... , n ∈ Z, 3.4 More Than One Rational End Invariant 32 have |yn | > 2, then Ω2 = {X} There will be no more rational end invariants by Theorem 3.1.3., and no irrational end invariants by Prop 3.4.1 ... are one- one correspondence between these pairs of concepts We will use one or another for the convenience of discussion Chapter One Rational End Invariant 3.1 End Invariants We will introduce end ... our Markoff maps refer to 0 -Markoff maps, unless otherwise stated 3.2 Rational and Irrational Rotations 24 We have an important theorem from [1] and [12] Theorem 3.1.3 Let φ be a given Markoff...