# Translation number of a monotone real map that commutes with x ↦ x + 1#

Let f : ℝ → ℝ be a monotone map such that f (x + 1) = f x + 1 for all x. Then the limit $$\tau(f)=\lim_{n\to\infty}{f^n(x)-x}{n}$$ exists and does not depend on x. This number is called the translation number of f. Different authors use different notation for this number: τ, ρ, rot, etc

In this file we define a structure CircleDeg1Lift for bundled maps with these properties, define translation number of f : CircleDeg1Lift, prove some estimates relating f^n(x)-x to τ(f). In case of a continuous map f we also prove that f admits a point x such that f^n(x)=x+m if and only if τ(f)=m/n.

Maps of this type naturally appear as lifts of orientation preserving circle homeomorphisms. More precisely, let f be an orientation preserving homeomorphism of the circle $S^1=ℝ/ℤ$, and consider a real number a such that ⟦a⟧ = f 0, where ⟦⟧ means the natural projection ℝ → ℝ/ℤ. Then there exists a unique continuous function F : ℝ → ℝ such that F 0 = a and ⟦F x⟧ = f ⟦x⟧ for all x (this fact is not formalized yet). This function is strictly monotone, continuous, and satisfies F (x + 1) = F x + 1. The number ⟦τ F⟧ : ℝ / ℤ is called the rotation number of f. It does not depend on the choice of a.

## Main definitions #

• CircleDeg1Lift: a monotone map f : ℝ → ℝ such that f (x + 1) = f x + 1 for all x; the type CircleDeg1Lift is equipped with Lattice and Monoid structures; the multiplication is given by composition: (f * g) x = f (g x).
• CircleDeg1Lift.translationNumber: translation number of f : CircleDeg1Lift.

## Main statements #

We prove the following properties of CircleDeg1Lift.translationNumber.

• CircleDeg1Lift.translationNumber_eq_of_dist_bounded: if the distance between (f^n) 0 and (g^n) 0 is bounded from above uniformly in n : ℕ, then f and g have equal translation numbers.

• CircleDeg1Lift.translationNumber_eq_of_semiconjBy: if two CircleDeg1Lift maps f, g are semiconjugate by a CircleDeg1Lift map, then τ f = τ g.

• CircleDeg1Lift.translationNumber_units_inv: if f is an invertible CircleDeg1Lift map (equivalently, f is a lift of an orientation-preserving circle homeomorphism), then the translation number of f⁻¹ is the negative of the translation number of f.

• CircleDeg1Lift.translationNumber_mul_of_commute: if f and g commute, then τ (f * g) = τ f + τ g.

• CircleDeg1Lift.translationNumber_eq_rat_iff: the translation number of f is equal to a rational number m / n if and only if (f^n) x = x + m for some x.

• CircleDeg1Lift.semiconj_of_bijective_of_translationNumber_eq: if f and g are two bijective CircleDeg1Lift maps and their translation numbers are equal, then these maps are semiconjugate to each other.

• CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq: let f₁ and f₂ be two actions of a group G on the circle by degree 1 maps (formally, f₁ and f₂ are two homomorphisms from G →* CircleDeg1Lift). If the translation numbers of f₁ g and f₂ g are equal to each other for all g : G, then these two actions are semiconjugate by some F : CircleDeg1Lift. This is a version of Proposition 5.4 from Étienne Ghys, Groupes d'homeomorphismes du cercle et cohomologie bornee.

## Notation #

We use a local notation τ for the translation number of f : CircleDeg1Lift.

## Implementation notes #

We define the translation number of f : CircleDeg1Lift to be the limit of the sequence (f ^ (2 ^ n)) 0 / (2 ^ n), then prove that ((f ^ n) x - x) / n tends to this number for any x. This way it is much easier to prove that the limit exists and basic properties of the limit.

We define translation number for a wider class of maps f : ℝ → ℝ instead of lifts of orientation preserving circle homeomorphisms for two reasons:

• non-strictly monotone circle self-maps with discontinuities naturally appear as Poincaré maps for some flows on the two-torus (e.g., one can take a constant flow and glue in a few Cherry cells);
• definition and some basic properties still work for this class.

## TODO #

Here are some short-term goals.

• Introduce a structure or a typeclass for lifts of circle homeomorphisms. We use Units CircleDeg1Lift for now, but it's better to have a dedicated type (or a typeclass?).

• Prove that the SemiconjBy relation on circle homeomorphisms is an equivalence relation.

• Introduce ConditionallyCompleteLattice structure, use it in the proof of CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq.

• Prove that the orbits of the irrational rotation are dense in the circle. Deduce that a homeomorphism with an irrational rotation is semiconjugate to the corresponding irrational translation by a continuous CircleDeg1Lift.

## Tags #

circle homeomorphism, rotation number

### Definition and monoid structure #

structure CircleDeg1Liftextends :

A lift of a monotone degree one map S¹ → S¹.

• toFun :
• monotone' : Monotone self.toFun
• map_add_one' : ∀ (x : ), self.toFun (x + 1) = self.toFun x + 1
Instances For
theorem CircleDeg1Lift.map_add_one' (self : CircleDeg1Lift) (x : ) :
self.toFun (x + 1) = self.toFun x + 1
Equations
@[simp]
theorem CircleDeg1Lift.coe_mk (f : ) (h : ∀ (x : ), f.toFun (x + 1) = f.toFun x + 1) :
{ toOrderHom := f, map_add_one' := h } = f
@[simp]
theorem CircleDeg1Lift.coe_toOrderHom (f : CircleDeg1Lift) :
f.toOrderHom = f
theorem CircleDeg1Lift.mono (f : CircleDeg1Lift) {x : } {y : } (h : x y) :
f x f y
@[simp]
theorem CircleDeg1Lift.map_add_one (f : CircleDeg1Lift) (x : ) :
f (x + 1) = f x + 1
@[simp]
theorem CircleDeg1Lift.map_one_add (f : CircleDeg1Lift) (x : ) :
f (1 + x) = 1 + f x
theorem CircleDeg1Lift.ext_iff {f : CircleDeg1Lift} {g : CircleDeg1Lift} :
f = g ∀ (x : ), f x = g x
theorem CircleDeg1Lift.ext ⦃f : CircleDeg1Lift ⦃g : CircleDeg1Lift (h : ∀ (x : ), f x = g x) :
f = g
Equations
@[simp]
theorem CircleDeg1Lift.coe_mul (f : CircleDeg1Lift) (g : CircleDeg1Lift) :
(f * g) = f g
theorem CircleDeg1Lift.mul_apply (f : CircleDeg1Lift) (g : CircleDeg1Lift) (x : ) :
(f * g) x = f (g x)
@[simp]
theorem CircleDeg1Lift.coe_one :
1 = id
instance CircleDeg1Lift.unitsHasCoeToFun :
CoeFun fun (x : ) =>
Equations
@[simp]
theorem CircleDeg1Lift.units_inv_apply_apply (f : ) (x : ) :
f⁻¹ (f x) = x
@[simp]
theorem CircleDeg1Lift.units_apply_inv_apply (f : ) (x : ) :
f (f⁻¹ x) = x

If a lift of a circle map is bijective, then it is an order automorphism of the line.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem CircleDeg1Lift.coe_toOrderIso (f : ) :
(CircleDeg1Lift.toOrderIso f) = f
@[simp]
theorem CircleDeg1Lift.coe_toOrderIso_symm (f : ) :
(CircleDeg1Lift.toOrderIso f).symm = f⁻¹
@[simp]
theorem CircleDeg1Lift.coe_toOrderIso_inv (f : ) :
(CircleDeg1Lift.toOrderIso f)⁻¹ = f⁻¹
theorem CircleDeg1Lift.coe_pow (f : CircleDeg1Lift) (n : ) :
(f ^ n) = (⇑f)^[n]

### Translate by a constant #

The map y ↦ x + y as a CircleDeg1Lift. More precisely, we define a homomorphism from Multiplicative ℝ to CircleDeg1Liftˣ, so the translation by x is translation (Multiplicative.ofAdd x).

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem CircleDeg1Lift.translate_apply (x : ) (y : ) :
(CircleDeg1Lift.translate (Multiplicative.ofAdd x)) y = x + y
@[simp]
theorem CircleDeg1Lift.translate_inv_apply (x : ) (y : ) :
(CircleDeg1Lift.translate (Multiplicative.ofAdd x))⁻¹ y = -x + y
@[simp]
theorem CircleDeg1Lift.translate_zpow (x : ) (n : ) :
@[simp]
theorem CircleDeg1Lift.translate_pow (x : ) (n : ) :
@[simp]
theorem CircleDeg1Lift.translate_iterate (x : ) (n : ) :

### Commutativity with integer translations #

In this section we prove that f commutes with translations by an integer number. First we formulate these statements (for a natural or an integer number, addition on the left or on the right, addition or subtraction) using Function.Commute, then reformulate as simp lemmas map_int_add etc.

theorem CircleDeg1Lift.commute_nat_add (f : CircleDeg1Lift) (n : ) :
Function.Commute f fun (x : ) => n + x
theorem CircleDeg1Lift.commute_add_nat (f : CircleDeg1Lift) (n : ) :
Function.Commute f fun (x : ) => x + n
theorem CircleDeg1Lift.commute_sub_nat (f : CircleDeg1Lift) (n : ) :
Function.Commute f fun (x : ) => x - n
theorem CircleDeg1Lift.commute_add_int (f : CircleDeg1Lift) (n : ) :
Function.Commute f fun (x : ) => x + n
theorem CircleDeg1Lift.commute_int_add (f : CircleDeg1Lift) (n : ) :
Function.Commute f fun (x : ) => n + x
theorem CircleDeg1Lift.commute_sub_int (f : CircleDeg1Lift) (n : ) :
Function.Commute f fun (x : ) => x - n
@[simp]
theorem CircleDeg1Lift.map_int_add (f : CircleDeg1Lift) (m : ) (x : ) :
f (m + x) = m + f x
@[simp]
theorem CircleDeg1Lift.map_add_int (f : CircleDeg1Lift) (x : ) (m : ) :
f (x + m) = f x + m
@[simp]
theorem CircleDeg1Lift.map_sub_int (f : CircleDeg1Lift) (x : ) (n : ) :
f (x - n) = f x - n
@[simp]
theorem CircleDeg1Lift.map_add_nat (f : CircleDeg1Lift) (x : ) (n : ) :
f (x + n) = f x + n
@[simp]
theorem CircleDeg1Lift.map_nat_add (f : CircleDeg1Lift) (n : ) (x : ) :
f (n + x) = n + f x
@[simp]
theorem CircleDeg1Lift.map_sub_nat (f : CircleDeg1Lift) (x : ) (n : ) :
f (x - n) = f x - n
theorem CircleDeg1Lift.map_int_of_map_zero (f : CircleDeg1Lift) (n : ) :
f n = f 0 + n

### Pointwise order on circle maps #

noncomputable instance CircleDeg1Lift.instLattice :

Monotone circle maps form a lattice with respect to the pointwise order

Equations
@[simp]
theorem CircleDeg1Lift.sup_apply (f : CircleDeg1Lift) (g : CircleDeg1Lift) (x : ) :
(f g) x = max (f x) (g x)
@[simp]
theorem CircleDeg1Lift.inf_apply (f : CircleDeg1Lift) (g : CircleDeg1Lift) (x : ) :
(f g) x = min (f x) (g x)
theorem CircleDeg1Lift.iterate_mono {f : CircleDeg1Lift} {g : CircleDeg1Lift} (h : f g) (n : ) :
(⇑f)^[n] (⇑g)^[n]
theorem CircleDeg1Lift.pow_mono {f : CircleDeg1Lift} {g : CircleDeg1Lift} (h : f g) (n : ) :
f ^ n g ^ n

### Estimates on (f * g) 0#

We prove the estimates f 0 + ⌊g 0⌋ ≤ f (g 0) ≤ f 0 + ⌈g 0⌉ and some corollaries with added/removed floors and ceils.

We also prove that for two semiconjugate maps g₁, g₂, the distance between g₁ 0 and g₂ 0 is less than two.

theorem CircleDeg1Lift.dist_map_zero_lt_of_semiconj {f : CircleDeg1Lift} {g₁ : CircleDeg1Lift} {g₂ : CircleDeg1Lift} (h : Function.Semiconj f g₁ g₂) :
dist (g₁ 0) (g₂ 0) < 2
theorem CircleDeg1Lift.dist_map_zero_lt_of_semiconjBy {f : CircleDeg1Lift} {g₁ : CircleDeg1Lift} {g₂ : CircleDeg1Lift} (h : SemiconjBy f g₁ g₂) :
dist (g₁ 0) (g₂ 0) < 2

### Limits at infinities and continuity #

theorem CircleDeg1Lift.tendsto_atBot (f : CircleDeg1Lift) :
Filter.Tendsto (⇑f) Filter.atBot Filter.atBot
theorem CircleDeg1Lift.tendsto_atTop (f : CircleDeg1Lift) :
Filter.Tendsto (⇑f) Filter.atTop Filter.atTop

### Estimates on (f^n) x#

If we know that f x is ≤/</≥/>/= to x + m, then we have a similar estimate on f^[n] x and x + n * m.

For ≤, ≥, and = we formulate both of (implication) and iff versions because implications work for n = 0. For < and > we formulate only iff versions.

theorem CircleDeg1Lift.iterate_le_of_map_le_add_int (f : CircleDeg1Lift) {x : } {m : } (h : f x x + m) (n : ) :
(⇑f)^[n] x x + n * m
theorem CircleDeg1Lift.le_iterate_of_add_int_le_map (f : CircleDeg1Lift) {x : } {m : } (h : x + m f x) (n : ) :
x + n * m (⇑f)^[n] x
theorem CircleDeg1Lift.iterate_eq_of_map_eq_add_int (f : CircleDeg1Lift) {x : } {m : } (h : f x = x + m) (n : ) :
(⇑f)^[n] x = x + n * m
theorem CircleDeg1Lift.iterate_pos_le_iff (f : CircleDeg1Lift) {x : } {m : } {n : } (hn : 0 < n) :
(⇑f)^[n] x x + n * m f x x + m
theorem CircleDeg1Lift.iterate_pos_lt_iff (f : CircleDeg1Lift) {x : } {m : } {n : } (hn : 0 < n) :
(⇑f)^[n] x < x + n * m f x < x + m
theorem CircleDeg1Lift.iterate_pos_eq_iff (f : CircleDeg1Lift) {x : } {m : } {n : } (hn : 0 < n) :
(⇑f)^[n] x = x + n * m f x = x + m
theorem CircleDeg1Lift.le_iterate_pos_iff (f : CircleDeg1Lift) {x : } {m : } {n : } (hn : 0 < n) :
x + n * m (⇑f)^[n] x x + m f x
theorem CircleDeg1Lift.lt_iterate_pos_iff (f : CircleDeg1Lift) {x : } {m : } {n : } (hn : 0 < n) :
x + n * m < (⇑f)^[n] x x + m < f x

### Definition of translation number #

An auxiliary sequence used to define the translation number.

Equations
• f.transnumAuxSeq n = (f ^ 2 ^ n) 0 / 2 ^ n
Instances For

The translation number of a CircleDeg1Lift, $τ(f)=\lim_{n→∞}\frac{f^n(x)-x}{n}$. We use an auxiliary sequence \frac{f^{2^n}(0)}{2^n} to define τ(f) because some proofs are simpler this way.

Equations
• f.translationNumber = limUnder Filter.atTop f.transnumAuxSeq
Instances For
theorem CircleDeg1Lift.transnumAuxSeq_def (f : CircleDeg1Lift) :
f.transnumAuxSeq = fun (n : ) => (f ^ 2 ^ n) 0 / 2 ^ n
theorem CircleDeg1Lift.translationNumber_eq_of_tendsto_aux (f : CircleDeg1Lift) {τ' : } (h : Filter.Tendsto f.transnumAuxSeq Filter.atTop (nhds τ')) :
f.translationNumber = τ'
theorem CircleDeg1Lift.translationNumber_eq_of_tendsto₀ (f : CircleDeg1Lift) {τ' : } (h : Filter.Tendsto (fun (n : ) => (⇑f)^[n] 0 / n) Filter.atTop (nhds τ')) :
f.translationNumber = τ'
theorem CircleDeg1Lift.translationNumber_eq_of_tendsto₀' (f : CircleDeg1Lift) {τ' : } (h : Filter.Tendsto (fun (n : ) => (⇑f)^[n + 1] 0 / (n + 1)) Filter.atTop (nhds τ')) :
f.translationNumber = τ'
theorem CircleDeg1Lift.transnumAuxSeq_zero (f : CircleDeg1Lift) :
f.transnumAuxSeq 0 = f 0
theorem CircleDeg1Lift.transnumAuxSeq_dist_lt (f : CircleDeg1Lift) (n : ) :
dist (f.transnumAuxSeq n) (f.transnumAuxSeq (n + 1)) < 1 / 2 / 2 ^ n
theorem CircleDeg1Lift.tendsto_translationNumber_aux (f : CircleDeg1Lift) :
Filter.Tendsto f.transnumAuxSeq Filter.atTop (nhds f.translationNumber)
theorem CircleDeg1Lift.tendsto_translationNumber_of_dist_bounded_aux (f : CircleDeg1Lift) (x : ) (C : ) (H : ∀ (n : ), dist ((f ^ n) 0) (x n) C) :
Filter.Tendsto (fun (n : ) => x (2 ^ n) / 2 ^ n) Filter.atTop (nhds f.translationNumber)
theorem CircleDeg1Lift.translationNumber_eq_of_dist_bounded {f : CircleDeg1Lift} {g : CircleDeg1Lift} (C : ) (H : ∀ (n : ), dist ((f ^ n) 0) ((g ^ n) 0) C) :
f.translationNumber = g.translationNumber
@[simp]
theorem CircleDeg1Lift.translationNumber_eq_of_semiconjBy {f : CircleDeg1Lift} {g₁ : CircleDeg1Lift} {g₂ : CircleDeg1Lift} (H : SemiconjBy f g₁ g₂) :
g₁.translationNumber = g₂.translationNumber
theorem CircleDeg1Lift.translationNumber_eq_of_semiconj {f : CircleDeg1Lift} {g₁ : CircleDeg1Lift} {g₂ : CircleDeg1Lift} (H : Function.Semiconj f g₁ g₂) :
g₁.translationNumber = g₂.translationNumber
theorem CircleDeg1Lift.translationNumber_mul_of_commute {f : CircleDeg1Lift} {g : CircleDeg1Lift} (h : Commute f g) :
(f * g).translationNumber = f.translationNumber + g.translationNumber
@[simp]
theorem CircleDeg1Lift.translationNumber_units_inv (f : ) :
(↑f⁻¹).translationNumber = -(↑f).translationNumber
@[simp]
theorem CircleDeg1Lift.translationNumber_pow (f : CircleDeg1Lift) (n : ) :
(f ^ n).translationNumber = n * f.translationNumber
@[simp]
theorem CircleDeg1Lift.translationNumber_zpow (f : ) (n : ) :
(↑(f ^ n)).translationNumber = n * (↑f).translationNumber
@[simp]
theorem CircleDeg1Lift.translationNumber_conj_eq (f : ) (g : CircleDeg1Lift) :
(f * g * f⁻¹).translationNumber = g.translationNumber
@[simp]
theorem CircleDeg1Lift.translationNumber_conj_eq' (f : ) (g : CircleDeg1Lift) :
(f⁻¹ * g * f).translationNumber = g.translationNumber
theorem CircleDeg1Lift.dist_pow_map_zero_mul_translationNumber_le (f : CircleDeg1Lift) (n : ) :
dist ((f ^ n) 0) (n * f.translationNumber) 1
theorem CircleDeg1Lift.tendsto_translation_number₀' (f : CircleDeg1Lift) :
Filter.Tendsto (fun (n : ) => (f ^ (n + 1)) 0 / (n + 1)) Filter.atTop (nhds f.translationNumber)
theorem CircleDeg1Lift.tendsto_translation_number₀ (f : CircleDeg1Lift) :
Filter.Tendsto (fun (n : ) => (f ^ n) 0 / n) Filter.atTop (nhds f.translationNumber)
theorem CircleDeg1Lift.tendsto_translationNumber (f : CircleDeg1Lift) (x : ) :
Filter.Tendsto (fun (n : ) => ((f ^ n) x - x) / n) Filter.atTop (nhds f.translationNumber)

For any x : ℝ the sequence $\frac{f^n(x)-x}{n}$ tends to the translation number of f. In particular, this limit does not depend on x.

theorem CircleDeg1Lift.tendsto_translation_number' (f : CircleDeg1Lift) (x : ) :
Filter.Tendsto (fun (n : ) => ((f ^ (n + 1)) x - x) / (n + 1)) Filter.atTop (nhds f.translationNumber)
theorem CircleDeg1Lift.translationNumber_translate (x : ) :
theorem CircleDeg1Lift.translationNumber_le_of_le_add (f : CircleDeg1Lift) {z : } (hz : ∀ (x : ), f x x + z) :
f.translationNumber z
theorem CircleDeg1Lift.le_translationNumber_of_add_le (f : CircleDeg1Lift) {z : } (hz : ∀ (x : ), x + z f x) :
z f.translationNumber
theorem CircleDeg1Lift.translationNumber_le_of_le_add_int (f : CircleDeg1Lift) {x : } {m : } (h : f x x + m) :
f.translationNumber m
theorem CircleDeg1Lift.translationNumber_le_of_le_add_nat (f : CircleDeg1Lift) {x : } {m : } (h : f x x + m) :
f.translationNumber m
theorem CircleDeg1Lift.le_translationNumber_of_add_int_le (f : CircleDeg1Lift) {x : } {m : } (h : x + m f x) :
m f.translationNumber
theorem CircleDeg1Lift.le_translationNumber_of_add_nat_le (f : CircleDeg1Lift) {x : } {m : } (h : x + m f x) :
m f.translationNumber
theorem CircleDeg1Lift.translationNumber_of_eq_add_int (f : CircleDeg1Lift) {x : } {m : } (h : f x = x + m) :
f.translationNumber = m

If f x - x is an integer number m for some point x, then τ f = m. On the circle this means that a map with a fixed point has rotation number zero.

theorem CircleDeg1Lift.floor_sub_le_translationNumber (f : CircleDeg1Lift) (x : ) :
f x - x f.translationNumber
theorem CircleDeg1Lift.translationNumber_le_ceil_sub (f : CircleDeg1Lift) (x : ) :
f.translationNumber f x - x
theorem CircleDeg1Lift.map_lt_of_translationNumber_lt_int (f : CircleDeg1Lift) {n : } (h : f.translationNumber < n) (x : ) :
f x < x + n
theorem CircleDeg1Lift.map_lt_of_translationNumber_lt_nat (f : CircleDeg1Lift) {n : } (h : f.translationNumber < n) (x : ) :
f x < x + n
f x < x + f.translationNumber + 1
theorem CircleDeg1Lift.lt_map_of_int_lt_translationNumber (f : CircleDeg1Lift) {n : } (h : n < f.translationNumber) (x : ) :
x + n < f x
theorem CircleDeg1Lift.lt_map_of_nat_lt_translationNumber (f : CircleDeg1Lift) {n : } (h : n < f.translationNumber) (x : ) :
x + n < f x
theorem CircleDeg1Lift.translationNumber_of_map_pow_eq_add_int (f : CircleDeg1Lift) {x : } {n : } {m : } (h : (f ^ n) x = x + m) (hn : 0 < n) :
f.translationNumber = m / n

If f^n x - x, n > 0, is an integer number m for some point x, then τ f = m / n. On the circle this means that a map with a periodic orbit has a rational rotation number.

theorem CircleDeg1Lift.forall_map_sub_of_Icc (f : CircleDeg1Lift) (P : ) (h : xSet.Icc 0 1, P (f x - x)) (x : ) :
P (f x - x)

If a predicate depends only on f x - x and holds for all 0 ≤ x ≤ 1, then it holds for all x.

theorem CircleDeg1Lift.translationNumber_lt_of_forall_lt_add (f : CircleDeg1Lift) (hf : ) {z : } (hz : ∀ (x : ), f x < x + z) :
f.translationNumber < z
theorem CircleDeg1Lift.lt_translationNumber_of_forall_add_lt (f : CircleDeg1Lift) (hf : ) {z : } (hz : ∀ (x : ), x + z < f x) :
z < f.translationNumber
theorem CircleDeg1Lift.exists_eq_add_translationNumber (f : CircleDeg1Lift) (hf : ) :
∃ (x : ), f x = x + f.translationNumber

If f is a continuous monotone map ℝ → ℝ, f (x + 1) = f x + 1, then there exists x such that f x = x + τ f.

theorem CircleDeg1Lift.translationNumber_eq_int_iff (f : CircleDeg1Lift) (hf : ) {m : } :
f.translationNumber = m ∃ (x : ), f x = x + m
theorem CircleDeg1Lift.continuous_pow (f : CircleDeg1Lift) (hf : ) (n : ) :
Continuous (f ^ n)
theorem CircleDeg1Lift.translationNumber_eq_rat_iff (f : CircleDeg1Lift) (hf : ) {m : } {n : } (hn : 0 < n) :
f.translationNumber = m / n ∃ (x : ), (f ^ n) x = x + m
theorem CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq {G : Type u_1} [] (f₁ : ) (f₂ : ) (h : ∀ (g : G), (f₁ g).translationNumber = (f₂ g).translationNumber) :
∃ (F : CircleDeg1Lift), ∀ (g : G), Function.Semiconj F (f₁ g) (f₂ g)

Consider two actions f₁ f₂ : G →* CircleDeg1Lift of a group on the real line by lifts of orientation preserving circle homeomorphisms. Suppose that for each g : G the homeomorphisms f₁ g and f₂ g have equal rotation numbers. Then there exists F : CircleDeg1Lift such that F * f₁ g = f₂ g * F for all g : G.

This is a version of Proposition 5.4 from Étienne Ghys, Groupes d'homeomorphismes du cercle et cohomologie bornee.

theorem CircleDeg1Lift.units_semiconj_of_translationNumber_eq {f₁ : } {f₂ : } (h : (↑f₁).translationNumber = (↑f₂).translationNumber) :
∃ (F : CircleDeg1Lift), Function.Semiconj F f₁ f₂

If two lifts of circle homeomorphisms have the same translation number, then they are semiconjugate by a CircleDeg1Lift. This version uses arguments f₁ f₂ : CircleDeg1Liftˣ to assume that f₁ and f₂ are homeomorphisms.

theorem CircleDeg1Lift.semiconj_of_isUnit_of_translationNumber_eq {f₁ : CircleDeg1Lift} {f₂ : CircleDeg1Lift} (h₁ : IsUnit f₁) (h₂ : IsUnit f₂) (h : f₁.translationNumber = f₂.translationNumber) :
∃ (F : CircleDeg1Lift), Function.Semiconj F f₁ f₂

If two lifts of circle homeomorphisms have the same translation number, then they are semiconjugate by a CircleDeg1Lift. This version uses assumptions IsUnit f₁ and IsUnit f₂ to assume that f₁ and f₂ are homeomorphisms.

theorem CircleDeg1Lift.semiconj_of_bijective_of_translationNumber_eq {f₁ : CircleDeg1Lift} {f₂ : CircleDeg1Lift} (h₁ : ) (h₂ : ) (h : f₁.translationNumber = f₂.translationNumber) :
∃ (F : CircleDeg1Lift), Function.Semiconj F f₁ f₂

If two lifts of circle homeomorphisms have the same translation number, then they are semiconjugate by a CircleDeg1Lift. This version uses assumptions bijective f₁ and bijective f₂ to assume that f₁ and f₂ are homeomorphisms.