UNIVERSIDADE FEDERAL DE SÃO CARLOS
CENTRO DE CIÊNCIAS EXATAS E DE TECNOLOGIA
PROGRAMA DE PÓS GRADUAÇÃO EM MATEMÁTICA
❑❛r❡♥ ❘❡❣✐♥❛ P❛♥③❛r✐♥
❆♣❧✐❝❛çõ❡s ❡♥tr❡
3
✲✈❛r✐❡❞❛❞❡s ❙♦❧ ❡
♥ú♠❡r♦s ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ❝♦✐♥❝✐❞ê♥❝✐❛s
❙ã♦ ❈❛r❧♦s ✲ ❙P
❆❣♦st♦ ❞❡ ✷✵✶✼
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
❑❛r❡♥ ❘❡❣✐♥❛ P❛♥③❛r✐♥
❖r✐❡♥t❛❞♦r✿ Pr♦❢ ❉r✳ ❉❛♥✐❡❧ ❱❡♥❞rús❝♦❧♦
❆♣❧✐❝❛çõ❡s ❡♥tr❡
3
✲✈❛r✐❡❞❛❞❡s ❙♦❧ ❡
♥ú♠❡r♦s ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ❝♦✐♥❝✐❞ê♥❝✐❛s
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐✲ ❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✳
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡✱ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s✳ ❙❡♠ ❊❧❡✱ ♥❛❞❛ t❡r✐❛ s✐❞♦ ♣♦ssí✈❡❧✳
❊♠ ❡s♣❡❝✐❛❧✱ ❛❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ❱❡♥❞rús❝♦❧♦ ♣♦r t♦❞♦ ❛♣♦✐♦✱ ❛♠✐③❛❞❡✱ ♣❛❝✐ê♥❝✐❛✱ ♣r♦♥t✐❞ã♦ ❡ ❞❡❞✐❝❛çã♦✳
❆♦ Pr♦❢✳ ❉r✳ ❲❛❧❞❡❝❦ ❙❝❤üt③❡r ♣❡❧❛ ❢✉♥❞❛♠❡♥t❛❧ ❛❥✉❞❛ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✷✳✹✳✶✳
➚ ♣r♦❢✳ ❉r❛✳ ❆❧✐❝❡ ❑✐♠✐❡ ▼✐✇❛ ▲✐❜❛r❞✐ q✉❡ s❡ ♣r♦♥t✐✜❝♦✉ ❛ ♠❡ ❛❥✉❞❛r q✉❛♥❞♦ ♣r❡❝✐s❡✐✳
❆♦s ♠❡✉s ♣❛✐s ❡ ❛✈ó✱ ♣♦r t♦❞❛ ♦r❛çã♦✱ ❛♣♦✐♦ ❡ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧✳ ➚s ♠✐♥❤❛s ✐r♠ãs✱ ❡♠ ❡s♣❡❝✐❛❧ à ❑❛r❛♥✐✱ ♣♦r t♦❞❛ ❛ ♣❛❝✐ê♥❝✐❛ ❡ ❛♠✐③❛❞❡✳ ❊ ❛♦ ♠❡✉ ♥♦✐✈♦ ▲❡❛♥❞r♦✳ ❙❡♠ ❛ ❛❥✉❞❛ ❡ ♦ s✉♣♦rt❡ ❞❡ ✈♦❝ês✱ t❡r✐❛ s✐❞♦ ✐♠♣♦ssí✈❡❧✳
❘❡s✉♠♦
❙❡❥❛MA✉♠ t♦r✉s ❜✉♥❞❧❡ s♦❜r❡S1 ♦❜t✐❞♦ ✉s❛♥❞♦ ❝♦♠♦ ❛♣❧✐❝❛çã♦ ❞❡ ❝♦❧❛❣❡♠
✉♠❛ ♠❛tr✐③ ❞❡ ❆♥♦s♦✈ A✳ ◆❡st❡ tr❛❜❛❧❤♦ ❞✐s❝✉t✐♠♦s ❛♣❧✐❝❛çõ❡s ❡♥tr❡ ❞♦✐s t♦r✉s ❜✉♥❞❧❡s MA ❡ MD ❡ ❝❛❧❝✉❧❛♠♦s ♦ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ❝♦✐♥❝✐❞ê♥❝✐❛s ❞❡ ❛❧❣✉♠❛s ❢❛♠í❧✐❛s ❞❡
❆❜str❛❝t
▲❡tMA❜❡ t❤❡ t♦r✉s ❜✉♥❞❧❡ ♦✈❡rS1♦❜t❛✐♥❡❞ ✉s✐♥❣ ❛s ❣❧✉✐♥❣ ♠❛♣ ❛♥ ❆♥♦s♦✈
♠❛tr✐① A✳ ■♥ t❤✐s ✇♦r❦ ✇❡ ❞✐s❝✉ss ♠❛♣s ❢r♦♠ MA t♦ MD ❛♥❞ ❝♦♠♣✉t❡ t❤❡ ❝♦✐♥❝✐❞❡♥❝❡
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ Pr❡❧✐♠✐♥❛r❡s ✸
✶✳✶ ❚❡♦r✐❛ ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ❈♦✐♥❝✐❞ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❚♦r✉s ❜✉♥❞❧❡s s♦❜r❡ S1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✷ ❆♣❧✐❝❛çõ❡s ❡♥tr❡ t♦r✉s ❜✉♥❞❧❡s ✼
✷✳✶ ❆♣❧✐❝❛çõ❡s ❞❡ MAr ❡♠ MA ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✷✳✷ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✸ ❆♣❧✐❝❛çõ❡s ❞❡ MAr ❡♠ MAs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✹ ❆♣❧✐❝❛çõ❡s ❞❡ MA ❡♠ MD ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✸ ❆✉t♦✲❛♣❧✐❝❛çõ❡s ❡♥tr❡ s❛✜r❛s ✸✵
✸✳✶ ❖ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ❛✉t♦✲❛♣❧✐❝❛çõ❡s ❡♥tr❡ s❛✜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
■♥tr♦❞✉çã♦
❊♠ ✶✾✷✼✱ ❏✳ ◆✐❡❧s❡♥ ❞❡✜♥✐✉ ♣❛r❛ ✉♠❛ ❛♣❧✐❝❛çã♦ f ❞❛❞❛✱ ✉♠ ♥ú♠❡r♦ N(f)
♦ q✉❛❧ ❝❤❛♠❛♠♦s ❞❡ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥ ❞❡f✳ ❊st❡ ♥ú♠❡r♦ é ✉♠ ❧✐♠✐t❛♥t❡ ✐♥❢❡r✐♦r ♣❛r❛
♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ✜①♦s ❞❡ t♦❞❛s ❛s ❛♣❧✐❝❛çõ❡s ❤♦♠♦tó♣✐❝❛s ❛ f✳ ■ss♦ ❞❡✉ ♦r✐❣❡♠ ❛
t♦❞❛ ✉♠❛ t❡♦r✐❛ q✉❡✱ ❞❡s❞❡ ❡♥tã♦✱ ✈❡♠ s✐❞♦ ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ✈ár✐♦s ♣❡sq✉✐s❛❞♦r❡s ❡ ❢♦✐ ❣❡♥❡r❛❧✐③❛❞❛ ♣❛r❛ ❝♦✐♥❝✐❞ê♥❝✐❛s✳
◗✉❛♥❞♦ tr❛❜❛❧❤❛♠♦s ❝♦♠ t❡♦r✐❛ ❞❡ ◆✐❡❧s❡♥✱ ❞✉❛s q✉❡stõ❡s ❜ás✐❝❛s s❡ ❧❡✲ ✈❛♥t❛♠✿ ❛ ♣r✐♠❡✐r❛ é s❡ ♦ ❧✐♠✐t❛♥t❡ ✐♥❢❡r✐♦r ♦❜t✐❞♦ ♥❡ss❛ t❡♦r✐❛ é ♦ ♠❡❧❤♦r ♣♦ssí✈❡❧ ❡ ❛ s❡❣✉♥❞❛ s❡ r❡❢❡r❡ à q✉❡stã♦ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ♦✉ s❡❥❛✱ ❝♦♠♦ ❝❛❧❝✉❧❛r ♦s ♥ú♠❡r♦s ❞❡ ◆✐❡❧s❡♥✳ ❊st❡ tr❛❜❛❧❤♦ s❡ ✐♥s❡r❡ ♥♦ ❝♦♥t❡①t♦ ❞♦ ❡st✉❞♦ ❞❡ ❛♣❧✐❝❛çõ❡s ❡♥tr❡ ✈❛r✐❡❞❛❞❡s ❢❡❝❤❛❞❛s ❡ ❞♦ ❝á❧❝✉❧♦ ❞❡ ♥ú♠❡r♦s ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ❝♦✐♥❝✐❞ê♥❝✐❛s✳
❆ ❢❛♠í❧✐❛ ❞❛s ✈❛r✐❡❞❛❞❡s ❞❡ ❞✐♠❡♥sã♦ 3 ❝♦♠ ❣❡♦♠❡tr✐❛ ❙♦❧ ♣♦ss✉✐ ❞✉❛s
s✉❜❢❛♠í❧✐❛s✳ ❯♠❛ ❞❡❧❛s ❝♦♥s✐st❡ ❞♦s t♦r✉s ❜✉♥❞❧❡s ❝♦♠ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ❝♦❧❛❣❡♠ ❆♥♦s♦✈✱ ❡ ❛ ♦✉tr❛ ❝♦♥té♠ ♦s t♦r✉s s❡♠✐✲❜✉♥❞❧❡s ✭t❛♠❜é♠ ❝❤❛♠❛❞♦s ❞❡ ✈❛r✐❡❞❛❞❡s s❛✜r❛s✮✭✈❡r ❬▼♦✱ ❇●❱❪✮✳
❆♣❧✐❝❛çõ❡s ❡♥tr❡ t♦r✉s ❜✉♥❞❧❡s s♦❜r❡S1 ❡ ❛ t❡♦r✐❛ ❞❡ ◆✐❡❧s❡♥ ❞❡ss❡s ❡s♣❛ç♦s
tê♠ s✐❞♦ ❡st✉❞❛❞♦ ♣♦r ❞✐✈❡rs♦s ❛✉t♦r❡s ✭♣♦r ❡①❡♠♣❧♦ ❬❙❛✱ ❙❲❲✱ ●❲✱ ❱✐✱ ❏▲❪✮✳ ❊♠ ❛❧❣✉♥s ❞❡ss❡s tr❛❜❛❧❤♦s✱ ♦s ❛✉t♦r❡s ❡st❛✈❛♠ ♠❛✐s ♣r❡♦❝✉♣❛❞♦s ❝♦♠ ❛ ❞❡s❝r✐çã♦ ❞❛s ♣♦ssí✈❡✐s ❛♣❧✐❝❛çõ❡s ❡♥tr❡ ❡ss❡s ❡s♣❛ç♦s ✭❡s♣❡❝✐❛❧♠❡♥t❡ ❛s ❛♣❧✐❝❛çõ❡s ♥ã♦ tr✐✈✐❛✐s✮ ❡ ❡♠ ♦✉tr♦s ❡❧❡s t❡♥t❛r❛♠ ❝❛❧❝✉❧❛r ♦ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s✳
❱❛♠♦s ❞❡♥♦t❛r ♣♦r T ♦ ❚♦r♦ ♦❜t✐❞♦ ❝♦♠♦ ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ R×R
Z×Z✳ ❈♦♥✲ s✐❞❡r❡♠♦s A ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❞♦ ❚♦r♦✱ ✐♥❞✉③✐❞♦ ♣♦r ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡♠ R×R q✉❡ ♣r❡s❡r✈❛ Z×Z✳ P♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r A ❝♦♠ ✉♠❛ ♠❛tr✐③ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ✐♥t❡✐r♦s ❡
❞❡t❡r♠✐♥❛♥t❡1 ♦✉−1✳
❈♦♥str✉í♠♦s MA= T ×R
((x, y), t)∼(An(x, y), t−n) ♦♥❞❡ n ∈ Z✱ ♦ q✉❛❧ é ✉♠
t♦r✉s ❜✉♥❞❧❡ s♦❜r❡S1✳ ❙❡ A é ✉♠❛ ♠❛tr✐③ ❞❡ ❆♥♦s♦✈ ✭✐st♦ é✱ det(A) = 1 ❡|tr(A)|>2♦✉
det(A) =−1 ❡ tr(A)6= 0✮✱ ❡♥tã♦MA é ✉♠❛3✲✈❛r✐❡❞❛❞❡ ❝♦♠ ❣❡♦♠❡tr✐❛ ❙♦❧✳
■♥tr♦❞✉çã♦ ✷
❙❡❣✉✐♥❞♦ ❛❧❣✉♠❛s ✐❞❡✐❛s ❞❡ ❬●❲❪ ❞✐s❝✉t✐♠♦s ❛♣❧✐❝❛çõ❡s ❡♥tr❡ ❛s ✈❛r✐❡❞❛❞❡s
MA❡MD ❡ ❝❛❧❝✉❧❛♠♦s ♦ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ❝♦✐♥❝✐❞ê♥❝✐❛s ❞❡ss❛s ❛♣❧✐❝❛çõ❡s t❛♥t♦ ♣❛r❛
♦ ❝❛s♦ ♦r✐❡♥tá✈❡❧ ❝♦♠♦ ♣❛r❛ ♦ ❝❛s♦ ♥ã♦✲♦r✐❡♥tá✈❡❧✳ ❈♦♥s❡❣✉✐♠♦s ❝♦♥❝❧✉✐r ❝♦♠♣❧❡t❛♠❡♥t❡ ♦s ❝❛s♦sMAr →MA❡MAr →MAs ❛♣r❡s❡♥t❛❞♦s ♥❛s s❡çõ❡s ❞♦ ❈❛♣ít✉❧♦ ✷ ❡ ♠❛✐s ❛❧❣✉♠❛s
s✐t✉❛çõ❡s✳ ❖ ❝❛s♦ ❣❡r❛❧ ✭s❡çã♦ ✷✳✹✮ s❡❣✉❡ ❡♠ ❡st✉❞♦✳ ❆❧é♠ ❞✐ss♦✱ ✉s❛♥❞♦ ♦ ❢❛t♦ q✉❡ ❛s ✈❛r✐❡❞❛❞❡s s❛✜r❛s ✭t♦r✉s s❡♠✐✲❜✉♥❞❧❡s✮ t❡♠ ✉♠ r❡❝♦❜r✐♠❡♥t♦ ❞✉♣❧♦ ♣♦r t♦r✉s ❜✉♥❞❧❡s✱ t❛♠❜é♠ ❝❛❧❝✉❧❛♠♦s ♦ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ❛✉t♦✲❛♣❧✐❝❛çõ❡s ❞❡ss❛s ✈❛r✐❡❞❛❞❡s✳
❊st❡ tr❛❜❛❧❤♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❡♠ três ❝❛♣ít✉❧♦s✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s✱ ✐♥❝❧✉✐♥❞♦ ✉♠❛ ✐♥tr♦❞✉çã♦ à t❡♦r✐❛ ❞❡ ◆✐❡❧✲ s❡♥ ♣❛r❛ ❝♦✐♥❝✐❞ê♥❝✐❛s ❡ ❞♦ ✜❜r❛❞♦ q✉❡ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ t♦r✉s ❜✉♥❞❧❡✳ ◆♦ ❝❛♣ít✉❧♦ ✷ ❞❡s❝r❡✈❡♠♦s ❛s ♣♦ssí✈❡✐s ❛♣❧✐❝❛çõ❡s ❡♥tr❡ ❞♦✐s t♦r✉s ❜✉♥❞❧❡s MA❡ MD✱ ❝♦♠❡ç❛♥❞♦ ❝♦♠ ♦
❈❛♣ít✉❧♦ ✶
Pr❡❧✐♠✐♥❛r❡s
✶✳✶✳ ❚❡♦r✐❛ ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ❈♦✐♥❝✐❞ê♥❝✐❛s
◆❡st❛ s❡çã♦ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ à t❡♦r✐❛ ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ❝♦✐♥❝✐✲ ❞ê♥❝✐❛s✱ ❝♦❧♦❝❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s q✉❡ ❥✉❧❣❛♠♦s s❡r❡♠ ♠❛✐s ✐♠♣♦rt❛♥t❡s ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ s❡rã♦ ♥❡❝❡ssár✐♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳
❙❡❥❛♠f, g :M →N ❛♣❧✐❝❛çõ❡s ❡♥tr❡ ✈❛r✐❡❞❛❞❡s ❢❡❝❤❛❞❛s ✭❝♦♠♣❛❝t❛s ❡ s❡♠
❜♦r❞♦✮ ❡ ❞❡ ♠❡s♠❛ ❞✐♠❡♥sã♦✳
❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❯♠❛ ❝♦✐♥❝✐❞ê♥❝✐❛ ❞❡ f ❡ g é ✉♠ ♣♦♥t♦ x∈M t❛❧ q✉❡ f(x) = g(x)✳
í♥❞✐❝❡ ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛ ❉❡♥♦t❛r❡♠♦s ♣♦r Coin(f, g) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s
❝♦✐♥❝✐❞ê♥❝✐❛s ❞❡M✱ ♦✉ s❡❥❛✱
Coin(f, g) ={x∈M;f(x) =g(x)}.
❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❉❛❞♦s ❞♦✐s ♣♦♥t♦sx1, x2❡♠Coin(f, g)✱ ❞✐③❡♠♦s q✉❡x1❡x2 sã♦ ◆✐❡❧s❡♥
❡q✉✐✈❛❧❡♥t❡s ❝♦♠ r❡❧❛çã♦ ❛ f ❡ g s❡ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ γ : [0,1]→M t❛❧ q✉❡ γ(0) =x1✱
γ(1) =x2 ❡ f ◦γ é ❤♦♠♦tó♣✐❝❛ ❛ g◦γ r❡❧❛t✐✈❛♠❡♥t❡ ❛♦s ♣♦♥t♦s ✜♥❛✐s✳
❊ss❛ r❡❧❛çã♦ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡ ❛ss✐♠✱ ♣♦❞❡♠♦s ♣❛rt✐❝✐♦♥❛r ♦ ❝♦♥❥✉♥t♦ Coin(f, g) ❡♠ ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ss❛ r❡❧❛çã♦✱ ❝❤❛♠❛❞❛s ❞❡ ❝❧❛ss❡s ❞❡
❝♦✐♥❝✐❞ê♥❝✐❛s ♦✉ ❝❧❛ss❡s ❞❡ ◆✐❡❧s❡♥ ❞♦ ♣❛r(f, g)✳
❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ♥♦çã♦ ❞❡ í♥❞✐❝❡ ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛✳ ❱❛♠♦s ❝✐t❛r ❛ ❞❡✜♥✐çã♦ ❞❡ ❬❱❦❪ ♣❛r❛ ❛♣❧✐❝❛çõ❡s ❡♥tr❡ ✈❛r✐❡❞❛❞❡s ♦r✐❡♥tá✈❡✐s ❡✱ ❡♠ s❡❣✉✐❞❛✱ ❛ ❞❡✜♥✐çã♦ ❞♦ ✧í♥❞✐❝❡✧ ♣❛r❛ ✈❛r✐❡❞❛❞❡s ♥ã♦✲♦r✐❡♥tá✈❡✐s✳ ❊ss❛ ❣❡♥❡r❛❧✐③❛çã♦ ❢♦✐ ❢❡✐t❛ ❡♠ ❬❉❏❪✳
❉❡✜♥✐çã♦ ✶✳✶✳✸✳ ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s ❢❡❝❤❛❞❛s✱ ♦r✐❡♥tá✈❡✐s ❡ ❞❡ ♠❡s♠❛ ❞✐♠❡♥sã♦✱ W ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡ M ❡ f, g : W → N ❛♣❧✐❝❛çõ❡s t❛✐s q✉❡ C = Coin(f, g)
✶✳✶✳ ❚❡♦r✐❛ ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ❈♦✐♥❝✐❞ê♥❝✐❛s ✹
s❡❥❛ ❝♦♠♣❛❝t♦✳ ❉❡✜♥✐♠♦s ♦ í♥❞✐❝❡ ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛ ❞♦ ♣❛r (f, g) ❡♠ W✱ ind(f, g : W)✱
❝♦♠♦ s❡♥❞♦ ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞❛❞♦ ♣❡❧❛ ✐♠❛❣❡♠ ❞❛ ❝❧❛ss❡ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ M ❛tr❛✈és ❞❛
❝♦♠♣♦s✐çã♦
Hn(M) → Hn(M, M−V) → Hn(W, W −V)
→ Hn(N ×N, N×N −∆(N))≈Z ♦♥❞❡ ❛ s❡❣✉♥❞❛ ❛♣❧✐❝❛çã♦ é ❛ ❡①❝✐sã♦✱ ❛ t❡r❝❡✐r❛ é ✐♥❞✉③✐❞❛ ♣♦r (f, g)(x) = (f(x), g(x))✱ ❡
V s❛t✐s❢❛③ Coin(f, g)⊂V ⊂V¯ ⊂W✳
❖ í♥❞✐❝❡ ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛ s❛t✐s❢❛③ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❡♥tr❡ ❡❧❛s✱ ❛ ❛❞✐✲ t✐✈✐❞❛❞❡ ❬s❡ W é ✉♠❛ ✉♥✐ã♦ ✜♥✐t❛ ❞❡ ❝♦♥❥✉♥t♦s ❛❜❡rt♦s Wi✱ i = 1, . . . , r✱ ❡ C é ✉♠❛
✉♥✐ã♦ ❞✐s❥✉♥t❛ ❞❡ ❝♦♥❥✉♥t♦s ❝♦♠♣❛❝t♦sCi ❝♦♠ Ci ⊂ Wi✱ ❡♥tã♦ ind(f, g:W) =ind(f1, g1:
W1) +. . .+ind(fr, gr :Wr), ♦♥❞❡ (fj, gj) = (f|Wj, g|Wj)❪ ✱ ❛ ❧♦❝❛❧✐③❛çã♦ ❬ind(f, g :W) =
ind(f|W′, g|W′ : W′) ♣❛r❛ t♦❞♦ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ W′ ⊂ W ❝♦♥t❡♥❞♦ C❪ q✉❡ ♣❡r♠✐t❡ ♥♦s r❡❢❡r✐r♠♦s ❛♦ í♥❞✐❝❡ ❝♦♠♦ ❵♦ í♥❞✐❝❡ ❡♠C✬ ❛♦ ✐♥✈és ❞❡ ❵♦ í♥❞✐❝❡ ❡♠W✬✱ ❡ ❛ ✐♥✈❛r✐â♥❝✐❛ ♣♦r
❤♦♠♦t♦♣✐❛ ❬s❡ft, gt:W →N✱0≤t≤1✱ sã♦ ❤♦♠♦t♦♣✐❛s t❛✐s q✉❡K ={x∈W; ❡①✐st❡ t∈
[0,1] ❝♦♠ ft(x) = gt(x)} é ❝♦♠♣❛❝t♦✱ ❡♥tã♦ ind(Coin(f0, g0)) =ind(Coin(f1, g1))❪✳
P❛r❛ ❣❡♥❡r❛❧✐③❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ í♥❞✐❝❡ ♣❛r❛ ✈❛r✐❡❞❛❞❡s ♥ã♦✲♦r✐❡♥tá✈❡✐s✱ ♦s ❛✉t♦r❡s ❞❡ ❬❉❏❪ ✐♥tr♦❞✉③✐r❛♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ s❡♠✐✲í♥❞✐❝❡ ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ◆✐❡❧s❡♥✳ P❛r❛ ✐ss♦✱ ❞❡✜♥✐r❛♠ ✉♠❛ r❡❧❛çã♦ ❞❡ r❡❞✉③✐❜✐❧✐❞❛❞❡ ♥❛ ❝❧❛ss❡ ❞❡ ◆✐❡❧s❡♥ ❡✱ ❝♦♠ ❡ss❛ r❡❧❛çã♦✱ ✜③❡r❛♠ ✉♠❛ ❝✐sã♦ ♥❡st❛ ❝❧❛ss❡ ❡♠ ♣❛r❡s ❞❡ ♣♦♥t♦s✳ ❆♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s q✉❡ r❡st❛r❛♠✱ ❝❤❛♠❛r❛♠ ❞❡ s❡♠✐✲í♥❞✐❝❡✳ Pr♦✈❛r❛♠ q✉❡ ❡ss❡ ♥ú♠❡r♦ é ✉♠ ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦ ❡ ✐♥✈❛r✐❛♥t❡ ♣♦r ❤♦♠♦t♦♣✐❛✱ ❡ ♣♦rt❛♥t♦ ♣✉❞❡r❛♠ ✉sá✲❧♦✱ ❝♦♠♦ ♥♦ ❝❛s♦ ♦r✐❡♥tá✈❡❧✱ ♣❛r❛ ♦❜t❡r ♦ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♦s r❡s✉❧t❛❞♦s ❞❡♠♦♥str❛❞♦s ❢♦r❛♠ ♦s q✉❡ ❝✐t❛♠♦s ❛❜❛✐①♦✳
❖ ❝♦♥t❡①t♦ ❞♦ tr❛❜❛❧❤♦ ❬❉❏❪ é ♣❛r❛ ✈❛r✐❡❞❛❞❡s ❢❡❝❤❛❞❛s✱ s✉❛✈❡s✱ ❝♦♥❡①❛s ❡ ❞❡ ♠❡s♠❛ ❞✐♠❡♥sã♦✳ P♦ré♠✱ ❡♠ ❬❏❡✷❪ ❢♦✐ ❣❡♥❡r❛❧✐③❛❞♦ ♣❛r❛ ✈❛r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s✳ ❆ r❡❧❛çã♦ ❞❡ r❡❞✉③✐❜✐❧✐❞❛❞❡ ♣❛rt❡ ❞❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿
❉❡✜♥✐çã♦ ✶✳✶✳✹ ✭✭✶✳✷✮✱ ❬❉❏❪✮✳ ❉✐③❡♠♦s q✉❡x ❡ y s❡ r❡❞✉③❡♠ ✉♠ ❛♦ ♦✉tr♦ s❡✱ ❡ s♦♠❡♥t❡
s❡✱ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ ω ❞❡ x ❛ y t❛❧ q✉❡ f ω ≃ gω ❡ t❛❧ q✉❡ ω r❡✈❡rt❡ ❛ ♦r✐❡♥t❛çã♦ ♥♦
✶✳✷✳ ❚♦r✉s ❜✉♥❞❧❡s s♦❜r❡ S1 ✺
❈♦♥s✐❞❡r❡ A⊂Coin(f, g)✳ P♦❞❡♠♦s ❡s❝r❡✈❡r
A={a1, b1, . . . , ak, bk;c1, . . . , cs},
♦♥❞❡ ai, bi s❡ r❡❞✉③❡♠ ✉♠ ❛♦ ♦✉tr♦ ❡ ✐ss♦ ♥ã♦ ❛❝♦♥t❡❝❡ ♣❛r❛ ♦s ♣❛r❡s ci, cj✱ i 6= j✳ ❈❤❛✲
♠❛♠♦s ❞❡ ❧✐✈r❡s ♦s ❡❧❡♠❡♥t♦s {c1, . . . , cs} ♥❡ss❛ ❞❡❝♦♠♣♦s✐çã♦✳
▲❡♠❛ ✶✳✶✳✺ ✭✭✶✳✸✮✱❬❉❏❪✮✳ ❖ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❧✐✈r❡s ✐♥❞❡♣❡♥❞❡ ❞❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡
A✳
❆ss✐♠✱ ♦s ❛✉t♦r❡s ❞❡✜♥✐r❛♠ ♦ s❡♠✐✲✐♥❞í❝❡ ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ◆✐❡❧s❡♥ A ♣❛r❛
✉♠ ♣❛r ❞❡ ❛♣❧✐❝❛çõ❡s (f, g) : M → N✱ ❝♦♠♦ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❧✐✈r❡s ❡♠ q✉❛❧q✉❡r
❞❡❝♦♠♣♦s✐çã♦ ❡ ❞❡♥♦t❛r❛♠✲♥♦ ♣♦r |ind|(f, g : A)✳ ❖ r❡s✉❧t❛❞♦ ❛❜❛✐①♦ r❡❧❛❝✐♦♥❛ í♥❞✐❝❡ ❡
s❡♠✐✲í♥❞✐❝❡ ❡ ❛ ❞❡✜♥✐çã♦ ❡♠ s❡❣✉✐❞❛✱ s❡ r❡❢❡r❡ às ❝❧❛ss❡s ❡ss❡♥❝✐❛✐s ❡ ❛♦ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥✳
▲❡♠❛ ✶✳✶✳✻ ✭✭✶✳✻✮✱❬❉❏❪✮✳ ❙❡❥❛♠ (f, g) : M → N ✉♠ ♣❛r ❞❡ ❛♣❧✐❝❛çõ❡s ❡♥tr❡ ✈❛r✐❡❞❛❞❡s
♦r✐❡♥tá✈❡✐s✱ ❡ A⊂Coin(f, g)✉♠❛ ❝❧❛ss❡ ❞❡ ◆✐❡❧s❡♥✳ ❊♥tã♦ |ind|(f, g:A) =|ind(f, g:A)| ♦♥❞❡ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡♥♦t❛ ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞♦ í♥❞✐❝❡ ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛ ✉s✉❛❧✳
❉❡✜♥✐çã♦ ✶✳✶✳✼✳ ❙❡❥❛ (f, g) : M → N ✉♠ ♣❛r ❞❡ ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s✳ ❉✐③❡♠♦s q✉❡
✉♠❛ ❝❧❛ss❡ ❞❡ ◆✐❡❧s❡♥ é ❡ss❡♥❝✐❛❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ s❡✉ í♥❞✐❝❡ ✭♦✉ s❡♠✐✲í♥❞✐❝❡✱ ♣❛r❛ ♦ ❝❛s♦ ♥ã♦✲♦r✐❡♥tá✈❡❧✮ é ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ ❖ ♥ú♠❡r♦ ❞❡ ❝❧❛ss❡s ❞❡ ◆✐❡❧s❡♥ ❡ss❡♥❝✐❛✐s é ❝❤❛♠❛❞♦ ❞❡ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥ ❡ ❞❡♥♦t❛❞♦ ♣♦rN(f, g)✳
❚❡♠♦s q✉❡ N(f, g) é ✉♠ ✐♥✈❛r✐❛♥t❡ ❤♦♠♦tó♣✐❝♦✱ ✜♥✐t♦ ❡ é ✉♠ ❧✐♠✐t❛♥t❡
✐♥❢❡r✐♦r ♣❛r❛ ♦ ❝♦♥❥✉♥t♦Coin(f′, g′) ❞❛s ❛♣❧✐❝❛çõ❡sf′ ❡ g′ ❤♦♠♦tó♣✐❝❛s ❛ f ❡ g✱ r❡s♣❡❝t✐✲ ✈❛♠❡♥t❡✳
✶✳✷✳ ❚♦r✉s ❜✉♥❞❧❡s s♦❜r❡
S
1◆❡st❛ s❡çã♦ ❢❛r❡♠♦s ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ❞♦ ✜❜r❛❞♦ q✉❡ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ t♦r✉s ❜✉♥❞❧❡✱ ♦ q✉❛❧ ♣♦ss✉✐ ✜❜r❛ ❚♦r♦ ❡ ❜❛s❡S1✳ ➱ ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ♠❛✐s ❞❡t❛❧❤❡s ❞❡st❡
✶✳✷✳ ❚♦r✉s ❜✉♥❞❧❡s s♦❜r❡ S1 ✻
❱❛♠♦s ❞❡♥♦t❛r ♣♦r T ♦ ❚♦r♦ ♦❜t✐❞♦ ❝♦♠♦ ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ R×R
Z×Z✳ ❈♦♥✲ s✐❞❡r❡♠♦s A ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❞♦ ❚♦r♦✱ ✐♥❞✉③✐❞♦ ♣♦r ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡♠ R×R q✉❡ ♣r❡s❡r✈❛ Z×Z✳ P♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r A ❝♦♠ ✉♠❛ ♠❛tr✐③ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ✐♥t❡✐r♦s ❡
❞❡t❡r♠✐♥❛♥t❡1 ♦✉ −1✳ ❙❡ T →M →p S1 é ✉♠ ✜❜r❛❞♦ ❝♦♠ ❜❛s❡ S1 ❡ ✜❜r❛ ❚♦r♦✱ ❡♥tã♦ ♦
❡s♣❛ç♦ t♦t❛❧M é ❞❛❞♦ ♣♦r✿
M =MA= T ×R
((x, y), t)∼(An(x, y), t−n),
♦♥❞❡n ∈Z✳
❆ ❛♣❧✐❝❛çã♦ ❞❡ ♣r♦❥❡çã♦✱p✱ é ❞❛❞❛ ♣♦rp[((x, y), t)] = [t]∈ R Z ≃
[0,1] 0∼1 ≃S
1✳
❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❯♠❛ ♠❛tr✐③A ❡♠ GL(2,Z)é ❝❤❛♠❛❞❛ ✉♠❛ ♠❛tr✐③ ❞❡ ❆♥♦s♦✈ s❡ ✉♠❛
❞❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ❢♦r s❛t✐s❢❡✐t❛✿
✭✐✮ det(A) = 1 ❡ |tr(A)|>2❀
✭✐✐✮ det(A) =−1 ❡ tr(A)6= 0✳
❙❡Aé ✉♠❛ ♠❛tr✐③ ❞❡ ❆♥♦s♦✈✱ ❡♥tã♦ MA é ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❝♦♠ ❣❡♦♠❡tr✐❛
❈❛♣ít✉❧♦ ✷
❆♣❧✐❝❛çõ❡s ❡♥tr❡ t♦r✉s ❜✉♥❞❧❡s
❖ ♥♦ss♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ é ❝♦♥s❡❣✉✐r ❝❛❧❝✉❧❛r ♦ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥ ♣❛r❛ ✉♠ ♣❛r ❞❡ ❛♣❧✐❝❛çõ❡s (f, g) ❡♥tr❡ ❞♦✐s t♦r✉s ❜✉♥❞❧❡s MA ❡ MD ❝♦♠ A ❡ D ♠❛tr✐③❡s ❞❡
❆♥♦s♦✈ q✉❛✐sq✉❡r ❡♠ GL(2,Z)✳ ❈♦♥s✐❞❡r❛♠♦s (f′, g′) : T → T ❛ ❛♣❧✐❝❛çã♦ ✐♥❞✉③✐❞❛ ♥❛ ✜❜r❛ t❛❧ q✉❡ f′
# = B✱ g#′ = C ❡ ( ¯f ,g¯) : S1 → S1 ❛ s✉❛ ✐♥❞✉③✐❞❛ ♥❛ ❜❛s❡✳ ❆ ♣r✐♥❝í♣✐♦
❝♦♠❡ç❛♠♦s ❛♥❛❧✐s❛♥❞♦ s✐t✉❛çõ❡s ❝♦♠ ❛❧❣✉♠❛s ❤✐♣ót❡s❡s r❡str✐t✐✈❛s✱ ❛s q✉❛✐s ❛♣r❡s❡♥t♦ ♥❛s s❡çõ❡s s❡❣✉✐♥t❡s✳ ❖ ♦❜❥❡t✐✈♦ ♥ã♦ ❢♦✐ ❛❧❝❛♥ç❛❞♦ ❡♠ s✉❛ t♦t❛❧✐❞❛❞❡✱ ❛✐♥❞❛ ❡st❛♠♦s ❝♦♠ ❛❧❣✉♥s ❝❛s♦s ❡♠ ❛❜❡rt♦ q✉❡ s❡❣✉❡♠ ❡♠ ❡st✉❞♦✳
✷✳✶✳ ❆♣❧✐❝❛çõ❡s ❞❡
M
Ar❡♠
M
A❱❛♠♦s ❝♦♠❡ç❛r ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛ ♠❛tr✐③ ❞❡ ❝♦❧❛❣❡♠ ❞♦ ❞♦♠í♥✐♦ é ✉♠❛ ♣♦tê♥❝✐❛ ❞❛ ♠❛tr✐③ ❞❡ ❝♦❧❛❣❡♠ ❞♦ ❝♦♥tr❛❞♦♠í♥✐♦✳ P❛r❛ ❡ss❡ ❝❛s♦ ♠♦str❛r❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r✱ ❞❛❞❛ ♣❡❧❛ ♠❛tr✐③ B ❞❡ s✉❛ ✐♥❞✉③✐❞❛ ♥♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧✱ ♣♦ss✉✐ ✉♠❛
❢♦r♠❛ ❡s♣❡❝í✜❝❛ s❡ deg ¯f =±r✱ ♦♥❞❡ r é t❛❧ ♣♦tê♥❝✐❛✱ ❡ q✉❡ é ♥✉❧❛ s❡ deg ¯f 6=±r✱ ♦ q✉❡
♥♦s ❞✐③ q✉❛✐s sã♦ ❛s ♣♦ssí✈❡✐s ❛♣❧✐❝❛çõ❡s ❞❡MAr ❡♠ MA✳
❙❡❥❛♠
f, g:MAr = T ×R
((x, y), t)∼((Ar)n(x, y), t−n) →MA=
T ×R
((x, y), t)∼(An(x, y), t−n)✱
❝♦♠ n∈Z✳
❚❡♠♦s q✉❡ ❛s ❛♣❧✐❝❛çõ❡sf ❡ g sã♦ ❤♦♠♦tó♣✐❝❛s à ❛♣❧✐❝❛çõ❡s q✉❡ ♣r❡s❡r✈❛♠
✜❜r❛ ❬❬❏❡✷❪✱✭✺✳✹✮❪ ❡✱ ♣♦rt❛♥t♦✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦✿
T
g′
f′
/
/MAr
g f
/
/S1
¯
g
¯
f
T //MA //S1
✭✷✳✶✮
✷✳✶✳ ❆♣❧✐❝❛çõ❡s ❞❡ MAr ❡♠ MA ✽
❙❡ f′ : T → T ❡ f¯: S1 →S1 sã♦ t❛✐s q✉❡ f′
# =B ❡ deg ¯f =k ❡♥tã♦✱ ♣❡❧♦
❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦ ❛❝✐♠❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡rf :MAr →MA ❝♦♠♦
f[((x, y), t)] = [(B(x, y) +η[((x, y), t)], kt)],
♦♥❞❡η :MAr →T é t❛❧ q✉❡η[((x, y),0)] = (0,0) ❡ [(,)] ❞❡♥♦t❛ ❛ ❝❧❛ss❡ ♥♦ q✉♦❝✐❡♥t❡✳
◆♦t❡♠♦s q✉❡ ❝♦♠♦ηé ♥✉❧❛ ❡♠t= 0✱ ❛ ❛♣❧✐❝❛çã♦ηé ❤♦♠♦tó♣✐❝❛ ❛ ❛♣❧✐❝❛çã♦
❝♦♥st❛♥t❡ ❡♠ ❝❛❞❛ ✐♥st❛♥t❡ t✱ ♦✉ s❡❥❛✱ ηt : T → T é ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ♥✉❧❛✱ ❛♣❡s❛r ❞❡ η:MAr →T ♥ã♦ ♦ s❡r✳ ❚❡♠♦s q✉❡✱ ❛ ♠❡♥♦s ❞❡ ❤♦♠♦t♦♣✐❛✱ ❛ ❢✉♥çã♦ η s❡ ❢❛t♦r❛ ♣♦r S1✳
❉❡ss❡ ♠♦❞♦✱ ❡❧❛ ♥ã♦ ✐♥✢✉❡♥❝✐❛ ♥♦ ❝á❧❝✉❧♦ ❞♦s ♥ú♠❡r♦s ❞❡ ◆✐❡❧s❡♥ ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛s✱ ❥á q✉❡ t❛❧ ❝♦♥t❛ é ❢❡✐t❛ ♥♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛s ✜❜r❛s✱ ♦♥❞❡ηt é ♥✉❧❛✳
❆❧é♠ ❞✐ss♦✱ f s❡ ❧❡✈❛♥t❛ ❛ ✉♠ r❡❝♦❜r✐♠❡♥t♦ f˜ : T ×R → T × R ❞❛❞♦ ♣♦r f˜((x, y), t) = (f′
[t](x, y), kt)✱ ♦♥❞❡ f′ : T → T é ❞❛❞❛ ♣♦r f′(x, y) = (x0, y0) s❡
f[((x, y),0)] = [((x0, y0),0)]✳ ❚❛❧ f′ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣♦✐s [((x1, y1),0)] = [((x2, y2),0)]
s❡✱ ❡ s♦♠❡♥t❡ s❡✱(x1, y1) = (x2, y2)✳
❉❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ q✉❡ ❞❡✜♥❡ ❛ ✈❛r✐❡❞❛❞❡✱ ♦❜t❡♠♦s q✉❡((x, y),0)∼
(A−r(x, y),1).
❆ss✐♠✱ f˜((x, y),0) = (f′
[0](x, y),0) = (B(x, y),0) ❡ f˜(A−r(x, y),1) =
(f′
[1](A−r(x, y)), k) = (B(A−r(x, y)), k) ∼ (AkBA−r(x, y),0)✳ ❊✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❞❡✲
✈❡♠♦s t❡r B =AkBA−r✱ ♦✉ s❡❥❛✱
BAr =AkB.
❖❜s❡r✈❡♠♦s ❛✐♥❞❛✱ q✉❡ ❛ ❛♣❧✐❝❛çã♦ f ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ✭s❡ ♣❛ss❛ ❛♦ q✉♦✲
❝✐❡♥t❡✮✱ ♣♦✐s✿ ❞❛❞♦ (x, y)∈ T✱ t❡♠♦s q✉❡ [((x, y),0)] = [(A−r(x, y),1)]✱ ❧♦❣♦ ❞❡✈❡♠♦s t❡r f[((x, y),0)] = f[(A−r(x, y),1)]✳
P♦✐s ❜❡♠✱ f[((x, y),0)] = [(B(x, y) + η((x, y),0),0)] = [(B(x, y),0)] ❡
f[(A−r(x, y),1)] = [(BA−r(x, y) +η(A−r(x, y),1), k)] = [(B(A−r(x, y)), k)]✳ ❱❡❥❛♠♦s q✉❡
❡ss❡s ❞♦✐s ♣♦♥t♦s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿
(BA−r(x, y), k)∼(Ak(BA−r(x, y), k−k)) = (AkBA−r(x, y),0).
▼❛s✱ ❝♦♠♦ AkB = BAr✱ ❡♥tã♦ AkBA−r(x, y) = BArA−r(x, y) = B(x, y)✱
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✾
❞❡✜♥✐❞❛✳
❯♠ r❛❝✐♦❝í♥✐♦ t♦t❛❧♠❡♥t❡ ❛♥á❧♦❣♦ ♣♦❞❡ s❡r ❢❡✐t♦ ♣❛r❛ ❛ ❛♣❧✐❝❛çã♦g✳
❆ ♣❛rt✐r ❞❡ ❛❣♦r❛✱ ❞❡♥♦t❛r❡♠♦sf′
#=B ❡ g#′ =C✳
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛
P♦❞❡♠♦s ❞❡t❡r♠✐♥❛r q✉❛❧ ♠❛tr✐③ é ✐♥❞✉③✐❞❛ ❡♠ ❝❛❞❛ ✜❜r❛✱ ♦ q✉❡ ♥♦s ❛❥✉❞❛rá ♥♦ ❝á❧❝✉❧♦ ❞♦ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥ ❞♦ ♣❛r (f, g)✳ ❖ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ✉s❛r ♠❡❝❛♥✐s♠♦s ❥á
❝♦♥❤❡❝✐❞♦s ♣❛r❛ ❢❛❝✐❧✐t❛r ♦s ❝á❧❝✉❧♦s ❞❡ss❡ ♥ú♠❡r♦ ❡ ✉♠ ❞❡❧❡s ♥♦s ♣❡r♠✐t❡ ❢❛③❡r t♦❞♦s ♦s ❝á❧❝✉❧♦s ♥❛s ✜❜r❛s ❛tr❛✈és ❞♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ♠❛tr✐③❡s ❛♣r♦♣r✐❛❞❛s✳ ❆ss✐♠✱ ♥♦s é ✐♥t❡r❡ss❛♥t❡ ❞❡t❡r♠✐♥❛r t❛✐s ♠❛tr✐③❡s✳
P❛r❛ ✐ss♦✱ ✈❛♠♦s ✉s❛r ❛s r❡❧❛çõ❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ q✉❡ ❞❡✜♥❡♠ ❛s ✈❛r✐❡❞❛❞❡s✳ P❛r❛ s✐♠♣❧✐✜❝❛r✱ ❞❛q✉✐ ❡♠ ❞✐❛♥t❡ ✐❞❡♥t✐✜❝❛r❡♠♦s ❛ ❛♣❧✐❝❛çã♦ f ❝♦♠ ❛ ❛♣❧✐✲
❝❛çã♦ ❧✐♥❡❛r ❞❛❞❛ ♣❡❧❛ ♠❛tr✐③ ❞❛ s✉❛ ✐♥❞✉③✐❞❛ ♥♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧✳
❈♦♠♦ deg ¯f =k ❡ ❧❡♠❜r❛♥❞♦ q✉❡ η é ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ♥✉❧❛ ❡♠ ❝❛❞❛ ✐♥s✲
t❛♥t❡t✱ t❡♠♦s q✉❡✿
˜
f((x, y),0) = (B(x, y),0)✐♠♣❧✐❝❛ q✉❡ ❡♠ t= 0✱ f′
[0] é ❞❛❞❛ ♣❡❧❛ ♠❛tr✐③B❀
˜
f
(x, y),1 k
=
B(x, y), k· 1
k
= (B(x, y),1)∼(AB(x, y),0)✱ ✐♠♣❧✐❝❛ q✉❡
❡♠ t= 1
k✱ f
′
[1
k] é ❞❛❞❛ ♣❡❧❛ ♠❛tr✐③
AB❀
❊ ❛ss✐♠ ❝♦♥s❡❝✉t✐✈❛♠❡♥t❡ ❛té ❛s ❞✉❛s ú❧t✐♠❛s✱ ♦♥❞❡
˜
f
(x, y),k−1 k
= (B(x, y), k−1)∼(Ak−1B(x, y),0)✱ ✐♠♣❧✐❝❛ q✉❡ ❡♠
t= k−1
k ✱f
′
[k−1
k ] é ❞❛❞❛ ♣❡❧❛ ♠❛tr✐③
Ak−1B❀ ❡
˜
f
(x, y),k k
= ˜f((x, y),1) = (B(x, y), k)∼(AkB(x, y),0) ✐♠♣❧✐❝❛ q✉❡ ❡♠
t= 1✱ f′
[1] é ❞❛❞❛ ♣❡❧❛ ♠❛tr✐③AkB✳
▼❛s✱ ♣❛r❛ q✉❡ ❛ ❝♦❧❛❣❡♠ ✜q✉❡ ❜❡♠ ❞❡✜♥✐❞❛✱ ❞❡✈❡♠♦s t❡r q✉❡ ❛s ❛♣❧✐❝❛çõ❡s
f′
[1] ❡ f[0]′ ❞❡✈❡♠ s❡r ❞❛❞❛s ♣❡❧❛ ♠❡s♠❛ ♠❛tr✐③✱ ♦ q✉❡ ❞❡ ❢❛t♦ ❛❝♦♥t❡❝❡✿ ♣♦r ✉♠ ❧❛❞♦
♦❜t✐✈❡♠♦s q✉❡((x1, y1),0) ˜
f
→(B(x1, y1),0)❡✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ♦❜t✐✈❡♠♦s q✉❡((x0, y0),1) ˜
f
→
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✶✵
❈♦♠♦ ♦s ♣♦♥t♦s ❞♦ ❞♦♠í♥✐♦ ❞❡✈❡♠ s❡r ❡q✉✐✈❛❧❡♥t❡s✱ ❡♥tã♦ ♣❡❧❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛((Ar)−1(x, y),1)∼((x, y),0)✱ t❡r❡♠♦s q✉❡A−r(x
1, y1) = (x0, y0).
❆ss✐♠✱ AkB(x
0, y0) =AkB(A−r(x1, y1)) = BAr(A−r(x1, y1)) =B(x1, y1).
❊✱ ♣♦rt❛♥t♦✱ ((x0, y0),1) = (A−r(x1, y1),1) ˜
f
→(B(x1, y1),0)✱ ♦✉ s❡❥❛✱ ❛ ❛♣❧✐✲
❝❛çã♦f′
[1] t❛♠❜é♠ é ❞❛❞❛ ♣❡❧❛ ♠❛tr✐③ B✳
❆❣♦r❛✱ ❝♦♥s✐❞❡r❡♠♦sdeg ¯f =k ❡deg ¯g =l✳ ❈♦♠♦ ❥á ✈✐♠♦s✱ ♣♦❞❡♠♦s ❡s❝r❡✲
✈❡r f[((x, y), t)] = [(B(x, y) +η((x, y), t), kt)] ❡ g[((x, y), t)] = [(C(x, y) +η((x, y), t), lt)]✳
❙❡❥❛♠ t0 = 0; t1 =
1
k−l; t2 =
2
k−l; . . .; t|k−l|−1 =
|k−l| −1
k−l ♦s |k−l|
♣♦♥t♦s ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛ ❞♦ ♣❛r( ¯f ,¯g)❡ ❧❡♠❜r❡♠♦s q✉❡η é ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ♥✉❧❛ ❡♠ ❝❛❞❛
✐♥st❛♥t❡ t✳ ❊♥tã♦✱ f˜((x, y), t0) = (B(x, y),0) ❡ ˜g((x, y), t0) = (C(x, y),0) s✐❣♥✐✜❝❛ q✉❡ ❛s
❛♣❧✐❝❛çõ❡sf′
[t0] ❡ g
′
[t0] sã♦ ❞❛❞❛s ♣❡❧❛s ♠❛tr✐③❡s B ❡C✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❚❛♠❜é♠✱f˜((x, y), t1) =B(x, y), k
k−l
∼
AB(x, y), l k−l
❡g˜((x, y), t1✮
=
C(x, y), l k−l
s✐❣♥✐✜❝❛ q✉❡ f′
[t1] ❡ g
′
[t1] sã♦ ❞❛❞❛s ♣❡❧❛s ♠❛tr✐③❡s AB ❡ C✱ r❡s✲
♣❡❝t✐✈❛♠❡♥t❡✳ ❊ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ ❛té ♦ ú❧t✐♠♦ ♣♦♥t♦ ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛✱ ♦♥❞❡ ♦❜t❡♠♦s q✉❡ f˜((x, y), t|k−l|−1) = B(x, y),k(|k−l| −1)
k−l
∼
A|k−l|−1B(x, y),l(|k−l| −1)
k−l
❡
˜
g((x, y), t|k−l|−1) =
C(x, y),l(|k−l| −1) k−l
♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ f′
[t|k−l|−1] ❡ g
′
[t|k−l|−1] sã♦
❞❛❞❛s ♣❡❧❛s ♠❛tr✐③❡sA|k−l|−1B ❡ C✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ♥♦s ♠♦str❛ ❛s ❢♦r♠❛s ❡s♣❡❝í✜❝❛s ❞❛ ♠❛tr✐③B✱ ♦✉ s❡❥❛✱ t♦✲
❞❛s ❛s ♣♦ssí✈❡✐s ❛♣❧✐❝❛çõ❡s ♣❛r❛ ♦ ♥♦ss♦ ❝♦♥t❡①t♦✱ ❛s q✉❛✐s ❞❡♣❡♥❞❡♠ ❞♦ ❣r❛✉ ❞❛ ❛♣❧✐❝❛çã♦
¯
f :S1 →S1✳
❚❡♦r❡♠❛ ✷✳✷✳✶✳ ❙❡❥❛ f :MAr → MA ✉♠❛ ❛♣❧✐❝❛çã♦ ❡♥tr❡ ❙♦❧✲t♦r✉s ❜✉♥❞❧❡s MAr ❡ MA
❝♦♠ ♠❛tr✐③❡s ❞❡ ❆♥♦s♦✈ Ar ❡ A✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ A ∈ GL(2,Z)✱ r ∈ N✳ ❈♦♥s✐❞❡r❡
t❛♠❜é♠ f′ : T → T ❛ ❛♣❧✐❝❛çã♦ ✐♥❞✉③✐❞❛ ♥❛ ✜❜r❛ t❛❧ q✉❡ f′
# = B =
m n p q ✱ ❡ ¯
f :S1 →S1✳ ❙✉♣♦♥❤❛ q✉❡ Ar=
a′ b′
c′ d′
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✶✶ B = q+
a′ −d′
c′ p b′ c′ p p q
, s❡ deg ¯f =r
−q
a′−d′
c′ q− b′ c′ p p q
, s❡ deg ¯f =−r ❡ detA= 1 ♦✉
deg ¯f =−r;r ♣❛r ❡ detA=−1
−q b′ d′ q − c′ a′ q q
, s❡ deg ¯f =−r;r í♠♣❛r ❡ detA=−1,
❝♦♠ a′, d′ 6= 0
0 , s❡
deg ¯f =−r;r í♠♣❛r ❡ detA=−1,
❝♦♠ a′ = 0 ♦✉ d′ = 0; ♦✉ deg ¯f 6=±r
,
♦♥❞❡
a′−d′
c′ p, b′ c′ p,
a′−d′
c′ q, b′ d′ q, c′ a′
q∈Z.
❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ q✉❡ deg ¯f =r✳
P❡❧♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦ ✭✷✳✶✮ t❡♠♦s q✉❡BAr =ArB✱ ♦ q✉❡ é ❡q✉✐✈❛❧❡♥t❡
❛♦ s✐st❡♠❛
ma′+nc′ = ma′ +pb′
mb′ +nd′ = na′ +qb′
pa′+qc′ = mc′+pd′
pb′+qd′ = nc′+qd′
✱ ♥❛s ✈❛r✐á✈❡✐s m, n, p ❡ q✳ ❆ ♣r✐♠❡✐r❛ ❡ q✉❛rt❛
❧✐♥❤❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s❀ ❞❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛ ♦❜t❡♠♦s ✶ n =
b′
c′
p ❡ ❞❛ t❡r❝❡✐r❛ ❧✐♥❤❛ m =
(a′−d′)
c′
p+q✳ ❙✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ❡♥❝♦♥tr❛❞♦s ♣❛r❛m❡n♥❛ s❡❣✉♥❞❛ ❧✐♥❤❛✱ ✈❡♠♦s
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✶✷
q✉❡ ❡st❛ ❡q✉❛çã♦ é ❝❧❛r❛♠❡♥t❡ s❛t✐s❢❡✐t❛✳ P♦rt❛♥t♦✱
B = q+
a′−d′
c′ p b′ c′ p p q , ♦♥❞❡
a′−d′
c′ p, b′ c′
p∈Z.
❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡deg ¯f =−r✱ ❝♦♠det(A) = 1♦✉r♣❛r ❡det(A) =−1✳
❖ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦ ✭✷✳✶✮ ✐♠♣❧✐❝❛ q✉❡BAr=A−rB✱ ♦ q✉❡ é ❡q✉✐✈❛❧❡♥t❡
❛♦ s✐st❡♠❛ ❧✐♥❡❛r
ma′ +nc′ = md′−pb′
mb′+nd′ = nd′−qb′
pa′+qc′ = −mc′+pa′
pb′+qd′ = −nc′+qa′
✱ ♥❛s ✈❛r✐á✈❡✐s m, n, p ❡ q✳ ❆ s❡❣✉♥❞❛ ❡
t❡r❝❡✐r❛ ❧✐♥❤❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s❀ ❞❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♦❜t❡♠♦s ✷ m = −q ❡ ❞❛ q✉❛rt❛ ❧✐♥❤❛
n= (a
′−d′)
c′ q−
b′
c′p✳ ❙✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ❡♥❝♦♥tr❛❞♦s ♣❛r❛ m ❡ n ♥❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛✱ ✈❡♠♦s q✉❡ ❡st❛ ❡q✉❛çã♦ é ❝❧❛r❛♠❡♥t❡ s❛t✐s❢❡✐t❛✳ P♦rt❛♥t♦✱
B = −q
a′−d′
c′ q− b′ c′ p p q , ♦♥❞❡
a′−d′
c′ q, b′ c′
p∈Z.
❙❡deg ¯f =−r✱r í♠♣❛r ❡det(A) =−1✱ ❡♥tã♦ ♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦ ✭✷✳✶✮
✐♠♣❧✐❝❛ q✉❡BAr =A−rB✱ ♦ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ s✐st❡♠❛ ❧✐♥❡❛r
ma′+nc′ =−md′+pb′ mb′ +nd′ = −nd′+qb′
pa′+qc′ = mc′−pa′ pb′+qd′ = nc′−qa′
✱
♥❛s ✈❛r✐á✈❡✐sm, n, p❡q✳ ❉❛ ♣r✐♠❡✐r❛ ❡ q✉❛rt❛ ❧✐♥❤❛s ♦❜t❡♠♦s✸ m=−q✳ ❙✉❜st✐t✉✐♥❞♦ ♥❛
s❡❣✉♥❞❛ ❡ t❡r❝❡✐r❛ ❧✐♥❤❛s ♦❜t❡♠♦s✱ s❡a′ 6= 0❡d′ 6= 0✱n= b′
d′q❡p=−
c′
a′q✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
Ar=
a′ b′
0 d′
s❡r✐❛ t❛❧ q✉❡1 = det(A) = det(Ar) =a′d′ ⇒a′ =±1 =d′ ⇒ |tr(Ar)|= 2♦ q✉❡ ♥ã♦
♣♦❞❡ ❛❝♦♥t❡❝❡r✱ ♣♦✐s ♣♦r s❡r ❞❡ ❆♥♦s♦✈✱ t❡♠♦s q✉❡|tr(Ar)|>2✳
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✶✸ P♦rt❛♥t♦✱ B = −q b′ d′ q − c′ a′ q q , ♦♥❞❡ b′ d′ q, c′ a′
q∈Z.
❙❡ a′ = 0 ✭❡ ♣♦rt❛♥t♦ d′ 6= 0✱ ❥á q✉❡ tr(Ar) 6= 0✮✱ ♦❜t❡♠♦s ♦ s✐st❡♠❛ ❧✐♥❡❛r
nc′ = −md′ +pb′
mb′+nd′ = −nd′+qb′
qc′ = mc′
pb′+qd′ = nc′
✱ ♥❛s ✈❛r✐á✈❡✐s m, n, p ❡ q✳ ❉❛ ♣r✐♠❡✐r❛ ❡ q✉❛rt❛ ❧✐♥❤❛s
♦❜t❡♠♦sq =−m✳ ❉❛ t❡r❝❡✐r❛✱ q =m❀ ❧♦❣♦ ❝♦♥❝❧✉í♠♦s q✉❡q =m = 0✳ ❙✉❜st✐t✉✐♥❞♦ ♥❛
s❡❣✉♥❞❛ ❧✐♥❤❛✱ ♦❜t❡♠♦sn = 0 ❡ ✈♦❧t❛♥❞♦ ♥❛ ♣r✐♠❡✐r❛✱ ♦❜t❡♠♦sp= 0✳ P♦rt❛♥t♦✱ B = 0✳
P❛r❛ ♦ ❝❛s♦d′ = 0 ✭❡a′ 6= 0✮✱ ❜❛st❛ ♣r♦❝❡❞❡r ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✳ P♦r ✜♠✱ s✉♣♦♥❤❛♠♦s q✉❡ deg ¯f =k 6=±r✳
❈♦♠♦ A é ✉♠❛ ♠❛tr✐③ ❞❡ ❆♥♦s♦✈✱ ❡♥tã♦ A é ❞✐❛❣♦♥❛❧✐③á✈❡❧ ✹✱ ❧♦❣♦ ❡①✐st❡
P ∈GL(2,R) t❛❧ q✉❡ P−1AP = ¯A =
λ1 0
0 λ2
♦♥❞❡ λ1 ❡ λ2 sã♦ ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ A✳
❈♦♥s✐❞❡r❡♠♦sB¯ =P−1BP =
x y z w . ❊♥tã♦ ¯
BA¯r = ¯AkB¯ ⇔
x y z w λr 1 0
0 λr
2 = λk 1 0
0 λk
2 x y z w ⇔ ⇔ xλr
1 = λk1x (I)
yλr
2 = λk1y (II)
zλr
1 = λk2z (III)
wλr
2 = λk2w. (IV)
❊♠(I) t❡♠♦s✿
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✶✹
xλr1−xλk1 = 0⇒
xλk
1(λr1−k−1) = 0 ⇒
xλk1 = 0 ⇒ x= 0 ♦✉λk1 = 0 ♦✉
λr1−k−1 = 0 ⇒ λr1−k= 1
✱ s❡k < r;
xλr1(1−λk1−r) = 0 ⇒
xλr1 = 0 ⇒ x= 0 ♦✉λr1 = 0 ♦✉
1−λk1−r= 0 ⇒ λk1−r = 1
✱ s❡r < k.
▼❛s✱ λ|1k−r| = 1 ♥ã♦ ❛❝♦♥t❡❝❡✱ ♣♦✐s λ1 6=±1 ❥á q✉❡ A é ❞❡ ❆♥♦s♦✈ ❡k 6=r✳
❚❛♠❜é♠λk
1 = 0 ❡ λr1 = 0 ♥ã♦ ❛❝♦♥t❡❝❡♠✱ ♣♦✐s λ1 = 06 ❥á q✉❡ λ1λ2 = det ¯A=±1✳
P♦rt❛♥t♦✱ só ♣♦❞❡♠♦s t❡rx= 0✳
❖ ❝❛s♦(IV)é t♦t❛❧♠❡♥t❡ ❛♥á❧♦❣♦✱ ❧♦❣♦ w= 0✳
❊♠(II)✿
❈♦♠♦ ±1 = det ¯A = λ1λ2✱ ❡♥tã♦ λ1 = ±
1
λ2✳ ❆ss✐♠✱
yλr
2 = λk1y ⇒ yλr2 =
± 1
λ2
k
y⇒yλr2+k =±y ❡✱ ❝♦♠♦ λ2r+k 6=±1 ❡k 6=−r✱ ❡♥tã♦ y= 0✳
❉♦ ♠❡s♠♦ ♠♦❞♦ ♦❜t❡♠♦sz = 0✳
P♦rt❛♥t♦✱ B¯ =
0 0 0 0
❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱B = 0 0 0 0 ✳
❖❜s❡r✈❛çã♦ ✷✳✷✳✷✳ ❊ss❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ♠❛tr✐③ B ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ❞♦ ❛rt✐❣♦ ❬❙❲❲❪
q✉❛♥❞♦ f é ✉♠❛ ❛✉t♦✲❛♣❧✐❝❛çã♦✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ r= 1✳
❈❧❛r❛♠❡♥t❡✱f ❝♦❜r❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ r❡❝♦❜r✐♠❡♥t♦s✳ ❊ ✉♠❛ q✉❡stã♦ ♥❛t✉r❛❧
q✉❡ s✉r❣❡ é q✉❛✐s ❣r❛✉s sã♦ r❡❛❧✐③á✈❡✐s✳ ❖ tr❛❜❛❧❤♦ ❬❙❲❲❪ r❡s♣♦♥❞❡ ❡ss❛ q✉❡stã♦ ♣❛r❛ ❛✉t♦✲❛♣❧✐❝❛çõ❡s ❡♥tr❡ t♦r✉s ❜✉♥❞❧❡s ❡ s❛✜r❛s✳ ❖ t❡♦r❡♠❛ ❛❝✐♠❛ ♥♦s ♣❡r♠✐t✐✉ ✐♥✐❝✐❛r ✉♠ tr❛❜❛❧❤♦✱ ❛✐♥❞❛ ❡♠ ❛♥❞❛♠❡♥t♦✱ ♣❛r❛ r❡s♣♦♥❞❡r ❡ss❛ q✉❡stã♦ ♣❛r❛ ♦s ♦✉tr♦s ❝❛s♦s✱ ♦✉ s❡❥❛ ♣❛r❛ ❛♣❧✐❝❛çõ❡s q✉❛✐sq✉❡r ❡♥tr❡ t♦r✉s ❜✉♥❞❧❡s ❡ s❛✜r❛s✳
❖ ♣ró①✐♠♦ t❡♦r❡♠❛ ♥♦s ❞✐③ ❝♦♠♦ ❝❛❧❝✉❧❛r ♦ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥ ❞❡ ❛♣❧✐❝❛✲
çõ❡sf, g :MAr = T ×R
((x, y), t)∼((Ar)n(x, y), t−n) →MA=
T ×R
((x, y), t)∼(An(x, y), t−n)✱
♦♥❞❡ r ∈ N, n ∈ Z✳ ❊♠ s✉❛ ❞❡♠♦♥str❛çã♦✱ ✉s❛r❡♠♦s ♦ q✉❡ ❢♦✐ ❢❡✐t♦ ♥❛ ❙❡çã♦ ✷✳✷✱ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ♠❛tr✐③ B ❡ t❛♠❜é♠ ❞♦✐s r❡s✉❧t❛❞♦s ❞♦s ❛rt✐❣♦s ❬❏❡✶❪ ❡ ❬❏❡✷❪✱ ♦s q✉❛✐s
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✶✺
❊st❡ ♣r✐♠❡✐r♦ r❡s✉❧t❛❞♦✱ ❡♥❝♦♥tr❛❞♦ ❡♠ ❬❏❡✶❪✱ ♥♦s ❞✐③ ❝♦♠♦ ❝❛❧❝✉❧❛r ♦ ♥ú✲ ♠❡r♦ ❞❡ ◆✐❡❧s❡♥ s❡ ❡st✐✈❡r♠♦s tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ♦ ❚♦r♦✿
▲❡♠❛ ✷✳✷✳✸ ✭✭✼✳✸✮✱❬❏❡✶❪✮✳ ❙❡❥❛Tn ♦ ❚♦r♦ n✲❞✐♠❡♥s✐♦♥❛❧ ❡ f, g:Tn→Tn ✉♠❛ ❛♣❧✐❝❛çã♦
❝♦♥tí♥✉❛✳ ❙❡❥❛♠ B ❡ C ♠❛tr✐③❡s n×n r❡♣r❡s❡♥t❛♥❞♦ ♦s ❡♥❞♦♠♦r✜s♠♦s f#, g#:π1Tn→
π1Tn✳ ❊♥tã♦
N(f, g) = |L(f, g)|=|det(B−C)|.
❊st❡ s❡❣✉♥❞♦✱ ❡♥❝♦♥tr❛❞♦ ❡♠ ❬❏❡✷❪✱ ♥♦s ❞✐③ ✭❛❧é♠ ❞❡ ♦✉tr❛s ❝♦✐s❛s✮✱ q✉❡ ♣♦❞❡♠♦s s♦♠❛r ♦s ♥ú♠❡r♦s ❞❡ ◆✐❡❧s❡♥ ❞❛s ✜❜r❛s ♣❛r❛ ♦❜t❡r ♦ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥ ❞♦ ❡s♣❛ç♦ t♦t❛❧✿
▲❡♠❛ ✷✳✷✳✹ ✭✭✺✳✺✮✱❬❏❡✷❪✮✳ ❙❡❥❛♠ φ, φ′ ❞✐❢❡♦♠♦r✜s♠♦s ❞♦ ❚♦r♦ k✲❞✐♠❡♥s✐♦♥❛❧✳ ❈♦♥s✐❞❡r❡ ❛s ♠❛tr✐③❡sk×k✱ A❡D✱ r❡♣r❡s❡♥t❛♥❞♦φ#, φ′#:π1T →π1T ❡ s✉♣♦♥❤❛ q✉❡det(I−A)6= 0✳
❊♥tã♦ q✉❛❧q✉❡r ♣❛r ❞❡ ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s f, g : MA → MD é ❤♦♠♦tó♣✐❝♦ ❛ ✉♠ ♣❛r ❞❡
❛♣❧✐❝❛çõ❡s q✉❡ ♣r❡s❡r✈❛♠ ✜❜r❛
MA
p
˜
f ,g˜
/
/MD
p′
S1 f ,¯g¯ //S1
✭✷✳✷✮
❙❡ind( ¯f ,g¯) = 0✱ ❡♥tã♦ ( ¯f ,g¯)é ❤♦♠♦tó♣✐❝♦ ❛ ✉♠ ♣❛r ❧✐✈r❡ ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛s✱
❡ ♣♦rt❛♥t♦ (f, g) t❛♠❜é♠ é✱ ❡ N(f, g) = 0✳ ❙❡ ind( ¯f ,g¯) = k 6= 0 ❡♥tã♦ ✜①❡ ✉♠ ♣♦♥t♦ ❞❡
❝♦✐♥❝✐❞ê♥❝✐❛ b∈S1 ❡ ❞❡♥♦t❡ ♣♦r B ❡ C ❛s ♠❛tr✐③❡s r❡♣r❡s❡♥t❛♥❞♦ fb˜
# ❡ ˜gb#✳ ❊♥tã♦✱
N(f, g) =
|k|−1
X
i=0
N(fb, φ′igb) =
|k|−1
X
i=0
|det(B−DiC)|.
❆♣❡s❛r ❞♦ t❡♦r❡♠❛ s❡❣✉✐♥t❡ s❡r ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞♦ ▲❡♠❛ ✷✳✷✳✹✺✱
❛ ♥♦ss❛ ❞❡♠♦♥str❛çã♦ ❛♣r❡s❡♥t❛ ✉♠❛ ✈✐sã♦ ♠❛✐s ❣❡♦♠étr✐❝❛✱ ♣♦r ✐ss♦✱ ♦♣t❛♠♦s ♣♦r ❛♣r❡s❡♥tá✲ ❧❛ ♥❛ t❡s❡✳
✺❚❡♠♦s q✉❡det(I−Ar)6= 0✱ ❧♦❣♦ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ▲❡♠❛ ✷✳✷✳✹✳ ❊ss❡ ❧❡♠❛ ♥♦s ❞✐③ q✉❡ s❡N( ¯f ,g¯) = 0
✭♦✉ s❡❥❛ k = l✮✱ ❡♥tã♦ N(f, g) = 0 ❡ s❡ N( ¯f ,¯g) 6= 0✱ ❡♥tã♦ N(f, g) =
N( ¯f ,¯g)−1
X
i=0
|det(B−DiC)| ✭q✉❡ é
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✶✻
❚❡♦r❡♠❛ ✷✳✷✳✺✳ ❙❡❥❛♠ f, g : MAr →MA ❛♣❧✐❝❛çõ❡s ❡♥tr❡ ❙♦❧✲t♦r✉s ❜✉♥❞❧❡s MAr ❡ MA
❝♦♠ ♠❛tr✐③❡s ❞❡ ❆♥♦s♦✈ Ar ❡ A✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ A ∈ GL(2,Z)✱ r ∈ N
∗✳ ❈♦♥s✐❞❡r❡ t❛♠❜é♠ f′, g′ : T → T ❛♣❧✐❝❛çõ❡s ✐♥❞✉③✐❞❛s ♥❛ ✜❜r❛ t❛✐s q✉❡ f′
# = B ❡ g#′ = C✱ ❡
¯
f ,g¯:S1 →S1 t❛✐s q✉❡ deg ¯f =k ❡ deg ¯g =l✳ ❊♥tã♦
N(f, g) =
0 , s❡ k =l
|k−l|−1
X
i=0
det(As✐❣♥(k)iB −C)
, s❡ k 6=l
♦♥❞❡ s✐❣♥(k) :=
−1 , s❡ k < 0 1 , s❡ k > 0.
❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ♦❜s❡r✈❡♠♦s q✉❡ ❡①✐st❡♠ s❡✐s ❝❛s♦s ♣❛r❛ s❡r❡♠ ❛♥❛❧✐s❛❞♦s✿
✭❛✮ k =r ❡ l6=±r❀
✭❜✮ k =r ❡ l=r❀
✭❝✮ k 6=±r ❡ l6=±r❀
✭❞✮ k =−r ❡ l=r❀
✭❡✮ k =−r ❡ l=−r❀
✭❢✮ k =−r ❡ l6=±r✳
❚❛♠❜é♠✱ ♥♦s ✐t❡♥s (a),(d) ❡ (f) ❡①✐st❡♠ ♦s ❝❛s♦s s✐♠étr✐❝♦s✳
❙✉♣♦♥❤❛♠♦s q✉❡ Ar=
a′ b′
c′ d′
✳
✭❛✮ ❙❛❜❡♠♦s q✉❡ f¯é ❤♦♠♦tó♣✐❝❛ ❛ ✉♠❛ ❛♣❧✐❝❛çã♦ f¯¯(t) = rt ❡✱ ♣❡❧♦ ❞✐❛❣r❛♠❛ ❝♦♠✉✲
t❛t✐✈♦ ✭✷✳✶✮✱ t❡♠♦s q✉❡ BAr = ArB ♦♥❞❡ B =
q+
a′ −d′
c′
p
b′
c′
p
p q
✱
a′−d′
c′
p,
b′
c′
p∈Z ✭❚❡♦r❡♠❛ ✷✳✷✳✶✮✳ ❆❧é♠ ❞✐ss♦✱ C = 0 ✭❚❡♦r❡♠❛ ✷✳✷✳✶✮ ❡ g¯ é
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✶✼
❆ss✐♠✱ rt = lt( mod 1) ⇒ |r − l|t = 0( mod 1) ⇒ t0 = 0;t1 =
1
r−l;t2 =
2
r−l;. . .;t|r−l|−1 =
|r−l| −1
r−l sã♦ ♦s|r−l| ♣♦♥t♦s ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛ ❞❡( ¯f ,¯g)✳ ❈♦♠♦ N( ¯f ,¯g) =|r−l|✱ ❡♥tã♦ ♦s ♣♦♥t♦s ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛ ❡stã♦ ❡♠ ❝❧❛ss❡s ❞✐st✐♥t❛s✳
❈♦♠♦ ✈✐♠♦s ♥♦ ❝♦♠❡ç♦ ❞❡st❡ ❝❛♣ít✉❧♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r f[((x, y), t)] = [(B(x, y) +
η((x, y), t), rt)] ❡ g[((x, y), t)] = [(C(x, y) +η((x, y), t), lt)]✳ ❙❡❣✉✐♥❞♦ ♦ r❛❝✐♦❝í♥✐♦
❢❡✐t♦ ♥❛ s❡çã♦ ✷✳✷✱ ✈❛♠♦s ♦❜t❡r q✉❡ ❛s ❛♣❧✐❝❛çõ❡sf′
[t0], g
′
[t0], f
′
[t1], g
′
[t1], f
′
[t2], g
′
[t2], . . . , f
′
[t|r−l|−1]
❡ g′
[t|r−l|−1] sã♦ ❞❛❞❛s ♣❡❧❛s ♠❛tr✐③❡s B, C, AB, C, A
2B, C, . . . , A|r−l|−1B ❡ C r❡s♣❡❝✲
t✐✈❛♠❡♥t❡✳
❆ss✐♠✱
N(f, g) Lema=2.2.4 N(f′
[t0], g
′
[t0]) +N(f
′
[t1], g
′
[t1]) +. . .+N(f
′
[t|r−l|−1], g
′
[t|r−l|−1])
Lema2.2.3
= |det(B−C)|+|det(AB −C)|+. . .+|det(A|r−l|−1B −C)|
=
|r−l|−1
X
i=0
|det(AiB−C)|.
✭❜✮ ❡ ✭❡✮ ◆❡st❡s ❝❛s♦s✱ ♣♦❞❡♠♦s t♦♠❛r f¯¯❤♦♠♦tó♣✐❝❛ ❛ f¯✱ ♦♥❞❡ f¯¯é ✉♠❛ tr❛♥s❧❛çã♦ ♣♦rε ❞❡
¯
f (0 < ε <1/4)❞❡ ♠♦❞♦ q✉❡ f¯¯❡ ¯g ♥ã♦ t❡♥❤❛♠ ♠❛✐s ❝♦✐♥❝✐❞ê♥❝✐❛s✳ ❈♦♥s❡q✉❡♥t❡✲
♠❡♥t❡✱ N(f, g) = 0✳
✭❝✮ P♦❞❡♠♦s s❡❣✉✐r ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ❞♦ ✐t❡♠(a)❡ ♦❜t❡r ♦s|k−l|♣♦♥t♦s ❞❡ ❝♦✐♥❝✐❞ê♥✲ ❝✐❛ ❞❡( ¯f ,g¯)✳ ❙❡❣✉✐♥❞♦ ❛✐♥❞❛ ♥❡ss❡ r❛❝✐♦❝í♥✐♦✱ ✈❛♠♦s ♦❜t❡r ♦ ♥ú♠❡r♦ ❞❡ ◆✐❡❧s❡♥ ❞❡f
❡ g ❡♠ t❡r♠♦s ❞❡ det(AiB−C)✳ P♦ré♠✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✷✳✶✱ t❡♠♦s q✉❡ B = 0 =C
❡ ♣♦rt❛♥t♦ AiB−C = 0✱ ❞❡ ♦♥❞❡ ♦❜t❡♠♦s q✉❡ N(f, g) = 0✳
✭❞✮ ❆q✉✐ ♥ós t❡♠♦s q✉❡ f¯é ❤♦♠♦tó♣✐❝❛ ❛ ✉♠❛ ❛♣❧✐❝❛çã♦ f¯¯(t) = −rt✱ g¯ é ❤♦♠♦tó♣✐❝❛
❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ¯¯g(t) =rt✱ BAr =A−rB ❡ CAr =ArC ✭♣❡❧♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✶✽ B = −q
a′−d′
c′ q− b′ c′ p p q
, s❡ (det(A) = 1) ♦✉
(det(A) =−1 ❡r é ♣❛r)
−q b′ d′ q − c′ a′ q q
, s❡ det(A) =−1 ❡ r é í♠♣❛r
❡ C =
u+
a′ −d′
c′ t b′ c′ t t u , ❝♦♠
a′−d′
c′ q✱ b′ c′ p✱
a′−d′
c′ t✱ b′ c′ t✱ b′ d′ q✱ c′ a′
q ∈Z.✭✷✳✷✳✶✮✳
❆ss✐♠✱rt=−rt( mod 1)⇒ |2r|t = 0( mod 1)⇒t0 = 0;t1 =
1
|2r|;t2 =
2
|2r|;. . .;t|2r|−1 = |2r| −1
|2r| sã♦ ♦s|2r|♣♦♥t♦s ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛ ❞❡( ¯f ,g¯)✳ ❈♦♠♦N( ¯f ,g¯) = |2r|✱ ♦s ♣♦♥t♦s ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛ ❡stã♦ ❡♠ ❝❧❛ss❡s ❞✐st✐♥t❛s✳
❈♦♥s✐❞❡r❡♠♦sf[((x, y), t)] = [(B(x, y)+η((x, y), t),−rt)]❡g[((x, y), t)] = [(C(x, y)+
η((x, y), t), rt)]✳ ❊♥tã♦ ❛s ❛♣❧✐❝❛çõ❡sf′
[t0], g
′
[t0], f
′
[t1], g
′
[t1], . . . , f
′
[t|2r|−1]❡ g
′
[t|2r|−1] sã♦ ❞❛✲
❞❛s ♣❡❧❛s ♠❛tr✐③❡s B, C, A−1B, C, . . . A−(|2r|−1)B ❡ C✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❯t✐❧✐③❛♥❞♦ ♦s ▲❡♠❛s ✷✳✷✳✸ ❡ ✷✳✷✳✹ ♦❜t❡♠♦s q✉❡ N(f, g) =
|2r|−1
X
i=0
|det(A−iB−C)|✳
✭❢✮ ◆❡st❡ ❝❛s♦✱ t❡♠♦s|l+r|❝❧❛ss❡s ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛✳ ❆ss✐♠✱ ❛s ❛♣❧✐❝❛çõ❡sf′
[t0], g
′
[t0], f
′
[t1], g
′
[t1], . . .✱
f′
[t|l+r|−1] ❡g
′
[t|l+r|−1]sã♦ ❞❛❞❛s ♣❡❧❛s ♠❛tr✐③❡s B, C, A
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✶✾ ♣❡❝t✐✈❛♠❡♥t❡❀ ♦♥❞❡ B = −q
a′−d′
c′ q− b′ c′ p p q
, s❡ (det(A) = 1) ♦✉
(det(A) =−1 ❡r é ♣❛r )
−q b′ d′ q − c′ a′ q q
, s❡ det(A) =−1 ❡ r é í♠♣❛r
❡ C = 0✳
❆ss✐♠✱ N(f, g) = N(f′
[t0], g
′
[t0]) + . . . + N(f
′
[t|r+l|−1], g
′
[t|r+l|−1]) = |det(B − C)| +
|det(A−1B−C)|+. . .+|det(A−(|r+l|−1)B−C)|=
|r+l|−1
X
i=0
|det(A−iB −C)|.
❖❜s❡r✈❛çã♦ ✷✳✷✳✻✳ ❙♦❜ ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ✷✳✷✳✺✱ ❝♦♠ ✉♠ ♣♦✉❝♦ ❞❡ ❝á❧❝✉❧♦✱ ♣♦❞❡♠♦s ♠♦str❛r q✉❡ s❡ A∈SL(2,Z)✱ f, g :MAr → MA ❡ B ❡ C sã♦ ❞❛ ❢♦r♠❛ ❞♦ ❚❡♦r❡♠❛ ✷✳✷✳✶
❡♥tã♦✱ det(AiB −C) = det(B) + det(C)✳
❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ss❡ t❡♦r❡♠❛ ❡ ❞❛ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛✱ ♣♦❞❡♠♦s ❡♥✉♥❝✐❛r ♦s ❞♦✐s ♣ró①✐♠♦s ❈♦r♦❧ár✐♦s✳
❈♦r♦❧ár✐♦ ✷✳✷✳✼✳ ❙♦❜ ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ✷✳✷✳✺✱ s❡ A∈SL(2,Z)✱ ❡♥tã♦
N(f, g) =|k−l||det(B) + det(C)|.
P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ s❡❣✉♥❞♦ ❈♦r♦❧ár✐♦✱ ♣r❡❝✐s❛r❡♠♦s ❞♦ ▲❡♠❛ ❛❜❛✐①♦✱ ❝✉❥❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬❙❛❪✳
▲❡♠❛ ✷✳✷✳✽✳ ❙❡❥❛♠ A = εAn
0 ✉♠❛ ♠❛tr✐③ ❞❡ ❆♥♦s♦✈ ♦♥❞❡ A0 = Qsi=1
ai 1 1 0 é
✉♠❛ r❛✐③ ♣r✐♠✐t✐✈❛ ❞❡ A❀ C(A) = {B ∈ GL(2,Z)|BAB−1 = A} ❡ R(A) = {B ∈
✷✳✷✳ ❆ ✜❜r❛ ❡ ❛ s✉❛ ♠❛tr✐③ ✐♥❞✉③✐❞❛ ✷✵
✭✶✮ C(A) = {±Ai
0(i∈Z)} ∼=Z+Z2.
✭✷✮ ❙✉♣♦♥❤❛ q✉❡det(A) = 1✱ ❡ q✉❡ ♦ ♣❡rí♦❞♦ ♣r✐♠✐t✐✈♦ (a1, . . . , as)s❡❥❛ ✐♥✈❡rtí✈❡❧✱ ✐st♦ é✱
❡①✐st❡ ✉♠ ✐♥t❡✐r♦u(1≤u≤s−1)t❛❧ q✉❡(as, as−1, . . . , a1) = (au+1, . . . , as, a1, . . . , au)✳
❈♦❧♦q✉❡ P =
0 1
−1 0
Qs
i=u+1
ai 1 1 0
✳ ❊♥tã♦ R(A) = C(A)P✳
✭✸✮ ❙❡ ❛ ❝♦♥❞✐çã♦ (2) ♥ã♦ ❢♦r s❛t✐s❢❡✐t❛✱ ❡♥tã♦ R(A) é ✉♠ ❝♦♥❥✉♥t♦ ✈❛③✐♦✳
✭✹✮ ❯♠❛ ♠❛tr✐③ Q ❡♠ R(A) ♣♦ss✉✐ ♣❡rí♦❞♦ 2 ♦✉ 4✱ ❞❡♣❡♥❞❡♥❞♦ s❡det(Q) = −1 ♦✉ +1✳
❈♦r♦❧ár✐♦ ✷✳✷✳✾✳ ❙♦❜ ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ✷✳✷✳✺✱ s❡f, g :MA→MA sã♦ ❤♦♠❡♦♠♦r✲
✜s♠♦s✱ ❡♥tã♦ N(f, g) = 0 ♦✉ N(f, g) = 4✳
❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ♦❜s❡r✈❡♠♦s q✉❡ ❝♦♠♦ f ❡ g sã♦ ❤♦♠❡♦♠♦r✜s♠♦s✱ ❡♥tã♦ k=deg( ¯f) =±1 ❡l = deg(¯g) =±1✳
❙❡ det(A) = −1✱ ❡♥tã♦ ♣❡❧♦ ▲❡♠❛ ✷✳✷✳✽ ✭✸✮✱ t❡♠♦s q✉❡ ♥ã♦ ❡①✐st❡ ♠❛tr✐③
B ❝♦♠ det(B) = ±1 t❛❧ q✉❡ BA = A−1B✳ ❆ss✐♠✱ deg ¯f = 1✱ ♦✉ s❡❥❛✱ f¯ ✐♥❞✉③ ♦ ❤♦✲
♠♦♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ❡♠π1(S1)✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ g¯t❛♠❜é♠ ✐♥❞✉③ ♦ ❤♦♠♦♠♦r✜s♠♦
✐❞❡♥t✐❞❛❞❡ ❡♠π1(S1)✳ P♦rt❛♥t♦✱ N(f, g) = 0✳
❙❡ det(A) = 1✱ ❡♥tã♦ ♥♦s ❝❛s♦s k = 1 = l ❡ k = −1 = l✱ t❡♠♦s q✉❡ N(f, g) = 0✳ P❛r❛ k =−1❡ l= 1 ✭♦✉ ♥♦ ❝❛s♦ s✐♠étr✐❝♦✮ t❡♠♦s✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✷✳✷✳✼✱ q✉❡
N(f, g) = 2|det(B) + det(C)|✳ ❆ss✐♠✱
• N(f, g) = 0✱ q✉❛♥❞♦ det(B) = 1 ❡ det(C) =−1✱ ♦✉ det(B) =−1 ❡det(C) = 1❀
• N(f, g) = 4✱ q✉❛♥❞♦ det(B) = det(C)✳
❖❜s❡r✈❛çã♦ ✷✳✷✳✶✵✳ ◆♦ ❝❛s♦ ♦♥❞❡ f ❡ g sã♦ ❤♦♠❡♦♠♦r✜s♠♦s✱ t❡♠♦s q✉❡ Coin(f, g) =
F ix(f−1 ◦g) ❡✱ ♣♦rt❛♥t♦✱ ♦ ❈♦r♦❧ár✐♦ ✷✳✷✳✾ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❞♦ ❚❡♦r❡♠❛ ✷✳✷ ❞❡ ❬●❲❪✳
