By By (author) Felix Klein By (author) Alexander Ziwet

This lectures used to be learn from Aug. 28 to Sept. nine, 1893 at Northwestern universityEvanston , unwell.

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**Example text**

Since Γ is a dense subset of E ∞ , we therefore conclude that u has 22 Symbolic Dynamics a unique continuous extension on E ∞ . 13) that u is bounded and H¨ older continuous. Since g(γ)−f (γ)− R = u(γ) − u(σ(γ) for all γ ∈ Γ, this holds for all γ. The proof of the implication (2) ⇒ (3) is therefore complete. The implications (3) ⇒ (4) and (4) ⇒ (5) are obvious. (5) ⇒ (1). 3) that for every ω ∈ E ∗ , say ω ∈ En, Q−2 e−T exp (S + P(g) − P(f ))n ≤ µ ˜ f ([ω]) ≤ Q2 eT exp (S + P(g) − P(f ))n . 14) Suppose that S = P(f ) − P(g).

Proof. Fix g ∈ Kβ , ω ∈ E ∞ and suppose that Aiω1 = 1. Then |Ai,ρ (g)(ω)| = |ρ(iω) g(iω)| ≤ ρ ◦ i 0 g 0 ≤ ρ◦i β g β. 30) Fix now in addition τ ∈ E ∞ \ {ω} such that |ω ∧ τ | ≥ 1. Then |Ai,ρ (g)(ω) − Ai,ρ (g)(τ )| = |ρ(iω)g(iω) − ρ(iτ )g(iτ )| = |ρ(iω)(g(iω) − g(iτ )) + g(iτ )(ρ(iω) − ρ(iτ ))| ≤ ρ ◦ i 0 |g(iω) − g(iτ )| + g 0 |ρ(iω) − ρ(iτ )| ≤ ρ◦i β g βe −β|ω∧τ | + g β ρ◦i βe −β|ω∧τ | . 30), we conclude that Ai,ρ (g) ≤ 3 ρ ◦ i β g β . consequently Ai,ρ acts on the space Kβ , is continuous and Ai,ρ β ≤ 3 ρ ◦ i β .

Small that γη < 1 and choose a subfamily R of (E r )∗ consisting of mutually incomparable words and such that B ⊂ {[ω] : ω ∈ R} and ˜ ∩ [ω]) ≤ m ˜ {[ω] : ω ∈ R} ≤ η m(B). ˜ Then m(B) ˜ ≤ ω∈R m(B γ m([ω]) ˜ = γ m ˜ {[ω] : ω ∈ R} ≤ γη m(B) ˜ < m(B). ˜ This ω∈R contradiction ﬁnishes the proof of the complete ergodicity of m. ˜ There is a sort of converse to part of the preceding theorem. We need the following lemma ﬁrst. The next two results are due to Sarig [Sar]. 5 Suppose the incidence matrix A is irreducible and m ˜ is an invariant Gibbs state for the acceptable function f .