1.2. BASIC DEFINITIONS AND NOTATIONS 5 1.2. Basic definitions and notations A quasiperiodically forced (qpf) system is a continuous map of the form (1.1) T : T1 × X → T1 × X , (θ, x) → (θ + ω, Tθ(x)) with irrational driving frequency ω. At most times, we will restrict to the case where the driving space X = [a, b] is a compact interval and the fibre maps Tθ are all monotonically increasing on X. In this case we say T is a qpf monotone interval map. Some of the introductory examples will also be qpf circle homeomorphisms, but there the situation can often be reduced to the case of interval maps as well, for example when there exists a closed annulus which is mapped into itself. Two notations which will be used frequently are the following: Given any set A ⊆ T1 × X and θ ∈ T1, we let Aθ := {x ∈ X | (θ, x) ∈ A}. If X = R and ϕ, ψ : T1 → R are two measurable functions, then we use the notation (1.2) [ψ, ϕ] := {(θ, x) | ψ(θ) ≤ x ≤ ϕ(θ)} similarly for (ψ, ϕ), (ψ, ϕ], [ψ, ϕ). Due to the minimality of the irrational rotation on the base there are no fixed or periodic points for T , and one finds that the simplest invariant objects are invariant curves over the driving space (also invariant circles or invariant tori). More generally, a (T -)invariant graph is a measurable function ϕ : T1 → X which satisfies (1.3) Tθ(ϕ(θ)) = ϕ(θ + ω) ∀θ ∈ T1 . This equation implies that the point set Φ := {(θ, ϕ(θ)) | θ ∈ T1} is forward invariant under T . As long as no ambiguities can arise, we will refer to Φ as an invariant graph as well. There is a simple way of obtaining invariant graphs from compact invariant sets: Suppose A ⊆ T1 × X is T -invariant. Then (1.4) ϕ+(θ) A := sup{x ∈ X | (θ, x) ∈ A} defines an invariant graph (invariance following from the monotonicity of the fi- bre maps). Furthermore, the compactness of A implies that ϕ+ A is upper semi- continuous (see [37]). In a similar way we can define a lower semi-continuous graph ϕ− A by taking the infimum in (1.4). Particularly interesting is the case where A = ∩n∈NT n (T1 × X) (the so-called global attractor, see [34]). Then we call ϕ+ A (ϕ−) A the upper (lower) bounding graph of the system. There is also an intimate relation between invariant graphs and ergodic mea- sures. On the one hand, to each invariant graph ϕ we can associate an invariant ergodic measure by (1.5) µϕ(A) := m(π1(A ∩ Φ)) , where m denotes the Lebesgue measure on T1 and π1 is the projection to the first coordinate. On the other hand, if f is a qpf monotone interval maps then the converse is true as well: In this case, for each invariant ergodic measure µ there exists an invariant graph ϕ, such that µ = µϕ in the sense of (1.5). (This can be found in [38], Theorem 1.8.4 . Although the statement is formulated for continuous-time dynamical systems there, the proof literally stays the same.)

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