNPRnÑ^_N"pC^q¯jqe R?}R?e_\ pC`PrR?}R?e_\ ) fh] cepqnÑyR?^ yR?`PR?epC^_f`c ¢ `Ð PrcRÐ Rr#-¯jqesR?}yR?e_\ q ∈ N RN"p}R^_N"MpC^ π ([f, g](q))ε >=0π (D(f (Rq)) M/NPR?e+R>jqe+Rw pC`"r rg cR>¦P`PRSpInjq`=^pqn?^Ur:fh]_^+e_fcbc^_fhjq`µjq` N ¢UÐ `¿m"pCe_^_fhn?bcEpCeG¯jqesR?}R?e_\ 0 <¢ t < T pC`Pr¿fR?}R?e_\ q∈N Á¯uyÄ e (q) ∈ Int(A ) , aNPR?e+R Int(A ) rcR?`Pjq^+R] ^_NPRf`=^+R?e_fhjqe
(58) jC A ¢ M/Ncfh]õ¯jqhj~]~jqe f`P]_^pC`PnRwe+jqQ ^_NPRÎrcR]n?e_fm: ^_fhjq`»jCÎ]_Q'pCÍ^_fQSRpC^_^pCf`"pCchR]+R?^+]¯jqe&]_f`cwhR>Íf`cmcbc^s`Pjq`:rcR?yR?`PR?epC^+R^_Nce+RR>r:fQSR?`P]+fhjq`"pC njq`=^_e+jq ]_\:]_^+R?QS]/wf}R?`{=\ jqce_\½f`
(59) ¢ Z=f`PnR 3pC`"r pCe+Rf`PRpCe_\f`PrcR?mR?`PrcR?`=^pC^.R?}R?e_\mjqf`=^=^_N"R>\SnpC`IRbP]RrS^+j f`=^_e+j=r:bPnRfpUQSg R?^_e_fhnK[f,jq` g]N e+Rg=bcfe_f`c (f (q), g(q), [f, g](q)) ^j RKp jqe_^_NPjq`Pjqe_Q'pCG"pq]+fh].jC ¯jqe R?}R?e_\ q ∈ N ¢ Z=b"nN QSR?^_e_fhnwaNcfhnNfh]õbP]_b"pC\npChRrU^_NPR ¶C«¶ ~"²'ª?¬©>¯®6¨ :ª>©>¹¬ª+° T N >©_±q² õR?`Prcj~] af^_N pnjqQmchR?^+RSofhR?Q'pC`c`cfEpC` ]_^_e_bPn?^_bce+R¤Á]+RRwG¯jqesf`P]_^pC`PnRw w ¹Ä `Ð !pqnnmjqe+rPpC`PnR½afN^_N!^_NPR'`Pjq^pC^_fhjq`P]f`=^_e+j=r:bPnRrËpCj~}yRw,R'af/rcR?`Pjq^+RI=\ d (·, ·) ^_NPR ¢ f`Pr:bPnRr'r:fh]_^pC`PnRjq` N pC`Pr#w¯jqeR?}yR?e_\ q ∈ N pC`Pr ρ > 0 y=\ B (q, ρ) ^_NPR/"pC"jCGnR?`=^+R?e pC`Pr¡epqr:fbP] ρ ¢ q M ε. . . q.
(60) . . . . . . . . . . . . M ε. 1. 1. 1. 2. 2. 2. i. i. i. M ε. ∗. tf. T q. ∗. T q. T q. q. N. N. . ð !#". i.
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(62) ÅM/NPRs^+RpCe+rce+jqm{^_epk_Rn?^+jqe_\¡jC,]_fÏR . 1/ε. ¢. ½× c ¨ .ª'¨±w¬®ª+°»¬ 3¶w¬ &«¹¨-®ª (D ) :¶C«'¬ :ªU³-©_±³Pª>©?¬º¿± ½·+ª?6¨ µ·>©_¶® qª?¬ ª?¨ª>©_¶w¬¹¨ U¬ cª U§c·>¹¨c«=³-©_±·>´hª>²±q¨ E« È ½¶q¨-°¥±q¨P´¸º8 ±q©ª Cª>©>º ε > 0 ¶q¨-° M a¬ :ª>©+ª'ª qE«?¬¹« «§c® »¬ 3¶w¬ ¨ ±q©_°=ª>©¬Í±³-©_± Cª ª qª>©º ¬ 6«'´¶Cq«>¬.∈³-N©_±³Pª>©?¬º .ª ¹´6´/qˆ± ∈>¬ª?A¨Ë(M, ²6²ε)¯®¡±q¨ M ¬ :ª¤qˆ·+ª 3∈¶ A¯±q(M, ©{± ε)¶¿¬ª+¶q©_°w©_±³É¬©+¶ ª+®¬Í±q©>º ±q¨Æ¬ :ª §:®´ ¯°ª+¶q¨¿³´¶q¨Gª ª¡®¶q´6´¬ª+¶q©+°q©_±³É¬©_¶ ª+®¬Í±q©>º¥±S« ~ª 1/ε Í«ªª Îw§=©+ª 8¬ cª ·+¶q¨ ·+¶q¨ ¬©_¶ ª+®¬Í±q©>º½± (D ) :±C«ª®+±w¨"¬©_±q´ u.°=ª ¨ª+°·?º 0 ≤ t ≤ , ε −ε <t≤ , u(t) = <t≤ , ε «?¬ª+ª>©« (1, 0) ∈ U R ¬Í± (−1, 0) ∈ U R ½× c M , M ¶q©+ªU¬ α
(63) ͪ>²S¶q¨P¨P¯¶q¨{²S¶q¨P ?±q´°C«¶q¨-° P : M → M E«¶S´±®¶q´ 6«?±q²'ª?¬©>º¡¶w¬ª Cª?©º³c±q¹¨"¬a± M a¬ :ª>¨ ª Cª>©>º¤¶y°q²6««Í·>´hª¬©_¶ ª_®¬Í±q©º ?±q© (D ) 6«¬©_¶w¨c« ±q©>²Sª+°»·?º ¹¨!¶w¨ ¶y°q²E««Í·>´ª'¬©_¶ ª_®¬Í±q©º ?±w© ¨{³c¶q©>¬¯®?§3´¶w© P 6«±q¨"¬ÍP±{¶q:¨-U°¡M¬ cª →U§PU·>6¨PM« C³-©+±·>´hª>² ±q¨ M E« .¬ cª>¨ ¬ :(Dª«?¶w²S)ªE«¬©>§cª ?±q©¬ :ª U§c·?6¨c« ³-©_±·?´ª?² ±q¨ M. . . . . M ε. . . . . . . . . . . . . R2. . .
(64) . . . ε. (0,0). qˆ. . π 3ε 2π ε 7π 3ε. π 3ε 2π ε. . . −. q. . 2. (0,0). 2. . . 1. 2. . . . 1. ∗. 1. . 2. . . 1. M2 ε. 2 M1 ε. . 1. 2. . ù. .
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(68) aNcfhnNXpqn?^+] ¹e+RR?\pC`Prr:fh]+njq`=^_f`=bPjqbP]_\Γjq` X ¢ £ RÎaf:rcR?`Pjq^+R.=\ Π : X → M ^_NPRUnpC`Pjq`cfhnpCõmce+jCk_Rn?^_fhjq`¤jC X jq`=^+j M ¢ £ NPR?`{^_NPR]+Rn?^_fhjq`"pC n?bce_}wpC^_bce+Rfh]njq`P]_^pC`=^jq` :^_NPR?` M fh]]pCfhr¤p º³Pª>©·_±w´ ¯®²S¶q¨" ±q´°UpC`Pr X p º³Pª>©·+±q´ ¯® «³:¶®ª ¢ M . .
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(72) . +e jqQ `Pj~ jq`õ X afrcR?`Pjq^+RÎp KpqrPpCQ'pCe+r ]_bce)ÍpqnRw~^_N"pC^õfh]p KpqrPpCQ'pCe+rU^jCr:fQSR?`P]_fhjq`"pC Q'pC`cf¯jqhr ¢½Ð M = X/Γ fh]p{njqQmchR?^+RIofhR?Q'pC`c`cfEpC`!]_bce)ÍpqnRw ^_NPR?`ÔRaf.rcR?`Pjq^+R'=\ ^_NPRS]pCQSRhR?^_^+R?e+]
(73) pC`Pr ^_NPRSnjqe_e+R]_mjq`"r:f`c8jqck)Rn?^+] jq` pC`Pr M/Ncfh] fh]QSjq^_f}wpC^+Rr!=\µ^_fN"R'gÍpqKn?^^_mN"pC^ Π : πX → M ,R?f`c p¤hj=npC/fh]+jqQSR?^_e_\yXfhrcR?`=^_fM¦"R¢]]_bPnN jq:k_Rn?^+]"pC^ahRpq]_^&pC^phj=npCõhR?}R? ¢ ^.fh].R?ÍÑ=`Pj~`½^_N"pC^ X fh]r:f #R?QSjqe_mcNcfhnK^+j R ¢ l&`=\yRj=rcR]_fhnK]+R?wQSR?`=^/f` X fh].^_NPR Ð bc`cfhg=bPRhR?`cw^_N:ÍQf`cfQfÏf`cK^_epk)Rn?^+jqe_\R?^RR?`f^+] R>Ç:^_e+R?QSR] ¢
(74) Ð `Um"pCe_^_fhn?bcEpCeypC=njqQmchR?^+R yRj=rcR]_fhn]&pCe+RU]_fQmchR ¢ l ©_¶qº&fh]/pN"pC6ÍyRj=rcR]_fhnjq` X ¢ l «ª+®¬Í±w©Îfh]ap e+R?wfhjq`¡jC X jqbc`PrcRr½=\'^jr:fh]+^_f`Pn?^ ep\:]]_^pCe_^_f`cIpC^^_NPRU]pCQSRsmjqf`=^PaNcfhnN{fh]npChRr{^_NPR qª>©?¬ª ¡± S ¢ M/N"R°ª+¶q´ ·+±q§3¨-°y¶q©>ºjC crcR?`Pjq^+Rr¡=\ 3fh]rcR>¦P`PRr{pq]a^_NPRgbPjq^_fhR?`=^jC ^_NPR]+R?^ jCpC#ep\:]am"pCepCQSR?^+R?e_fÏRr¤=X\¡pCe+n>ÍhR?`cw^_N¤=\½X(∞) ^_NPRURgbcf}wpChR?`PnRe+R?EpC^_fhjq` . 2. . c1 ∼ c2 ⇐⇒ lim sup dX (c1 (t), c2 (t)) < ∞ .. /M NPRURgbPf}wpChR?`PnRUn?Epq]+]jC pm"pCepCQSR?^+R?e_fÏRr{ep\ fh]rcR?`Pjq^+Rr{=\ pC`Pr¡f^afh]npChRr ^_NPR¡ª?¨-°³c±q6¨-¬a± c ¢KÐ c : R → X fh]pSm"pCepCQSR>^cR>e+fÏ~Rr¥njqQmchR?^+Rc(+∞) yRj=rcR]_fhnwG^_NPR?` c(−∞) rcR?`Pjq^+R]^_N"RRg=bcf}qpChR?`PnR n?Epq]+]jC [0, ∞) 3 t 7→ c(−t) ¢ jqeUpIwf}yR?`¿mjqf`=^ p ∈ X ^_NPR?e+Rfh]UpIjq`PR>Í^+jCjq`PRSnjqe_e+R]_mjq`PrcR?`PnR ψ R?^ÌRR?` U X pC`Pr aNcfhnN¡pq]+]_fw`P].^+j ^_NPR&Rg=bcf}qpChR?`PnRKn?Epq]+]jC M/NPRX(∞) njqe_e+R]+mjq`PrcR?`PnR&rcR>¦P`"R?]/pv ^+∈jqmUjqhjqXw\Sjq` X(∞) yaNPfnNIfh]ÎnpChR[0,r'∞) ^_NPR3«Í³ t:ª>7→©+ªsexp(t, ¬Í±³c±q´± v)wº ¢¢ t→∞. p. p. . ð !#". p.
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(76). . ¬Í±q§3©-y¶w´±w¬¯¬. . /M NPR]_mcNPR?e+Rs^+jqmjqhjqw\R>Ç3^+R?`Prc]a^+j^_NPR]+jCnpChRrµ®±q¨ª ¬Í±³c±q´± wºsjq` X = X ∪ X(∞) ¢ M/NPR njq`PR^+jqmjqhjqw\fh]ayR?`PR?epC^+Rr¤=\I^_NPRUjqmR?`¤]+R?^+]jC X pC`Pr¡^_NPRU]+R?^+] ψp (U ) ∪ (∪t>0 exp(t, U )) ,. aNPR?e+R p ∈ X pC`Pr U f]pC`jqmR?`S]+R?^ jC U X ¢ TKjq^_fhnR^+N"pC^,^_NPRpqn?^_fhjq`Sjq` X jCGpC`R?hR?QSR?`=^ jC Γ N"pq]&p`"pC^_bcepC njq`=^_f`=bPjqbP]&R>Ç3^+R?`P]_fhjq`¤jq` X ¢ sf}yR?`µp½]+R?^ f` GRae_f^+R ∂Ω ¯jqeK^_NPRjqbc`PrPpCe+\»jC Ω f` X GaNcfhR Ω(∞) af rcR?`Pjq^+RU^_NPRsf`=^R>eΩ]+Rn?^_fhjqX`¤R?^RR?` X(∞) pC`Pr¡^_NPRUn?hjw]_bce+RUjC f` M/N"R½fh]+jqQSR?^_e_fhn^_epC`P])jqe_Q'pC^_fhjq`P]'jC X npC` R¡n?Epq]+]_f¦"Rr Ωf` ^+R?Xe_QS¢ ]jCK^_NPR]+jCnpChRr °q6«Í³-´¶y®ª>²'ª>¨"¬ >§=¨®¬±w¨ l&`{fh]jqQSR?^_e_\ fh]npChRr ª>´6´¸ä³G¬® f f^N"pq]/pC^hRpq]_^/jq`PR¦cÇcXRrI3mjqpf`=7→^fd` X(p, γp)º³Pª>¢ ©·+±q´ ¯®f ^_NPRKr:fh]_γmc:EpqXnR?QS→R?`=X^bc`Pn?^_fhjq`{pC^_^pCf`P] f^+]aQf`cfQbcQ f` X PpC`Pr{]_bPnN{Qf`cfQbcQ fh]]_^_e_fhn?^_\½mjw]+f^_f}yR ³c¶q©_¶·+±q´¸®jq^_NPR?e_afh]+R ¢ p. X. . .
(77)
(78) . fh]p½njqQmchR?^+RofhR?Q'pC`c`cfEpC`µ]_bce)ÍpqnRwG^_NPR?`¿^_NPRjq`c\8R?fmc^_fhnR?hR?QSR?`=^UjC hfÐ ]^_MNPRU=fhrcX/Γ R?`=^_f^\ ¢ R?^ G R pn?hjw]+Rr{yRj=rcR]_fhnUf` M ¢ fÇ¡jq`PR jC,f^+]f^+] Ge f` X pC`Pr{hR?Γ^ RpIm"pCepCQSR?^+R?e_fÏ~pC^_fhjq` jC =\8pCe+n>ÍhR?`cw^_N M/NPR?e+Rfh]sjq`PRfh]+jqQSR?^_e_\ c:R → X ]_bPnN ^+N"pC^ γ(c(0)) = c(T ) aNPR?e+R T fhG]e ^_NPRhR?`cw^_NÆjC ¢ G ¢ M/NPR?` c(t + T ) = c(t)γ ∈jqΓe R?}R?e_\ Á¹^_NPRmce+j=jC
(79) yj=R]pq]af`¡^_NPRsN=\3mR?e_jqfhnUnpq]+Rwc]RR zccM/NPRjqe+R?Q ¹Ä M/NPR fh]+jqQSR?^_te_\∈ γRfh]N=\3mR?e_jqfhn8Á]+RR : ¹ÄUpC`Pr Ge fh]UnpChRrÔpC`Ó¶
(80) q6«sjC γ ¢ lan?^_:b"¢ pC:¢ \ ¢ R?}R?e_\ N=\3mR?e_jqfhnfh]+jqQSR?^_e_\{N"pq]pC^KhRpq]+^Kjq`PRpÇ:fh] ¢&Ð `Pjq`PRjCÎ^_NPRN"pC6ÍmPhpC`"R?]sjqbc`PrcRr»=\ Ge fh] pC^P^_NPR?`õ"wf}yR?`»p`PR?fwN=jqe_NPj=j=r U jC c(−∞) pC`"r¤p`PR?fwN=jqe_NPj=j=r V jC c(+∞) f` :^_NPR?e+R R>Ç3fh]_^+] n ∈ N ]_bPnN{^_N"pC^ X ¡ ¢ ¡ ¢ Á Ä γ X \U ⊂V , γ X \V ⊂U, ¯jqeR?}yR?e_\ m ≥ n Ppq]amce+j~}Rr¡=\ apCQ'pC`c`¤f`
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(82) v cmce+j~}Rr^_N"pC^ pC:njqQmchR?^+RanjqQm"pqn?^ ofhR?Q'pC`c`cfEpC`]+bce)¯pqnR],njq`=^pCf`SpC^
(83) hRpq]_^,jq`PRn?hjw]+RryRj=rcR]_fhn ¢ l^+R?e ^_N"pC^M/NPjqe_R?e_y]+]jq` C ,R>Ç3^+R?`PrcRr8^_NPR e+R]_bc^&^+j½pC
(84) njqQmchR?^+RwGnjq`c`PRn?^+Rr¿ofhR?Q'pC`c`cfEpC` ]_bce)ÍpqnR]`PR?f^_NPR?eKNPjqQSRjqQSjqe_mcNcfEn^+jS^_NPR mcEpC`PR`Pjqe&^+j'^_NPRn?\=f`PrcR?e ¢ lK]KpSnjq`P]RgbPR?`PnRw f Γ fh]a`Pjq^n?\3n?fhnwc^_NPR?` M = X/Γ njq`=^pCf`P]Kpn?hjw]+Rr¡yRj=rcR]_fhn ¢ ¡× 6 Û ª?¬ Γ ·+ª¶ q©_±q§~³Ô± E«±q²'ª?¬©>¯ª«± X Î p ∈ X ¶w¨-°½®±q¨c«¯°ª>© Γ(p) ¬ :ª®´±q«§=©ª6¨ X ± ¬ :ª Γ ±q©·>¬± p :ª«ª?¬ L(Γ) = Γ(p) ∩ X(∞) °y±ª«¨±w¬a°ª³Pª>¨-° ±q¨ p.¶q¨-°'6«U®¶q´6´hª+°½¬ :ª ´ ¹²¹¬ «ª¬± Γ . . m. . −m.
(85) . . . . . ù.
(86) U§c·?6¨c«³-©_±·?´ª?² ±q¨8«§3©?¶y®ª«.
(87)
(88). ¡× 6 Û ª?¬ X ·ª{¶ ¶y°y¶q²S¶q©_°¿«§3© ¶y®ª{¶q¨° M = X/Γ ·ª{¶µ®±q²&³-´ª¬ª
(89) ¯ª ²S¶q¨P¨"¶q¨8«§3©?¶y®ª ª «?¶qºI¬ :¶w¬ E«± ¬ cª Ω«?¬ ~¹¨-°I L(Γ) = X(∞) ±w¬ cª>© 6«ª .ª «¶qº'¬ 3¶w¬ M E« ± U¬ :ªU«ª+®±q¨-° ~¹¨-M° ½× c Γ 6«K®?ºy®?´¸® ¬ :ª>¨ M 6«K± K¬ :ª&«ª+®±q¨-° 6¨-° ¨°ªª_° , Γ = {Id},¬ :ª?¨ 6«ª>²&³¬º 6«U¨-±q¨"¬©> ¶w´ ¬ :ª>¨¥¹¬.6« ª>¨ª?©_¶w¬ª+°¤·>º¡¶ º³Pª>©·+±q´ ¯®S±q©¶ ³:¶q©+¶·_±w´ ¯® L(Γ) 6«?±q²'ª?¬©>º s°ª>¨±w¬ª+°µΓ·>º γ γ 6« º³Pª>©·_±w´ ¯® ¬ :ª?¨ ¹¬K¬©_¶q¨c«>´¶w¬ª>«I¶w¬&´hª_¶q«>¬s±q¨Gª ª+±~°ª«¯® :ª?¨-®ª 6«a²S¶y°ªK± ¬ :ªK¬ αª>¨°³:±w6¨"¬¹«&± a«§c ® ª+±°ª>«¯® §=«>¬
(90) ®±q¨c«>°=ª>©¬ cªK±q©·>¹¬,± &¶ ³c±q¹¨"¬ ±qL(Γ) ¨¤¬ :
(91) ª ª+±~°ª«>® γ ,6¨c«?¬ª_¶°
(92) 6«³:¶q©+¶·+±q´ ¯® ¬ :ª>¨8¬ :ª>©+ªª q6«>¬¹«¶&³c±q6¨-¬ z ∈ X(∞) ¯ ® ½6«s6¨ C¶q©¯¶q¨"¬ §3¨-°ª?©U¬ :ª¶y®¬¯±q¨¿± άͱ =ª?¬ :ª>© ¬ ½¬ cª 3±w©_±C«Í³ cª>©+ª«®ª>¨"¬ª>©+ª+°¶w¬ Í«ªª ª>²²S¶ Î I¶q¨"ºK³:±q¹¨"¬ x γ∈ X ¶w¨-°I®+±w¨c«¯°ª>© ¶ 3±q©_±C«³ :ª>©ª U ®ª>¨"¬ª?©+ª_°¶wz¬ ¯ ® ½®±q¨"¬Í¶q¹¨c« :ª>¨ 6«K®±q¨"¬Í¶q¹¨ª+°6¨ U (∞) ¨8·+±w¬ ½®¶C«ª>« M 6«¨±w¬ ± K¬ :ª z x L(Γ) ©«>¬ ~6¨° :ª>©+ª ?±q©+ª Î M 6«U± ¬ :ª Ω«?¬ ~¹¨-° ά :ª>¨¿¹¬.®±q¨-¬Í¶q6¨P«U¶½®?´±C«ª_° ª+±°ª>«¯® ¡× 6 Û ª?¬ M ·ª¶
(93) ͪ>²S¶q¨P¨"¶q¨«§3© ¶y®ª ª«?¶qº¬ 3¶w¬G¬¹« ¶q§=««>¶q¨I®?§3© ¶w¬§3©+ª 6«s§3¨P ?±q©>²´¸º¨ª y¶w¬ Cª sup K < 0 K ÐR> Ç:fhK]_^+]Ufh]pIbcm"`cpCfejqpCe_QSQR?^+\»R?e_`PfÏR?RrµpC^_yf}RRj=rcjqR` ]_fhnX #^_NPR?`õ#jqeR?]_}bPR?ne_N¥\8^_^N"pCj¡^ R?hR?QSR?`=^+] x, ypC`PjCr X(∞) #^_NPR?e+R M/N"pC^Sfh]~ÎbP]_f`cµ^_NPR^+R?e_Qf`Pjqhjqw\Ëcf`=:^_e+Rj=r:→bPnRXr¼=\ R?e_hR?fc(+∞) `ÊpC`Pr =ÀæT&xR?ff` c(−∞) z =fh]'yp ¢. X 6«Í·>¹´ ¬ºS²S¶q¨P ±q´° ¢ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. . .
(94) . . .
(95) . s Rj=rcR]+fnSnj=jqe+r:f`"pC^+R] npC`¥Rwhjq"pC\¿rcR>¦P`PRr¥jq` X ¢ M/NPR?\»rcR?mR?`Prµjq`¥^_NPRnNPjqfhnRjC pC`¤R?hR?QSR?`=^ q jC U X PpC`Pr¤pCe+RUrcR>¦P`PRr{^_Nce+jqbcwN¤^_NPRQ'pCm ÁÍxwÄ φ : R −→ M (x, y) 7−→ π(e ◦ e ◦ e (q)) . R?^ B : R → R Rs^+NPR]jqbc^_fhjq`¤jC
(96) ^_NPRU]_\:]_^+R?Q pC`Pr B + KB = 0 , Á¯zyÄ B(x, 0) ≡ 1, B (x, 0) ≡ 0, f`½aNcfhnN^_NPR&f`PrcR>Ç y pCmcmRpCe_f`cf` B , B ]_^pC`Prc].¯jqe.^_NPR&m"pCe_^_fEpC#r:f #R?e+R?`=^_fEpC^_fhjq`¡af^_N e+R]_mRn?^^+j y ¢ T&jq^_fhnR^_N"pC^P]+f`PnR K ≤ 0 B fh]awhjq"pC\¡rcR>¦P`PRr8pC`Pr Á Ä B(x, y) ≥ 1 , Á
(97) yÄ yB (x, y) ≥ 0 , q. 2. yf. π g 2. xf. 2. y. yy. y. y. . ð !#". yy.
(98)
(99). . ¬Í±q§3©-y¶w´±w¬¯¬. . ¯jqeR?}yR?e_\. (x, y) ∈ R2. ¢ M/NPRUnj=jqe+r:f`"pC^+RR>Ç3mce+R]+]+fhjq`¯jqea^_NPRQSR?^_e_fhn m jq` X fh]awf}R?`¡=\ m(x, y) = B 2 (x, y)dx2 + dy 2 .. ÁÍZ3RRw:¯jqeaf`P]_^pC`PnRw
(100) ¢ ÄM/NPRbc`cf^cbc`Pr:hR ½µ. aNPR?e+R. µ. npC`{RfhrcR?`=^_f¦"Rr{af^_N. ¯. ¶. ¯ cos θ x, y, , sin θ ∈ R4 ¯¯ (x, y) ∈ R2 , θ ∈ S 1 B(x, y). Ð `¡^_NPRUnjjqer:f`"pC^+R] f (x, y, θ) =. UX. pC`Pr g pCe+Rwf}yR?`{=\. ¾. .. (x, y, θ) f. cos θ , sin θ, F (x, y) cos θ B(x, y). F (x, y) =. ¶T. Á
(101)
(102) Ä. g(x, y, θ) = (0, 0, 1)T ,. ,. Á
(103) yÄ. By (x, y) . B(x, y). g=bcf}qpChR?`=^+\ Á D ÄnpC`{Rae_f^_^+R?`8pq]ajqhj~] . X ε. cos θ , B y˙ = sin θ, θ˙ = u + F cos θ.. x˙ =. TKjq^_fhnR^_N"pC^"pq]af^npC`{RURpq]_f\rcRr:bPnRr¡¹e+jqQ Á¯zyÄ> F ]pC^_fh])¦"R]a^_NPRÂ/pCbPnN=\½mcejqchR>Q. Á
(104) vyÄ Á
(105) =Ä Á
(106) uyÄ. Á
(107) Ä M = X/Γ fh]UpnjqQmchR?^+R'ofhR?Q'pC`c`cfEpC`Ô]_bce)Ípqn?Rwõ^_NPR?`õ#¯jqe pC`=\¤¦cÇcRr q ∈ N #^+NPR Ð Q'pCm φ rcR>¦P`PRrÊpq]If` ÁÍxwÄ>afh]¡pµhj=npCsr:f #RjqQSjqe_mcNcfh]_Q pC^½R?}R?e_\Æmjqf`=^jC R ¢ M/NPR mcbc"pqnÑSjCG^_NPRQSR?^_e_fhn m ^_Nce+jqbcwN φ R?`Prcj~] R af^_N½pofhR?Q'pC`c`cfEpC`½]_^_e_bPn?^_bce+RwNcfhnN e+R?`PrcR?e+] R fh]+jqQSjqe_mcNcfhnU^+j X ¢ ½× c :ª³©+ª ¯±q§« ±q©>²§=´¶w¬¯±q¨Ë± '¬ :ª U§c·?6¨c« ³©_±·>´hª>² 6¨Æ®±±w©_°q¹¨-¶w¬ª«S«?¬6´6´ ²S¶ qª>«a«ª>¨c«ª
(108) cª>¨¡±q¨P´¸ºU´±~®+¶w´ ¹¨c«>¬ª+¶y°± q´±=·_¶w´ ª+±°=ª«¯®®±±q©+°q6¨-¶y¬ª«&¶q©+ªK°ª ¨Gª_° ±w¨ «>°ª>©¬ :ª/®¶C«ªa± ¶
(109) ͪ>²S¶q¨P¨P¯¶q¨«§3© ?¶y®ª M ¬ ³c±C««Í·>´¸º« q¨ ¶q©>ºq6¨ ®§=© ¶w¬§3©+ª
(110) ª+®¶q´6´ ¬ 3¶w¬a¶ 3¶q´ ³-´¶q¨ª± M 6«¶I«¹²&³-´ º®+±w¨P¨ª+®¬ª+°{±³Pª>¨¿«§P·«ª?¬± M ·_±w§=¨-°=ª_°¤·?º¶I«¹²&³-´ª ±³Pª>¨8®±q²&³-´hª?¬ª ª+±°=ª«¯® M ®+±q¨-¬Í¶q6¨P«K¶ 3¶q´ ³-´¶q¨ª H ¶q¨-° K 6«¨-±q¨w³:±C«>¹¬ Cª±q¨ H ¬ :ª?¨ H ¶y°q²¬¹«I¶¤«ºC«>¬ª>² ± ª+±~°ª«>®{®±~±q©_°q¹¨-¶w¬ª« >¨-°ªª_° ±q©I¶q¨Pº q ∈ N «§c ® ¬ 3¶w¬ ®¶q¨·ª&ª>¨-°± .ª_° ¬ ¬ :ªa²'ª?¬©>® φ m ¶q¨-°U¬ :ª U§P·>6¨P« φ (R×[0, ∞)) = H R×[0, ∞) Fy = −K − F 2 ,. F (x, 0) ≡ 0 .. 2. q. q. 2. . . 2. . . . . . . . . . q. . . . . . . . . ∗ q. . . ù.
(111) U§c·?6¨c«³-©_±·?´ª?² ±q¨8«§3©?¶y®ª«. v
(112). ³-©_±=·>´ª?² ±q¨ H ®¶q¨¼·ª½°ª>«?®?©>¯·ª+°µ·?º¿ª ?§c¶w¬±w¨c« "¹²6´¶q©>´ º q = (p, v) ∈ N ¶q¨° a, b, r > 0 ¶q©+ª«§c® ¿¬ 3¶y¬ K ≤ 0 ±q¨ B (p, r) ¶q¨° a + b < r ¬ :ª>¨!¬ :ªS©+ª_®¬Í¶q¨ q´hª ª>¨°y± .ª+° ¹¬ ,E«s¶¨-±q¨q³c±C«¬ qª>´¸º®?§=© Cª+°I®± Cª>©>6¨ '°y±q²S¶q¹¨ [−a, a] × [−b, b] ⊂ R ± &¶ ¨Gª> :·+±q© :±±~°± p y¶q¹¨ õ«>º«?¬ª>² φ m °ª«®?©>¯·ª«K¬ :ª §P·>¹¨c« ³-©_±=·>´ª?² ±q¨I«>§:® ¨-±w¨ a¶w¬Î©+ª_®¬Í¶q¨ q´hª ( -)(- (/(-Y(Ò8 £^+jR.]_^_ne+jqR]hR]/n?^_^,NPNPRsR?e+f`=R^+^_R?NPe_e+RR?]_E^pCpC^_fh^+jqR?`PQS]R?`=^+R?] ^jCR"R?^_`{NPR.^_NPQ'RpCmcf`e+jqe+mR]_jwbc]+R^+r¡]
(113) `Pmce+Rj~n}R]+R]rpCe_f`\^_pCNP`PR.r{`"]_Rb Ç:'^,]+n?RfhR?n?`=^_fh^jq`õnjqq`Pf`r:fjq^_e+fhjqrc`PR?]e ¯jqea^_NPRUVsÂaÂÉmce+jqmR?e_^\^+jNPjqhr ¢ . Û Û 6 Û ª¬ ·+ª¶ ®±q²&³-´hª?¬ª
(114) ®±q¨P¨Gª_®¬ª_°
(115) ͪ>²S¶q¨P¨P¯¶q¨«>§=© ¶y®ª cª §P·>¹¨c« ³-©_±=·>´ª?²±w¨ M 3¶C«½¬ :ª M Ƴ-©_±³Pª>©?¬º¿ ½¶w¬s´hª_¶q«>¬U±q¨Gª¡± ½¬ :ª ?±w´6´± ¹¨ µ®±q¨-°q¬±w¨c«'6« «¶w¬E« ª+
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