By R. C. Penner
Measured geodesic laminations are a common generalization of straightforward closed curves in surfaces, and so they play a decisive position in a number of advancements in two-and 3-dimensional topology, geometry, and dynamical platforms. This booklet offers a self-contained and finished remedy of the wealthy combinatorial constitution of the gap of measured geodesic laminations in a hard and fast floor. households of measured geodesic laminations are defined by way of specifying a teach song within the floor, and the distance of measured geodesic laminations is analyzed via learning houses of educate tracks within the floor. the cloth is built from first ideas, the recommendations hired are primarily combinatorial, and just a minimum history is needed at the a part of the reader. particularly, familiarity with hassle-free differential topology and hyperbolic geometry is thought. the 1st bankruptcy treats the elemental idea of teach tracks as found via W. P. Thurston, together with recurrence, transverse recurrence, and the specific development of a measured geodesic lamination from a measured educate tune. the next chapters improve convinced fabric from R. C. Penner's thesis, together with a normal equivalence relation on measured teach tracks and conventional versions for the equivalence sessions (which are used to research the topology and geometry of the gap of measured geodesic laminations), a duality among transverse and tangential buildings on a educate tune, and the specific computation of the motion of the mapping type workforce at the house of measured geodesic laminations within the floor.
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How far away can F and G be from a and still have a = F|G? As a rough rule of thumb, the larger the length of a, the closer F and G must be to a. AN INTRODUCTION TO THE THEORY OF SURREAL NUMBERS 40 Proof, this is Not (a) even if (a) => (b). Since all initial segments are dyadic fractions, clear. =«• not (b). ) Suppose a is not real. Then £(a) _> u>. If for all n less than u>, a has a fixed sign then the condition -n < a < n fails, as is clear from the ordering, so that case is clear. Now let a w be the initial segment of a of length a>.
Although we don't need the information i t is of passing interest to note how as a function of for satisfies being in the lower part does not. b°a varies a . F i r s t , by solving the defining equation 1-U-a^b i-at, b°a, we get b°a, = = h + ^ — * The first expression i implies that b and x a i a i b°a 1 is an increasing function of b iff a < a x and AN INTRODUCTION TO THE THEORY OF SURREAL NUMBERS 24 the second expression that b°a1 is an increasing function of a iff ab > 1. Hence b°ax is an increasing function of one of the variables iff the other variable is upper.
6. Let a = F|G be the canonical representation of a real number a which is not a dyadic fraction. Then for all positive dyadic r there exist b e F, c e G such that c-b _< r. Proof. Since there is no last + and no last - in a, then for all n there are elements b e F, c e G which agree in the first n terms. Thus c-b is bounded above by an expression of the form — + -^- ••• + +1 _< --=— • Since s can be made arbitrarily large by a suitable choice of n this proves the lemma. Note that it is easy to see that the requirement that a be real can be relaxed but this is of no special concern.
Combinatorics of train tracks by R. C. Penner