| This paper concerns a kind of δ-hyperbolic geodesic spaces. Analogous to the situation on the Cayley Graph of a group, we define Floyd metric on this kind of spaces, and prove the bi-Lipschitz equivalence of a sort of Floyd metric and the corresponding visual metric on these spaces.Starting from the basic properties of Gromov hyperbolic spaces, we state how the metric dv is defined, and iterate the reason why it can be replaced as a substitute of a particular kind of visual metric in our investigation in this paper. Then we introduce the conceptions and properties of Floyd metric and its boundary in Cayley graph of a group. Similar to the counterpart in the Cayley Graph, we define the Floyd metric and its boundary of any space subsequently. This definition is actually a generalization of what is defined on Cayley Graph.We then generalise a lemma in [31,32] and obtain Lemma3.0.1in Chaper3, which illustrates that a function on the distance between the base point and the geodesic controls the Floyd metric. Then by this lemma, we find out that there is a map from X∪(?)∞X to (X,df) under certain circumstances. This guarantees the reasonability of the following studies.In the forthcoming chapter, we provide a combinatorial model for the space we concern by introducing a discrete θ—net, its relevant graph Gθ, the metric on this graph, and the quasi-Floyd metric on the graph. On Gθ, by the similar analysis to [42], it is shown that a quasi-Floyd geodesic can be approximated by a local geodesic. And we also find out the relation between the metric on Gθ and the original metric on original space, as well as the relation between the quasi-Floyd metric on Gθ and the original Floyd metric.In the last part of this paper, combined the approximation method with what has been proved previously and another important fact, we compute and achieve a sufficient condition for the bi-Lipschitz equivalence of a sort of Floyd metric and the corresponding visual metric. |
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