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Title: Pairs of closed geodesics in metric graphs
Author: Al Abri, Al Jalila
ISNI:       0000 0004 6496 3834
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2017
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In this thesis we are interested in the problem of counting pairs of closed geodesics in metric graphs. We start with counting pairs of closed geodesics ordered by their word length on the metric graph and such that the difference of their geometric lengths is in a prescribed interval. Then we study a similar problem but where the interval is now allowed to shrink at a specific rate as the word length tends to infinity. Next we study a variant on this problem where we fix a set of generators for the fundamental group of the graph and then order the closed geodesics by the word length of the corresponding conjugacy class with respect to these generators. Again we may also allow the interval to shrink at an appropriate rate. We also study a restricted version of our first problem where we only count null homologous closed geodesics. The final counting problem we study differs from the previous ones by counting pairs of geodesic paths instead of pairs of closed geodesics. These geodesic paths are also ordered by their word lengths in the metric graph and again the difference between the geometric lengths of these geodesic paths lies in a fixed interval. The techniques we use in our study include coding metric graphs by subshifts of finite type and the concepts from the thermodynamic formalism that appear in the ergodic theory of these systems. In particular, we use the spectral properties of transfer operators and their relationship to pressure and entropy.
Supervisor: Not available Sponsor: Jāmiʻat al-Sulṭān Qābūs
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics