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▲✐♠✐t s❡ts ✐♥ t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡ ❝②❧✐♥❞r✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥s

❏❛♥ ❑✇✐❛t❦♦✇s❦✐ ❚❤❡ t❛❧❦ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆rt✉r ❙✐❡♠❛s③❦♦✳ ❇➛❞❧❡✇♦✱ ❏✉♥❡✱ ✷✵✶✽

❈②❧✐♥❞r✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥s

X ✕ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ ♠♦♥♦t❤❡t✐❝ ❣r♦✉♣✱ T : X − → X ✕ ❛ ♠✐♥✐♠❛❧ ❤♦♠❡♦♠♦r♣❤✐s♠❀ f : X − → R ✕ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ Tf : X × R − → X × R✱ Tf (x, r) = (Tx, r + f (x)) − ❛ ❝②❧✐♥❞r✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥ (Tf )n(x, r) = (T nx, r + f (n)(x))✱ ✇❤❡r❡ f (n)(x) =      f (x) + f (Tx) + . . . + f (T n−✶x), n ≥ ✶, ✵, n = ✵, −(T −✶x) − . . . − f (T −nx), n ≥ ✶. f (·)(·) : Z × X − → R✱ s❛t✐s✜❡s ❛ ❝♦❝②❝❧❡ ❝♦♥❞✐t✐♦♥✿ f (n+m)(x) = f (n)(x) + f (m)(T nx)

❚❤❡♦r❡♠ ✭❇❡s✐❝♦✈✐t❝❤ ✶✾✺✶✮

✶ ■❢ Tf ✐s t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡ ✭✐✳❡✳ ❛❞♠✐ts ❞❡♥s❡ ♦r❜✐ts✮✱ t❤❡♥

✐t ♣♦ss❡ss❡s ❛ ❜♦✉♥❞❡❞ ❛❜♦✈❡ ♦r❜✐t ✭❛♥❞ ❛ ❜♦✉♥❞❡❞ ❜❡❧♦✇ ♦r❜✐t✮❀

✐♥ ♣❛rt✐❝✉❧❛r✱ Tf ❝❛♥♥♦t ❜❡ ♠✐♥✐♠❛❧✳

✷ ❚❤❡r❡ ❡①✐sts ❛♥ ✐rr❛t✐♦♥❛❧ ❝✐r❝❧❡ r♦t❛t✐♦♥ ❛♥❞ ✐ts t♦♣♦❧♦❣✐❝❛❧❧②

tr❛♥s✐t✐✈❡ ❝♦❝②❝❧❡ ✇✐t❤ ❛ ❞✐s❝r❡t❡ ♦r❜✐t✳ ❚❤❡♦r❡♠ ✭●♦tts❝❤❛❧❦✲❍❡❞❧✉♥❞ ✶✾✺✺✱ ▲❡♠❛➠❝③②❦✲▼❡♥t③❡♥ ✷✵✵✷✮ ▲❡t µ ❜❡ t❤❡ ❍❛❛r ♠❡❛s✉r❡ ♦♥ X✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ tr✐❝❤♦t♦♠② ❤♦❧❞s✿

✶

• f dµ = ✵

≡ ❡✈❡r② ♦r❜✐t ✐s ❛ ❞✐s❝r❡t❡ s❡t❀ ✕ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ♦r❜✐t ✐s ✐ts❡❧❢ ❛ ♠✐♥✐♠❛❧ s❡t❀

✷ f ✐s ❛ ❝♦❜♦✉♥❞❛r②

≡ Tf ❛❞♠✐ts ❜♦✉♥❞❡❞ s❡♠✐✲♦r❜✐ts ✭✐✳❡✳ t❤❡r❡ ❡①✐sts ∃g∈C(X) f = g ◦ T − g✮❀ ✕ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ♠✐♥✐♠❛❧ s❡t ✐s ❛ s❤✐❢t❡❞ ❣r❛♣❤ ♦❢ g❀

✸ Tf ✐s t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡✳

P♦ss✐❜❧❡ ♣r♦❜❧❡♠s

✶ ❉♦ t❤❡r❡ ❡①✐st t♦♣✳ tr❛♥s✳ tr❛♥s❢♦r♠❛t✐♦♥s ✇✐t❤♦✉t ♠✐♥✐♠❛❧

s❡ts❄

✷ ●✐✈❡♥ ❛ ♠✐♥✐♠❛❧ r♦t❛t✐♦♥✱ ❝♦♥str✉❝t ✐ts t♦♣✳ tr❛♥s✳ ❝♦❝②❝❧❡

✇✐t❤♦✉t ❞✐s❝r❡t❡ ♦r❜✐ts✳

✸ ●✐✈❡♥ ❛ ♠✐♥✐♠❛❧ r♦t❛t✐♦♥✱ ❝♦♥str✉❝t ✐ts t♦♣✳ tr❛♥s✳ ❝♦❝②❝❧❡

✇✐t❤ ♣♦ss✐❜❧② ♠❛♥② ❞✐s❝r❡t❡ ♦r❜✐ts✳

✹ ❲❤❛t ❦✐♥❞ ♦❢ ♦r❜✐ts ❝❛♥ ❝♦❡①✐st ❢♦r ❛ t♦♣✳ tr❛♥s✳

tr❛♥s❢♦r♠❛t✐♦♥ ✭❧♦❝❛❧❧② ❝♦♠♣❛❝t σ✲❈❛♥t♦r s❡ts ❢♦r ✐♥st❛♥❝❡✮❄

✺ ❉♦❡s t❤❡r❡ ❡①✐st ❛ t♦♣✳ tr❛♥s✳ ❝♦❝②❧❡ ❤❛✈✐♥❣ ❡✐t❤❡r ❞❡♥s❡ ♦r

❞✐s❝r❡t❡ ♦r❜✐ts ♦♥❧②❄ ✕ ✭❖P❊◆✮

✻ ❲❤❛t ❛r❡ ♣♦ss✐❜❧❡ ♠✐♥✐♠❛❧ s❡ts ❢♦r ❝②❧✐♥❞r✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥s❄

✭❖P❊◆✳ ❊✈❡r② ♥♦♥ ❞✐s❝r❡t❡ ♠✐♥✐♠❛❧ s❡t ✐s ❛ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❈❛♥t♦r s❡t✳ ❖♥❧② ❞✐s❝r❡t❡ ♦r❜✐ts ❤❛✈❡ ❜❡❡♥ r❡❝♦❣♥✐③❡❞ s♦ ❢❛r ❛s ♠✐♥✐♠❛❧ s❡ts✳✮

❚❤❡♦r❡♠ ✭▼❛ts✉♠♦t♦✲❙❤✐s❤✐❦✉r❛ ✷✵✵✷✱ ▼❡♥t③❡♥✲❙✐❡♠❛s③❦♦ ✷✵✵✹✮ ▲❡t (X, T) ❜❡ ❡✐t❤❡r ❛♥ ✐rr❛t✐♦♥❛❧ ❝✐r❝❧❡ r♦t❛t✐♦♥ ♦r ❛♥ ♦❞♦♠❡t❡r✳ ❆ t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡✱ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ❝♦❝②❝❧❡ ♦✈❡r (X, T) ❛❞♠✐ts ♥♦ ♠✐♥✐♠❛❧ s❡ts✳ ❚❤❡♦r❡♠ ✭▼❡♥t③❡♥✱ ❙✐❡♠❛s③❦♦ ✷✵✵✹✮

✶ ❊✈❡r② ♠✐♥✐♠❛❧ r♦t❛t✐♦♥ ♦♥ ❛♥ ✐♥✜♥✐t❡ ❝♦♠♣❛❝t ♠❡tr✐❝ ❣r♦✉♣

❛❞♠✐ts ❛ t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡ ❝♦❝②❝❧❡✳

✷ ■❢ (X, T) ✐s ❡✐t❤❡r ❛♥ ✐rr❛t✐♦♥❛❧ ❝✐r❝❧❡ r♦t❛t✐♦♥ ♦r ❛♥ ♦❞♦♠❡t❡r

t❤❡♥ (X, T) ❛❞♠✐ts ❛ t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡ ❝♦❝②❝❧❡ t❤❛t ✐s ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✳

❚❤❡♦r❡♠ ✭❋r→❝③❡❦✲▲❡♠❛➠❝③②❦✱ ✷✵✵✾✮ ❊✈❡r② ✐rr❛t✐♦♥❛❧ ❝✐r❝❧❡ r♦t❛t✐♦♥ ❛❞♠✐ts ❛ t♦♣✳ tr❛♥s✳ tr❛♥s❢♦r♠❛t✐♦♥ ✇✐t❤ ❛ ❧♦t ♦❢ ❞✐s❝r❡t❡ ♦r❜✐ts✿ t❤❡ s✉❜s❡t D := {x ∈ S✶ : (x, ✵) ❤❛s ❛ ❞✐s❝r❡t❡ ♦r❜✐t} ✇✐t❤ t❤❡ ❍❛✉s❞♦r✛ ❞✐♠❡♥s✐♦♥ ✐♥ (✵, ✶

✷)✳

❚❤❡♦r❡♠ ✭❉②♠❡❦✱ ✷✵✶✸✱ ♣r❡♣r✐♥t✮ ❊✈❡r② ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥ ❛❞♠✐ts ❛ t♦♣✳ tr❛♥s✳ ❝♦❝②❝❧❡ ✇✐t❤ t❤❡ s❡t D ♦❢ ❢✉❧❧ ❍❛✉s❞♦r✛ ❞✐♠❡♥s✐♦♥✳ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖ ▲❡t ✉s ❞❡♥♦t❡ D−,− = {x ∈ X : ❧✐♠j→±∞ f (j)(x) = −∞}✱ D+,+ = {x ∈ X : ❧✐♠j→±∞ f (j)(x) = +∞}✱ D−,+ = {x ∈ X : ❧✐♠j→±∞ f (j)(x) = ±∞}✱ D+,− = {x ∈ X : ❧✐♠j→±∞ f (j)(x) = ∓∞} ❚❤❡♦r❡♠ ✭❑✇✐❛t❦♦✇s❦✐✱ ❙✐❡♠❛s③❦♦ ✷✵✶✵✮ ❊✈❡r② ♦❞♦♠❡t❡r ❛❞♠✐ts ❛ t♦♣✳ tr❛♥s✳ ❝♦❝②❝❧❡ ✇✐t❤ ❛❧❧ D′s ❤❛✈✐♥❣ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ ❝♦♥t✐♥✉✉♠✳

Lx,y = {s ∈ R : s = ❧✐♠ f (j)(x), T j(x) → y ✐♥ X}, x, y ∈ X. P♦✐♥❝❛ré ❛tt❡♠♣t❡❞ t♦ ❝❧❛ss✐❢② ♣♦ss✐❜❧❡ ❢♦r♠s ♦❢ S = L(✵,✵)✳ P♦✐♥❝❛ré ❝❧❛✐♠❡❞ t❤❛t

✶ S = ∅ ✕ ❛ ❞✐s❝r❡t❡ ♦r❜✐t❀ ✷ S = {✵} ✕ ❝♦❜♦✉♥❞❛r②❀ ✸ S = R ✕ ❞❡♥s❡ ♦r❜✐t❀ ✹ S = τZ,

τ > ✵✳

❚❤❡♦r❡♠ ✭❑r②❣✐♥✱ ✶✾✼✺✮

✶ ❊✐t❤❡r S = ∅ ♦r S ✐s ❛ ❝❧♦s❡❞ s❡♠✐✲❣r♦✉♣ ✐♥ R✳ ✷ ■❢ S ✐s ❛ ❣r♦✉♣✱ t❤❡♥ ❡✐t❤❡r S = {✵} ♦r S = R✳ ✸ ■❢ K ✐s ❛ ❝❧♦s❡❞ s❡♠✐✲❣r♦✉♣ ♦❢ R t❤❛t ✐s ♥♦t ❛ ❣r♦✉♣✱ t❤❡♥ ❢♦r

❡❛❝❤ ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥ ✇✐t❤ ✉♥❜♦✉♥❞❡❞ ♣❛rt✐❛❧ q✉♦t✐❡♥ts t❤❡r❡ ❡①✐sts ❛ ❝♦❝②❝❧❡ f : S✶ − → R ✇✐t❤ S = K✳ ❚❤❡♦r❡♠ ✭❑r②❣✐♥✱✶✾✼✽✮ ▲❡t ✉s ❛ss✉♠❡ t❤❛t f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❚❤❡♥

✶ S = {✵} ✕ ❝♦❜♦✉♥❞❛r②❀ ✷ S = R ✕ ❞❡♥s❡ ♦r❜✐t❀ ✸ S = R+❀ ✹ S = R−✳

f ❤❛s t❤❡ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✱ s♦ S = ∅ ✐s ❡①❝❧✉❞❡❞ ❜② ▼✳✲❙✳

❖❞♦♠❡t❡r

❙❡t λt ≥ ✷✱ p−✶ = ✶, pt = λtpt−✶✱ t = ✵, ✶, ✷, . . .✳ Z

• λ
• =
• x =

∞

• t=✵

xtpt−✶ : xt ∈ {✵, ✶, . . . , λt − ✶}

• ✭✶✮

❋♦r x ∈ Z

• λ
• ❛♥❞ s ≥ ✵ ✇❡ ✇✐❧❧ ✇r✐t❡

X−✶ = ✵, Xs =

s

• t=✵

xtpt−✶. ❙❡t Tx = x + ¯ ✶✱ ¯ ✶ = ✶ + ✵ + ✵ + . . .✳ T ✕ ❛ ♠✐♥✐♠❛❧ ❤♦♠❡♦♠♦r♣❤✐s♠ ♦❢ Z

• λ
• T ❛❝ts ❛s ❛ ❝②❝❧❡ ♦♥ ❧❡✈❡❧s ♦❢ ❡✈❡r② T (t)✳

❙♣❡❝✐❛❧ ❝♦❝②❝❧❡s ♦♥ ♦❞♦♠❡t❡rs

❚❛❦❡ at ∈ R+, at < ∞✱ ❛♥❞ τt : {✵, ✶, . . . λt − ✶} − → {±✶, ✵} ✇✐t❤

λt−✶

• i=✵

τt(i) = ✵ ❛♥❞ τt ≡ ✵. ❉❡✜♥❡ f (x) =

∞

• t=✵

τt(xt)at. f ∈ C

• Z(λ)
• ❀
• fdµ = ✵❀

✐❢ ∃x∈Z(λ)(f (j)(x)) ✐s ✉♥❜♦✉♥❞❡❞✱ t❤❡♥ Tf ✐s t♦♣✳ tr❛♥s✳❀ ✐❢ Tf ❤❛s ❞✐s❝r❡t❡ ♦r❜✐ts✱ t❤❡♥ Tf ✐s t♦♣✳ tr❛♥s✳ ❛♥❞ f ❤❛s t❤❡ ✉♥❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✳

❉❡✜♥❡ ❛ ❝②❧✐♥❞r✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥ Tf : Z(λ) × R − → Z(λ) × R ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ Tf (x, r) = (Tx, r + f (x)). ❋♦r j ∈ Z ✇❡ ❤❛✈❡ T j

f (x, r) = (T j, r + f (j)(x)),

✇❤❡r❡ f (j)(x) =            j−✶

k=✵ f (T kx) :

j ≥ ✶ ✵ : j = ✵ − j

k=✶ f (T −kx) :

j > ✵. ✭✷✮

■♥ ♦r❞❡r t♦ ✐♥✈❡st✐❣❛t❡ ♣♦s✐t✐✈❡ s❡♠✐✲♦r❜✐ts ♦❢ Tf ✱ ✇❡ ❞❡✜♥❡ Lx,y = {s ∈ R : (y, s) ∈ ω(x, ✵)} = {s ∈ R; s = ❧✐♠ f (j)(x), j → y ✐♥ Z(λ)} ❢♦r x, y ∈ Z(λ)✳ ❍❡r❡ ω(x, r) = {(y, r + s) : y = ❧✐♠k→+∞ T jk(x), s = ❧✐♠k→+∞ f (jk)(x) ❢♦r s♦♠❡ jk→+∞} ❞❡♥♦t❡s t❤❡ ω✲❧✐♠✐t s❡t ♦❢ (x, r)✳ ❲❡ ✇✐❧❧ ✇r✐t❡ Lx := Lx,x✳ ❲❡ ✇✐❧❧ ❛❧s♦ ❝♦♥s✐❞❡r s♦♠❡ ♦t❤❡r ❢✉♥❝t✐♦♥s ❣❡♥❡r❛t❡❞ ❜② τt✳ ◆❛♠❡❧②✱ τ (s)

t

= τ (s)

t

(✵) =    ✵ : s = ✵; s−✶

k=✵ τt(k)

: s = ✶, . . . , λt. ✭✸✮ P❛rt✐❝✉❧❛r❧②✱ ✇❡ ❤❛✈❡ τ (λt)

t

= ✵ ❢♦r ❡✈❡r② t ≥ ✵✳

❙♦♠❡ ❢♦r♠✉❧❛s ❢♦r f (j)(x)✱ j > ✵✳

▲❡t Jt =

t

• s=✵

jsps−✶, ✵ ≤ js ≤ λs − ✶, jt > ✵. ✭✹✮ ❲❡ ❤❛✈❡ f (Jt)(✵) =

t

• s=✵
• ps−✶τ (js)

s

+ Js−✶τs(js)

• as + Jt

∞

• s=t+✶

τs(✵)as. ✭✺✮

◆♦✇✱ ✜① ❛♥ ❛r❜✐tr❛r② ❡❧❡♠❡♥t x = ∞

t=✵ xtpt−✶ ∈ Z(λ) ❛♥❞ ❛

♥❛t✉r❛❧ ♥✉♠❜❡r Jt ❞❡✜♥❡❞ ✐♥ ✭✹✮✳ ▲❡t l := x + Jt =

∞

• s=✵

ls · ps−✶ ∈ Z(λ). ❙✐♥❝❡ Jt ✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✱ ✇❡ ♠❛② ✜♥❞ t❤❡ s♠❛❧❧❡st ♥❛t✉r❛❧ ♥✉♠❜❡r v ≥ t s✉❝❤ t❤❛t lv > xv ❛♥❞ ls = xs ❢♦r s > v✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ f (Jt)(x) =

v

• s=✵
• ps−✶τ (ls)

s

+ Ls−✶τs(ls)

• as

−

v

• s=✵
• ps−✶τ (xs)

s

+ Xs−✶τs(xs)

• as

+ (Lv − Xv)

∞

• s=v+✶

τs(xs)as. ✭✻✮

❊①❛♠♣❧❡ ✶

▲❡t at = ✶/pt−✶, t = ✵, ✶, . . .✳ ❚❤❡♥ t❤❡ ❢♦r♠✉❧❛ ✭✻✮ t❛❦❡s t❤❡ ❢♦r♠ ❢♦r x = ✵ f (Jt)(✵) =

t

• s=✵
• τ (js)

s

+ Js−✶ ps−✶ τs(js)

• + Jt

pt

∞

• s=t+✶

pt ps−✶ τs(✵). ✭✼✮ ❲❡ ❛ss✉♠❡ ❛❞❞✐t✐♦♥❛❧❧② t❤❛t λt = λ, τt = τ✱ τ(✵) = ✶✱ τ(λ − ✶) = −✶ ❛♥❞ t❤❡ ♠❛♣ τ : Λ → {−✶, ✵, ✶}, t ≥ ✵✱ s❛t✐s✜❡s τ (s) ∈ {✵, ✶} ❢♦r ❡✈❡r② s = ✵, ✶, . . . , λ. ✭✽✮ ◆♦✇✱ t❤❡ ❢♦r♠✉❧❛ ✭✼✮ ❤❛s ❛ ❢♦r♠ f (Jt)(✵) =

t

• s=✵
• τ (js)

s

+ Js−✶τ(js)

• +

Jt · λ λ − ✶, ✭✾✮ ✇❤❡r❡ Js−✶ = js−✶

λ + · · · + j✵ λs ✱ s = ✶, ✷, . . .✳

■t ✐s ♥♦t ❞✐✣❝✉❧t t♦ s❡❡ t❤❛t ❛ss✉♠♣t✐♦♥ ✭✽✮ ✐♠♣❧✐❡s t❤❛t (τ (s), τ(s)) ❝❛♥ ♣♦ss✐❜❧② ❛❞♠✐t ♦♥❧② ♦♥❡ ♦❢ ❢♦✉r ✈❛❧✉❡s✱ ♥❛♠❡❧② (✵, ✵), (✵, ✶), (✶, ✵) ❛♥❞ (✶, −✶)✳ ▲❡t ✉s ❞✐✈✐❞❡ t❤❡ s❡t Λ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✈❛❧✉❡ ♦❢ ❛ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s (τ (s), τ(s))✱ s ∈ Λ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ♣✉t Λa,b = {s ∈ Λ : (τ (s), τ(s)) = (a, b)}. ❲❡ ❛ss✉♠❡ t❤❛t #Λ(a,b) ≥ ✷ ❢♦r ❡❛❝❤ (a, b)✳ ❉❡♥♦t✐♥❣ F(Jt) :=

t

• s=✵
• τ (js) +

Js−✶ · τ( Js−✶)

• ✇❡ ❝❛♥ ✇r✐t❡

f (Jt)(✵) = F(Jt) + Jt · λ λ − ✶. ✭✶✵✮

❚❛❦✐♥❣ ✐♥t♦ ❝♦♥s✐❞❡r❛t✐♦♥ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ s❡t λ = {✵, ✶, . . . , λ − ✶} ✐♥t♦ s❡ts Λ(✵,✵), Λ(✵,✶), Λ(−✶,✶), Λ(✶,p) ✇❡ ❤❛✈❡ F(Jt) = τ (j✵) +

• js∈Λ(✵,✶)

✶≤s≤t

( Js−✶) +

• js∈Λ(✶,−✶)

✶≤s≤t

(✶ − Js−✶) +#{✶ ≤ s ≤ t; js ∈ Λ(✶,✵)}. ✭✶✶✮

■♥ ♦r❞❡r t♦ ✐♥✈❡st✐❣❛t❡ ❛❝❝✉♠✉❧❛t✐♦♥ ♣♦✐♥ts ♦❢ t❤❡ s❡q✉❡♥❝❡ (f (Jt)(✵))✱ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ ❛♥❛❧②③❡ ❜❧♦❝❦s ♦❢ ❝♦❡✣❝✐❡♥ts B = Bt = (j✵, j✶, . . . , jt) ∈ Λ × . . . × Λ

• t+✶

♦❢ t❤❡ ♥✉♠❜❡r Jt = t

u=✵ juλu✱ ✇❤❡r❡ Jt → ∞✱ ✇❤✐❧❡ t → ∞ ❛❧♦♥❣ t❤❡

s✉❜s❡q✉❡♥❝❡ ♦❢ t❤❡ ♥❛t✉r❛❧ ♥✉♠❜❡rs✳

• ✐✈❡♥ s✉❝❤ ❛ ❜❧♦❝❦ B = Bt✱ ✇❡ ✇✐❧❧ ❞❡♥♦t❡

J(B) =

t

• u=✵

juλu. ■❢ (a, b) ∈ {(✵, ✵), (✵, ✶), (✶, ✵), (✶, −✶)}✱ t❤❡♥ ❜② ❛ ❝❧✉st❡r ♦✈❡r (a, b) ✇❡ ♠❡❛♥ ❛ ❜❧♦❝❦ C = (c✶ . . . cd) ⊂ B s✉❝❤ t❤❛t ci ∈ Λa,b ❢♦r ❡❛❝❤ i = ✶, . . . d✳ ❲❡ ✇✐❧❧ r❡♣r❡s❡♥t ❜❧♦❝❦s Bt ❞❡s❝r✐❜✐♥❣ ♥✉♠❜❡rs Jt ❛s ❛ s❡q✉❡♥❝❡s ♦❢ ❝❧✉st❡rs ♦✈❡r ♣❛✐rs (a, b)✳

▲❡t Bt = C✶ . . . Ck, k = k(t) ✭✶✷✮ ❜❡ s✉❝❤ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ s♦♠❡ ❜❧♦❝❦ B✳ ▲❡t ✉s ❛ss✉♠❡ ❛ ❢❡✇ ❢✉rt❤❡r ♥♦t❛t✐♦♥s✿ it(a, b) = i(a, b) ✕ t❤❡ ♥✉♠❜❡r ♦❢ ❝❧✉st❡rs ✐♥ ✭✶✷✮ t❤❛t ❛r❡ ♦✈❡r (a, b)❀ lt(✶, ✵) = l(✵, ✶) ✕ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐♦♥s s ✐♥ Bt s✉❝❤ t❤❛t js ∈ Λ✶,✵

s ✱ s = ✶, . . . , t❀

l∗t(✵, ✶) = l∗(✵, ✶) ✕ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐♦♥s s ✐♥ Bt s✉❝❤ t❤❛t js ∈ Λ✵,✶

s

\ {✵}❀ l∗t(✶, −✶) = l∗(✶, −✶) ✕ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐♦♥s s ✐♥ Bt s✉❝❤ t❤❛t js ∈ Λ✶,−✶

s

\ {λ − ✶}✳

Pr♦♣♦s✐t✐♦♥ ▲❡t Bt ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❜❧♦❝❦s ♦❢ t❤❡ ❢♦r♠ ✭✶✷✮ s✉❝❤ t❤❛t J(Bt) → ∞✳ ❚❤❡♥ f (J(Bt))(✵) ≤ K < ∞ ❢♦r s♦♠❡ K ∈ R ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡❛❝❤ ♦❢ t❤❡ s❡q✉❡♥❝❡s (it(a, b)), (lt(✶, ✵)), (l∗t(✵, ✶)), (l∗t(✶, −✶)) ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❛❜♦✈❡✳ ❘❡❝❛❧❧ t❤❛t t❤❡ t → ∞ ♠❡❛♥s t❤❛t t ❣♦❡s t♦ t❤❡ ✐♥✜♥✐t② ❛❧♦♥❣ s♦♠❡ s✉❜s❡q✉❡♥❝❡ ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs✳

❚❤❡ s❡ts Lx ❛♥❞ Lxy✳

❲❡ st❛rt ✇✐t❤ ❛ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ s❡ts L✵ ❛♥❞ L✵y✱ y ∈ Z(λ)✳ ❉❡✜♥❡ Cant(Λ(✵,✵)) =

• x =

∞

• i=✶

ci λi , ci ∈ Λ(✵,✵)

• ,

Cantl(Λ(a,b)) =

• x =

l

• i=✶

ci λi , ci ∈ Λ(a,b)

• , l = ✶, ✷, . . . ,

S(Λ(✵,✶)) = {c✶ + · · · + cn; ci ∈ Λ(✵,✶), ci > ✵, n = ✶, ✷, . . .} S′(Λ(✶,−✶)) = {(λ − ✶ − c✶) + · · · + (λ − ✶ − cn); ci ∈ Λ(✶,−✶), ci < (λ − ✶), n = ✶, ✷, . . .} Cant∗

s (Λ(✵,✶))

= {x =

i≥✵ ci λi , ci ∈ Λ(✵,✶), i≥✶

ci = s}, s ∈ S(Λ(✵,✶)), s > ✵ Cant∗

s′(Λ(✶,−✶)) =

{x =

i≥✶ ci λi , ci ∈ Λ(✶,−✶), i≥✶

(λ − ✶ − ci) = s′}, s′ ∈ S′(Λ(✶,−✶)), s′ > ✵.

Pr♦♣♦s✐t✐♦♥ ❚❤❡ s❡t L✵ ✐s ❛ ♥♦♥✲❡♠♣t②✱ ❝❧♦s❡❞ s❡♠✐✲❣r♦✉♣ ♦❢ R+✳ L✵ ❝♦♥t❛✐♥s t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜s❡ts✿

{

λ λ−✶ · Cant(Λ(✵,✵))}✱

{

λ λ−✶ · Cantl(Λ(✵,✵)), l = ✶, ✷, . . .}✱

{l +

λ λ−✶ · Cantl(Λ(✶,✵)), l = ✶, ✷, . . .}✱

{

s λ−✶, s ∈ S(Λ(✵,✶))}✱

{

λ λ−✶ · Cantl(Λ(✵,✵)) + λ λ−✶ · Cant∗ s (Λ(✵,✶)), l = ✶, ✷, . . . , s ∈

S(Λ(✵,✶))}✱ { s′

λ−✶ + λ λ−✶ · Cants′(Λ(✶,−✶)), s′ ∈ S′(Λ(✶,−✶))}✱

❛♥❞ ❛❧❧ t❤❡✐rs ▼✐♥❦♦✇s❦✐ s✉♠s✳

❘❡♠❛r❦ ❊✐t❤❡r L✵ ✐s ❛ s❡t ♦❢ ✜rst ❝❛t❡❣♦r② ♦r L✵ ❝♦♥t❛✐♥s ❛ r❛② [a, ∞]✱ a > ✵✳ L✵ ✐s ❛ ✜rst ❝❛t❡❣♦r② ✐✛ Λ(✵,✵) Λ = {✵, ✶, . . . , λ − ✶}, ✇❤❡r❡ Λ(✵,✵) ✐s t❤❡ s♠❛❧❧❡st s✉❜❣r♦✉♣ ♦❢ Λ ❣❡♥❡r❛t❡❞ ❜② Λ(✵,✵)✳ ❚♦ ❣✐✈❡ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ s❡ts Lx ❛♥❞ Lxy✱ ✇❡ ❞❡✜♥❡ t✇♦ s✉❜s❡ts ∆✵ ❛♥❞ ∆✶ ♦❢ Z(λ) ✳ ∆✵ = {x =

∞

• t=✵

xtλt s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts t✵ ≥ ✵ s❛t✐s❢②✐♥❣✿ ❡✐t❤❡r xt = ✵, t ≥ t✵ ♦r xt = (λ − ✶), t ≥ t✵ ♦r xt ∈ Λ(✵,✵) ❢♦r t ≥ t✵}, ∆✶ = {x =

∞

• t=✵

xtλt; s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts t✵ ≥ ✵, s❛t✐s❢②✐♥❣ xt ∈ Λ(✶,✵) ❢♦r t ≥ t✵}.

Pr♦♣♦s✐t✐♦♥ ■t ❤♦❧❞s L✵y = ∅ ✐❢ y ∈ Z(λ) \ ∆✵✱ L✵y = L✵ + F(y)✱ F(y) > ✵ ✐❢ y ∈ ∆✵✱ Ly = L✵ − ry✱ ❢♦r s♦♠❡ ry > ✵ ✐❢ y ∈ ∆✵✱ Ly = [−∞, ry]✱ ❢♦r s♦♠❡ ry > ✵ ✐❢ y ∈ ∆✶✱ Ly = [−∞, ∞] ❢♦r y ∈ ∆✵ ∪ ∆✶✱ ✐✳❡✳ Tf ✲♦r❜✐t ♦❢ (y, ✵) ✐s ❞❡♥s❡ ✐♥ X × R✳

❊①❛♠♣❧❡ ✷

◆♦✇✱ ❧❡t λ = ✷, τ(✵) = ✶, τ(✶) = −✶ ❛♥❞ ❧❡t r❡❛❧ ♥✉♠❜❡rs {at} s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥s✿

∞

• t=✵

at < ∞,

∞

• t=✵

(✷t · at) = ∞, (✷t · at) ց ✵. ❘❡❝❛❧❧ t❤❛t f (x) =

∞

• t=✵

atτ(xt)✱ x =

∞

• t=✵

xt · ✷t ∈ Z(λ)✳ ❆❣❛✐♥✱ ✇❡ ❞❡✜♥❡ s♣❡❝✐❛❧ s✉❜s❡ts ∆✵ ❛♥❞ ∆✶ ♦❢ Z(λ)✳ ❚♦ ❞♦ ✐t✱ ✇❡ r❡♣r❡s❡♥t t❤❡ s❡q✉❡♥❝❡ (x✵, x✶, . . .) ❜② ❛ s❡q✉❡♥❝❡ ♦❢ t❤❡ s❡r✐❡s ♦❢ ③❡r♦s ♦r ♦♥❡s✳ ▲❡t N✵ = N✵(x) ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ t = ✵, ✶, . . . ✇❤✐❝❤ ❛r❡ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❛♥② s❡r✐❡s ♦❢ ③❡r♦s ♦r ♦♥❡s ✐♥ t❤❡ ❡①♣❛♥s✐♦♥ ♦❢ x✳

❉❡✜♥❡ ∆✵ =   x =

∞

• t=✵

(xt · ✷t),

• t∈N✵

(✷t · at) < ∞    . ◆❡①t✱ ✇❡ ❞❡✜♥❡ ❛ s❡t N✶ = N✶(x) ❛s ❛ s❡t ♦❢ ❛❧❧ t = ✵, ✶, . . . s✉❝❤ t❤❛t xt ❛♣♣❡❛rs ✐♥ ❛ s❡r✐❡s ♦❢ ③❡r♦s ♦r ♦♥❡s ✇✐t❤ t❤❡ ❧❡♥❣t❤ ≥ ✷✱ ❡①❝❡♣t ❢♦r t❤❡ ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥✳ ❚❤❡♥✱ ❞❡✜♥❡ ∆✶ ❛s ❢♦❧❧♦✇s ∆✶ =   x =

∞

• t=✶

(xt · ✷t),

• t∈N✶

(✷t · at) < ∞    .

Pr♦♣♦s✐t✐♦♥ ■t ❤♦❧❞s L✵ = [✵, ∞]✱ Lx = [−rx, ∞] ❢♦r s♦♠❡ rx > ✵✱ x ∈ ∆✵✱ Lx = [−∞, r′

x] ❢♦r s♦♠❡ r′ x > ✵✱ x ∈ ∆✶✱

Lx = [−∞, ∞] ❢♦r x / ∈ ∆✵ ∪ ∆✶✱ Lxy = ∅ ✐❢ ✭x ∈ ∆✵✱ y / ∈ ∆✵✮ ♦r ✭x ∈ ∆✶✱ y ∈ ∆✶✮✱ Lxy = ∅ ✐❢ x, y ∈ ∆✵ ♦r x, y ∈ ∆✶✳

❘❡❢❡r❡♥❝❡s

❆✳❙✳ ❇❡s✐❝♦✈✐t❝❤✱ ❆ ♣r♦❜❧❡♠ ♦♥ t♦♣♦❧♦❣✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡✳ ■■✱ Pr♦❝✳ ❈❛♠❜r✐❞❣❡ P❤✐❧♦s✳ ❙♦❝✳ ✹✼✱ ✭✶✾✺✶✮✱ ✸✽✲✹✺✳ ❑✳ ❋r→❝③❡❦✱ ▼✳ ▲❡♠❛➠❝③②❦ ❖♥ t❤❡ ❍❛✉s❞♦r✛ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ s❡t ♦❢ ❝❧♦s❡❞ ♦r❜✐ts ❢♦r ❛ ❝②❧✐♥❞r✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥✱ ◆♦♥❧✐♥❡❛r✐t② ✷✸ ✭✷✵✶✵✮✱ ♥♦✳ ✶✵✱ ✷✸✾✸✕✷✹✷✷✳ ❉✳❑✳ ●❛♥❣✉❧✐✱ ❈❤✳ ❉✉tt❛✱ ❙♦♠❡ r❡♠❛r❦s ♦♥ s✉❜s❡r✐❡s ♦❢ ❞❡❝❡r❣❡♥t s❡r✐❡s✱ ❙♦♦❝❤♦✇ ❏♦✉r♥❛❧ ♦❢ ▼❛t❡♠❛t✐❝s✱ ❱♦❧✉♠❡ ✸✶ ✭✷✵✵✺✮✱ ◆♦✳ ✹✱ ✺✸✼✲✺✹✷✳ ❲✳❍✳ ●♦tts❝❤❛❧❦✱ ●✳❆✳ ❍❡❞❧✉♥❞✱ ❚♦♣♦❧♦❣✐❝❛❧ ❉②♥❛♠✐❝s✱ ❆▼❙ ❈♦❧❧♦q✉✐✉♠ P✉❜❧✐❝❛t✐♦♥s✱ ✸✻ ✭✶✾✺✺✮✳ ❆✳❇✳ ❑r②❣✐♥✱ ❚❤❡ ω✲❧✐♠✐t s❡ts ♦❢ ❛ ❝②❧✐♥❞r✐❝❛❧ ❝❛s❝❛❞❡ ✭❘✉ss✐❛♥✮✱ ■③✈✳ ❘♦ss✳ ❆❦❛❞✳ ◆❛✉❦ ❙❡r✳ ▼❛t✳ ✸✾ ✭✶✾✼✺✮✱ ♥♦✳ ✹✱ ✽✼✾✲✽✾✽✳ ❆✳❇✳ ❑r②❣✐♥✱ ω✲❧✐♠✐t s❡ts ♦❢ s♠♦♦t❤ ❝②❧✐♥❞r✐❝❛❧ ❝❛s❝❛❞❡s ✭❘✉ss✐❛♥✮✱ ▼❛t✳ ❩❛♠❡t❦✐ ✷✸ ✭✶✾✼✽✮✱ ♥♦✳ ✻✱ ✽✼✸✕✽✽✹✳

❘❡❢❡r❡♥❝❡s

❏✳ ❑✇✐❛t❦♦✇s❦✐✱ ❆✳ ❙✐❡♠❛s③❦♦✱ ❉✐s❝r❡t❡ ♦r❜✐ts ✐♥ t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡ ❝②❧✐♥❞r✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥s✱ ❉✐s❝r❡t❡ ❈♦♥t✐♥✳ ❉②♥✳ ❙②st✳ ✷✼ ✭✷✵✶✵✮✱ ♥♦✳ ✸✱ ✾✹✺ ✕✾✻✶✳ ▼✳ ▲❡♠❛➠❝③②❦✱ ▼✳❑✳ ▼❡♥t③❡♥✱ ❚♦♣♦❧♦❣✐❝❛❧ ❡r❣♦❞✐❝✐t② ♦❢ r❡❛❧ ❝♦❝②❝❧❡s ♦✈❡r ♠✐♥✐♠❛❧ r♦t❛t✐♦♥s✱ ▼♦♥❛ts❤✳ ▼❛t❤✳ ✶✸✹✱ ✭✷✵✵✷✮✱ ✷✷✼✲✷✹✻✳ ❙✳ ▼❛ts✉♠♦t♦✱ ▼✳ ❙❤✐s❤✐❦✉r❛✱ ▼✐♥✐♠❛❧ s❡ts ♦❢ ❝❡rt❛✐♥ ❛♥♥✉❧❛r ❤♦♠❡♦♠♦r♣❤✐s♠s✱ ❍✐r♦s❤✐♠❛ ▼❛t❤✳ ❏✳ ✸✷ ✭✷✵✵✷✮✱ ✷✵✼✲✷✶✺✳ ▼✳❑✳ ▼❡♥t③❡♥✱ ❆✳ ❙✐❡♠❛s③❦♦✱ ❈②❧✐♥❞❡r ❝♦❝②❝❧❡ ❡①t❡♥s✐♦♥s ♦❢ ♠✐♥✐♠❛❧ r♦t❛t✐♦♥s ♦♥ ♠♦♥♦t❤❡t✐❝ ❣r♦✉♣s✱ ❈♦❧❧✳ ▼❛t❤✳ ✶✵✶ ✭✷✵✵✹✮✱ ✼✺✲✽✽✳ ❑✳ ◆✐❦♦❞❡♠✱ ❩✳ Pá❧❡s✱ ▼✐♥❦♦✇s❦✐ s✉♠s ♦❢ ❈❛♥t♦r✲t✉♣❡ s❡ts✱ ❈♦❧❧✳ ▼❛t❤✳ ❱❖▲✳ ✶✶✾ ✭✷✵✶✵✮✱ ◆✵✳ ✶✱ ✾✺✲✶✵✽✳

❘❡❢❡r❡♥❝❡s

❍✳ P♦✐♥❝❛ré✱ ❙✉r ❧❡s ❝♦✉r❜❡s ❞é✜♥✐❡s ♣❛r ✉♥❡ éq✉❛t✐♦♥ ❞✐✛ér❡♥t✐❡❧❧❡✱ ❏♦✉r♥❛❧ ❞❡ ♠❛t❤é♠❛t✐q✉❡s ♣✉r❡s ❡t ❛♣♣❧✐q✉é❡s✱ ✭■❱✮ ✷ ✭✶✽✽✻✮✱ ✶✺✶✲✷✶✼✳ ❚✳ ➆❛❧át✱ ❖♥ ❙✉❜s❡r✐❡s ♦❢ ❉✐✈❡r❣❡♥t ❙❡r✐❡s✱ ▼❛t❡♠❛t✐❝❦✞ ② ↔❛s♦♣✐s✱ ❱♦❧✳ ✶✽ ✭✶✾✻✽✮✱ ◆♦✳✹✱ ✸✶✷✲✸✸✽✳

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳