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

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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 circle_deg1_lift for bundled maps with these properties, define translation number of f : circle_deg1_lift, 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 #

• circle_deg1_lift: a monotone map f : ℝ → ℝ such that f (x + 1) = f x + 1 for all x; the type circle_deg1_lift is equipped with lattice and monoid structures; the multiplication is given by composition: (f * g) x = f (g x).
• circle_deg1_lift.translation_number: translation number of f : circle_deg1_lift.

Main statements #

We prove the following properties of circle_deg1_lift.translation_number.

• circle_deg1_lift.translation_number_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.

• circle_deg1_lift.translation_number_eq_of_semiconj_by: if two circle_deg1_lift maps f, g are semiconjugate by a circle_deg1_lift map, then τ f = τ g.

• circle_deg1_lift.translation_number_units_inv: if f is an invertible circle_deg1_lift 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.

• circle_deg1_lift.translation_number_mul_of_commute: if f and g commute, then τ (f * g) = τ f + τ g.

• circle_deg1_lift.translation_number_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.

• circle_deg1_lift.semiconj_of_bijective_of_translation_number_eq: if f and g are two bijective circle_deg1_lift maps and their translation numbers are equal, then these maps are semiconjugate to each other.

• circle_deg1_lift.semiconj_of_group_action_of_forall_translation_number_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 →* circle_deg1_lift). 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 : circle_deg1_lift. 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 : circle_deg1_lift.

Implementation notes #

We define the translation number of f : circle_deg1_lift 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.

References #

• Étienne Ghys, Groupes d'homeomorphismes du cercle et cohomologie bornee

TODO #

Here are some short-term goals.

• Introduce a structure or a typeclass for lifts of circle homeomorphisms. We use units circle_deg1_lift for now, but it's better to have a dedicated type (or a typeclass?).

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

• Introduce conditionally_complete_lattice structure, use it in the proof of circle_deg1_lift.semiconj_of_group_action_of_forall_translation_number_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 circle_deg1_lift.

Tags #

circle homeomorphism, rotation number

Definition and monoid structure #

Translate by a constant #

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.

Pointwise order on circle maps #

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.

Limits at infinities and continuity #

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.

Definition of translation number #