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\4DFUËJROcXHDWKAURK>ºTKAOQi>S/Ga_DFGaXHOYXIDWURSTi>i]OWJLKAEHS[º[KADQU`S&JLK

ξôœGIS&_S[E

t ¿FEHOQU`JLS

γ \4EHSTGH\4S&SfGa_DYó URSTURº[K]OYiAZS[UbXIS&\[DQU

t ^ô γ ∼ e^{χt˜ ¼ U} ZoDLJLD½™ŒDFEIZOQi>SJLK9JRS`URKAEHSiÇàSTGI_žDFURS[UbXHS

χ ^ÿK>i

GIS[¿QP`S[UbXISF–

,p®Få†È“±³ÜQ±Ç¯žÈ“®

­EF

 

"

õ

SfGa_DQU`S[UbXIS€\[OQEHOYXaXHS[EHKAGaXIK]\4DJLKNMN·FOY_`PRURDc¸HG

õ4ø>÷Yö

 IJ#

÷cø

I

õ4öÛö

#JG

õ

ö÷

LK

õ  ö õ

ξ∈ T^xM ^{ø|ø öõM  õoõ $ öõM}

χ(x, ξ) = lim

t→∞

1

t logkD^xΦ^tξk kξk

í´ ï

­ <­

<]¢ qpÈîÆQ²`ÎT¯φ²L°áÐT±ÇÆQ¯ų²L°c®

C+DQUžGaK]JLS[EHK]OYZD/KAi9\TOQGID½JRKœPRU|àDQEHŸRK>XHOp_žSTEIKADLJLK]\[O½JLK9_S[EHKAD”JRD

τ–eK>UËþbPRSfGšXHD½\TOQGID½iÇàOY_Ló _RiAKA\TOYºTK>DFURSˆXjOYUR¿FS[UbXISkÿkPRU/DQ_S[EjOcXHDQEHSˆi>KAURSfOYEHS6GIP

TxM˜
”PR_`_žDFURKAOQZDe_S[E?GaSTZ_Ri>K]\4K>XHÁ

\j—RS

DxΦ^{τ OYŸRŸRK]O} n = dim M OYPRXIDc¸cOYiADQEHK

λ1, . . . , λn

ôb\[DQU

|λ¹| > |λ²| > . . . > |λⁿ|^S

GIKAOQURD

u1, . . . , un

K`EHK]Ga_S4XIXIKA¸bKOYPLXHDc¸QS[XaXIDFEIKۘ1ghi>iADQEjO@_S[E9DF¿QURK

k > 0^Sˆ_S[E 1≤ i ≤ n^GIK

OḔEHÁ

DxΦ^kτui= λ^k_iui

S@_REISTU`JLSTU`JLDoK>i’iAK>ZK>XIS€JLSTi>i]OGaDQXaXIDbGaPž\[\4SfGIGIKADQURS

t = kτ, k∈ N

GIK’_RPR&\4DFU`\4iAP`JLSTEIS@\j—RS

χ(x, ui) = 1

τ log|λⁱ| =: χⁱ(x).

gˆUžOYiADQ¿FOQZS[UbXISFôRGaS

1≤ r ≤ nô`_REHSTGIDPRUB¸QS[XaXIDFEIS

ξ = crur+ . . . + cnun

ôGaS

cr6= 0

i]OEaóýSTGIKAZO\[DQZ_DQURSTUFXHS2JLDFZK>U`O&SeGaK“—`O

χ(x, ξ) = χr(x)˜@C+DFU`GIKAJLSTEIK]OYZD&OQi>iADQEjOoiAO GIS[¿QP`S[UbXIS€GISTþbPRS[U`ºTOoJLKNGIDYXaXHDFGI_`OYºTK

L1⊃ L²⊃ . . . ⊃ L^{n JLK} TxMôRJLS`URK>XHOJRO

L1=< u1, . . . , un>= TxM L2=< u2, . . . , un>

˜˜˜

Ln=< un>

í§ ï

< u1, . . . , uk > u1, . . . , uk

¼

GIK>DFURS€JLKNGIDYXaXHDFGI_`OYºTK

E = L1⊃ L²⊃ . . . ⊃ L^m, m≤ n = dim E

JLKvJLKAZS[U`GIK>DFURSJLST\[EISfGI\[S[UbXISFôžÿJLS4XIXHON

øöG

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E˜…gˆiAiADQEjO i]Om_REHST\[STJLSTUbXISBí§ ïhÿPRUžO/_žOYEIXIK]\4DQi]OYEHSQžiÓXHEHOQº[KADQURSJLK

TxM SEHKAGIPRi>XHOpKAUþbPRSfGšXHDp\TOQGID

χ(x, ξ) = χi(x) ^GIS ξ∈ Lⁱ\ Lⁱ⁺¹ í³JLDF_žD&Oá¸FS[E¡_žDbGšXHD

Ln+1 ={0}^{ï ˜}

­ <­ <× .4ÅòÆF²RÎT¯ ä®”È1®b°c²RÅÇ®

”S€iÛàDFEIŸ`KÓXjOURDQU½ÿ€_S[EHK>DLJLK]\[OoURDFUBÿ€_žDbGIGIK>Ÿ`K>iAS€EIKA_S4XISTEIS€KAi|EjOY¿FK>DFU`OYZS[UbXHD2_REHST\[STJLSTUbXISQô

_S[Ej\j—RÞ@iAOZOY_`_`Oi>KAURSfOYEHS

DxΦ^{t ¸cOJRO} TxM ^O TΦ^t(x)M6= T^xMôRZS[UbXIEHS€¿QiAKNOYPRXIDYó

¸cOYiADQEHK`GIDQURDeJRS`URK>XIKKAUFXHEIKAU`GIST\TOYZS[UbXISkGaDFi>D€_S[E¡ZOY_R_ShiAK>U`STOYEHKžJ`O€PRURDeGI_`OQº[KAD…i>KAURSfOYEHS

KAU½GIÿQ˜

^ àOQiÓXHEHO2_`OYEIXIS6ÿ@_žDbGIGIKAŸRK>iAS@PR¿QPžOYiAZoSTUbXIS@OQGHGaSTEIKAEISkiÇàSTGIK]GšXHS[URºfO2JLSTi>i]OR`i>XIEjOYº[KADQU`SkKAU/DF¿QURK

_RPRUbXHDJRK

x∈ M Oo_`OYEIXIKAEIS€JROoPRUžOJLS`U`K>ºTK>DFURS@_REIDc¸”¸”K]GaDFEIK]O…JLK

χ(x, ξ)ôž_RKBJLSTŸžDFi>SF–

χ(x, ξ) := lim sup

t→∞

1

tlogkDΦ^txξk ^íǾFï

\j—RS¡_žSTEIZS[XaXIS?JLKLEHKAUb¸”K]OYEHS9KAiL_REHDQŸRiAS[ZO6JLS[iAiÇàSTGIK]GšXHS[URºfOkJLSTiLi>KAZKÓXHS…íŒDFiÓXHEISòOkOá¸FS[E“DQZSTGHGID

iÛàKAEIEHKAi>ST¸áOQUbXIS@JLS[URDFZK>U`OYXIDFEIS

kξk^{ï ˜ ^} O2XHOQi>S…JLS`URKAº[KADQURS€GIS[¿FPRS€\j—RS

 χ(x, cξ) = χ(x, ξ)

χ(x, ξ + ξ⁰)≤ max {χ(x, ξ) + χ(x, ξ⁰)}

_S[E¡DQ¿QU`K

ξ, ξ⁰∈ T^xM ^S c6= 0˜†M1O…_REHKAZO2EHS[i]OYº[KADQU`SˆÿkŸžOYU`OQi>S>F_S[Eòi]OeGIST\4DFU`JROeŸ`OQGaXHO P`GHOYEHS?iAOeJLS`U`K>ºTK>DFURShJLKži>KAZ=GaPR_NôFGIST\4DFU`JLD€i]O…þbP`OYiASˆ_S[E+DQ¿QU`K

 > 0STGIKAGaXIS

T> 0^XHOQi>S

\j—RS

t⁻¹logkDΦ^txξk < χ(x, ξ) + 

_S[EˆDF¿QURK

t > T

ôRS…OYUžOYiADQ¿FOQZS[UbXIS6_žSTE

ξ⁰–9JROþbPRSTGaXHOJLK]GIPR¿QP`OQ¿QiAKAOQURºTOoGIS[¿QP`S6KAUL™³OYXaXIKÛô _S[E

t > T

ô

t⁻¹logkD^xΦ^t(ξ + ξ⁰)k ≤ t⁻¹log(kD^xΦ^tξk + kD^xΦ^tξ⁰k)

≤ t⁻¹log[2 max(kD^xΦ^tξk, kD^xΦ^tξ⁰k)]

= t⁻¹[max(logkD^xΦ^tξk, log kD^xΦ^tξ⁰k) + log 2]

≤ max(χ(x, ξ), χ(x, ξ⁰)) +  + t⁻¹log 2,

J`O\4PRK’i]OGaSf\4DFU`JROR˜
SkPRSfGšXHS@JRPRS€EISTiAOQº[KADQURKZDFGaXIEjOYU`Do\j—RS€_S[E?DF¿QURK

ϑ∈ R L(ϑ) :={ξ ∈ T^xM : χ(x, ξ)≤ ϑ}

ÿGaDQXaXIDbGa_žOYº[KAD/JRK

TxM^{˜ ¹}U_`OYEIXIK]\4DFiAOQEISoGaK†—`O½\j—RSGaS

ϑ⁰ < ϑ OQi>iADQEjO

L(ϑ⁰)⊂ L(ϑ)^ô

OYU`º[KÛôFGISˆSTGIKAGaXIS

ξ∈ L(ϑ)XHOYiASh\j—RS

χ(x, ξ) = ϑô”OYiAi>DFEHO

ξ /∈ L(ϑ⁰)ôbDc¸b¸FS[EHDkiÛàK>U`\[i>PžGaKADQURS ÿ&_REHDQ_`EIK]O½S

dim L(ϑ) > dim L(ϑ⁰)˜T”S&URSp\4DFU`\4iAP`JLSm\j—`S/OYi+¸cOYEHKAOQEIS&JLK

ξ 6= 0 ^K>U TxM i]O/þbP`OQUFXHKÓXjÁ

χ(x, ξ) OFGIGIPRZS

m≤ n ¸cOYiADQEHK“JRKAGaXIKAUbXIK

χ^∗₁(x) > . . . > χ^∗_m(x)^{˜ ¹}

GIDYXaXHDFGI_`OYºTK

Lk:= L(χ^∗_k), 1≤ k ≤ môR\[DFGaXIK>XIPRK]GI\[DQURDi]OR`iÓXHEHOQº[KADQURS€\[S[Ej\[OcXjOR–

TxM = L1(x) ⊃ . . . ⊃ L^m(x) ^\[DQU iAS½_REHDQ_REHK>S[XHÁ\j—RS

χ(x, ξ) = χ^∗_k(x) GIS/SWGIDQiADGIS

ξ ∈ Lk(x)\ L^k+1(x)^˜

gˆŸ`ŸRKAOQZDJRK>ZDFGaXIEjOcXHD2KAiNGIS[¿FPRS[UbXISF–

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GW#KXY

x∈ M^M



ùe÷Yø I ÷

G

÷ G õZ



ξ ^ TxM ø>÷@[\`÷

 ö

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NfY

ö#

m≤ n ^{I ÷cø#!G   ö  ö}

χ^∗₁(x) > χ^∗₂(x) > . . . > χ^∗_m(x);

 

ù2õ



öýõW\

 ÷ N øöG

÷gO

#!

õ

 TxM ^ m ^{# öÛö#J0h ÷gO M}

TxM := L1(x)⊃ L²(x)⊃ . . . ⊃ L^m(x), ^í³Àbï

ö ÷cøAõ

"$i

õ

ξ ∈ Lⁱ(x)\ Lⁱ⁺¹(x)⇒ χ(x, ξ) = χ^∗i(x), i = 1, . . . , m; ^íjQï

  

ù



÷ (e1, . . . , en) ^{\  ÷ekH÷  õQ } TxM ^{ö÷cø>õ "li õ h õ Gm#KXY} i = 1, . . . , m^{M ø  \cõ G#}

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  öý÷Yø>õmkj÷

 õ

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÷ Li(x)\ Lⁱ⁺¹(x) ^p νi= dim Li− dim Li+1

ù2qkøø

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÷

$ni

÷

Xn i=1

χ(x, ei)≤ Xn i=1

χ(x, fi),

#JI

õ (f1, . . . , fn) ^{p \  ÷rkj÷  õ[\`÷cø÷   } TxM^ù

Mœà“PRi>XIKAZDt_RP`UFXHDtGIS[¿FPRSmJROQi+™³OYXaXHDt\j—RS/iAO8Ÿ`OQGIS

(e1, . . . , en) —`O8_žSTEo\[DFGaXIEHPRº[KADQURSmK>i ZOY¿F¿QKADQE6UbP`ZoSTEID½_DFGHGaKAŸRKAi>S&JRKœ¸QS[XaXIDFEIKvU`S[¿QiAKœGI_`OYºTK9\4DFUtKAU`JLK]\4S&OQiÓXHD`ô1JRPRU`þbPRS&\4DFU

χ

Ÿ`OFGIGIDY”_S[EˆiÛàSfGaK]GaXIS[U`ºTOoJLK’XHOQi>S€ŸžOQGIS@GIK’EHK>U”¸”KAOoO½û¾áüۘ

,p®Få†È“±³ÜQ±Ç¯žÈ“®8¢Es

ø 

\cõ G#

νi= dim Li(x)− dim Lⁱ⁺¹(x) ^{p Qõ[öÇö}#ZDFiÓXHS[_RiAKA\[KÓXjÁ

 

χ^∗_i(x)^{ù U #!Y÷J #t o# øÓöGõ}

χi(x) := χ(x, ei),

#!I

õ (e1, . . . , en) ^{p kj÷  õt } TxM ^{"$#J ø>÷} hoGl#lhoG

õ[öC]TFõ

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 öõeõ

χ(x, ξ) ^{p FõNfYö# Q÷ _Cab ù}

wmx

 Y

õõ

Sp(x) :={χ¹(x), . . . , χn(x)},

p

Qõ[öÇö

#&GI_žS[XaXHEID

Fõ K ø

zy|{ w 

õ[ø h \  ö #

x∈ M^ù

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GI

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x GIK|_RPR&DQXaXHS[URSTEIS€PR¿FP`OYiAZS[UbXISe_REISTU`JLSTU`JLDKAU DQEjJLKAURS…\4K]OQGH\4P`URDJLS[K1¸áOQi>DFEIK|JRKAGaXIKAUbXIK

χ^∗_i(x), i = 1, . . . , mô`S…EIKA_S4XISTU`JLDFi>DPRUBU”PRZSTEID JLK|¸FDQi>XIS6_`OYEHK|OYiAi]OoGIP`OoZDQi>XIST_Ri>K]\4K>XHÁ

νi

˜

•

KAZOQURSpK>i?_REHDQŸ`i>STZOtJLK?þbP`OYU`JRDtKAiòi>KAZ-GIPR_ JLSTi>i]OJLS`U`K>ºTK>DFURS/_`EIDc¸”¸”KAGIDQEHK]O í³¾Fï

GIKAO2SS4XIXIKA¸áOQZS[UbXIS@PRUpi>KAZKÓXHS€S€þFP`K>U`JRK|GaK_RPRP`GHOYEHS6iAOoJLSžURK>ºTK>DFURSí

´ ï ˜ µ

U”PRU`\[KAOQZoD

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´

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(M, µ, Φ) ^{\ öýõ÷H ÷J M} M ^{÷ õ4öC] õ÷ ÷ ÷Qù}

qkøø

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÷ h õ G [\`÷

|#lK!^

x ∈ M ^{õ h õ} G#lK!^Z#

öÛö

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#

E ⊂ T^xM ^{õ  ö õ NfYö#€ø}

ø  öýõ

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Qõ

Of÷p÷

#lK!^

ξ∈ T^xM^ù

d†S[E?i]OJLK>ZDbGšXHEHOQº[KADQURS@GaKNEIKAU”¸bK]OoO/û¾cü

 0‚ ƒ

uaw’‡kˆz hRxœk`}šz|‚…„yˆz|{9uI}



rƒ

• ^{ì6iAK µ C¡M} χi(x) GaDFURDp\[DFGaXHOQUFXHKvJLS[ivZDYXHDY|PžGIOQU`JLDpi>SEHS[¿QDFi>SoJLKœ\4DFZ_žDbGaKAº[KADQURS JRS[i†&`PžGIGIDtSBJLSTi>i]OZOY_R_`OtXjOYUR¿FS[UbXIS½GaST¿QPRSB\j—RS

χ(Φ^t(x), DxΦ^tξ) = χ(x, ξ)

JRK>EHS4XIXHOYZSTUFXHS…JROYiAi]OoJRS`URKAº[KADQU`SQ˜

• d†STEBPRUGaK]GšXHS[ZO K>UbXHS[¿QEjOYŸ`K>iASQô@\j—RS—`O XIPLXIXIK2K€ZoDQXIKeþbP`OFGaK>óý_žSTEIKADLJLKA\[K€GaP XIDFEIK KAU”¸cOYEHKAOQUFXHKÇôL¿Fi>K

µ

C¡M GaDFURDoXIPLXIXIK|U”PRiAi>Kۘ

• d9EISTU`JLS[UžJLDPRU½¸QS[XaXIDFEIS6KAURK>ºTKAOQi>S€\TOQGIP`OYiAS€GaKXHEIDc¸cO

χ^∗₁(x)\4DQU½_REHDQŸ`OQŸRKAi>K>XHÁ

´ ˜

¹

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χ(x, ξ) = χ^∗₁(x)GaS@S6GIDQiAD2GIS

ξ∈ T^xM\ L²(x)^˜

V8OtGIKA\T\4DFZoSpOW_`OQEaXHS

L1(x) = TxM ¿Fi>KòOYi>XIEHK¡GIDYXIXIDbGa_`OQº[Kò\j—RS/\[DFGaXIK>XIPRK]GH\4DQU`D i]OT`i>XIEjOYº[KADQU`S/—`OQURURD \4DLJLKAZS[U`GIKADQURS½_DFGIKÓXHK>¸cOSBþFP`K>U`JRKˆZKAGIPREjOtJLKkM|S[ŸSTGI¿QP`S

n^óJLKAZS[U`GIK>DFU`OYiAS€U”PRiAiAO`ô`_REHSTGIDPRUB¸QS[XaXHDQEHS…\[OQGIP`OQi>SFô`\4DFUB_REIDFŸ`OYŸ`K>iAKÓXjÁ

´

GaKNXIEHDc¸cO

χ^∗₁(x)^˜

^ OYi6_RPRUbXHD JLK@¸”KAGaXHOØ\4DQZ_RPRXHOYºTK>DFU`OYiAS8OQU`\j—RS8GIKk_žOYEIXIK]GIGIS8\4DFU"PRU¸FS4XIXIDQEHS

ξ⁰

OQ_R_`OQEaXHS[URSTUFXHSe_S[E6STGIS[Z_RKAD/O

L2

OYiAi>DFEHOK†_REHKAZoK“S[EHEHDQEHKNU”PRZS[EHKA\[K¡íÇOYU`\j—`S2GIDQiAD

iÛàOYEHEIDQXIDQUžJROYZS[UbXHDbï’\[DQU`JLP`EIEHS[ŸRŸS[EHD?KAURS[¸”K>XHOYŸ`K>iAZS[UbXISòOYiL\[OQiA\[DQiADhJLK

χ^∗₁(x)^S9URDQU

JRK χ^∗₂(x)˜dSkPRSTGaXIDoÿkP`U&™³OcXIXID2™ŒDQU`J`OYZS[UbXHOQi>S6S@GšXjO2OQi>i]O…Ÿ`OFGaS6JLS[iAiÛàOQi>¿FDQEHKÓXHZoDoJLK

\TOYi]\4DFi>DJLS[¿Fi>K µ C¡Mœ˜

• ^LS µÿ?i]OkZK]GaP`EHO@JLK`MNS[ŸSTGI¿QPRSFôQOYiAiADQEjO Pn

i=1χi(x) = 0_S[EœþbP`OFGaKLDF¿QURK

x∈ M^˜

• LS€K>iNGIK]GšXHS[ZO

˙x = X(x)URDQUp—`O_RP`UFXHKNGaXHOYºTK>DFU`OYEHKÛôbOQi>iADQEjO

χ(x, X(x)) = 0^˜

• ^{^} OcXHD&PRU8GIK]GšXHS[ZO—`OQZoKAi>XIDQU`KAOQURD

(M, µ, Φ)^ôDc¸QS µÿ…i]OZKAGIPREjO&JLK“M|KADQPR¸”KAi>iAS…S

n ÿeK>i“U”PRZS[EHD&JLK1¿FEHOFJLKNJLK1i>KAŸS[EIXHÁRôiADmGa_S4XIXIEHDmJLST¿QiAK

µ

C¡Mؗ`O_S[E6DQ¿QU`K|_RPRUbXID

P`U`O_`OYEIXIK]\4DFiAOQEIS6GaKAZZS4XIEHK]OR–

Sp(x) ={χ¹(x), . . . , χn−1(x), χn(x),−χⁿ(x),−χⁿ−1(x), . . . ,−χ¹(x)},

JRDc¸QS@K|JLPRS€¸cOYiADQEHK|\4STUbXIEjOYiAK

χn(x) GaDFURDUbP`i>iAK’_žSTEˆi]O2_`EIDF_REIKAS4XjÁe_`EISf\4STJRS[UbXISF˜

¹ U2¿QSTURS[EjOYiASQôf_žSTE1P`UeGaK]GaXIS[ZOh—`OYZKAiÓXHDQURK]OYURDžôTiADkGI_`OQº[KADhJLS[iAiASœ™³OQGIKbURDQU…ÿ¡\4DFZo_žOcXaó

XHD`˜€d†DFGHGIDQURD&_S[EHmSfGIGIS[EHS2\[DQZ_`OYXaXIS2i>SGIPR_žSTE\4KvJLK†i>KA¸QSTi>iAD/JRS[iAiÇà—`OYZKAiÓXHDQURK]OYUžO

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