Continuum-wise hyperbolic homeomorphisms on surfaces

This paper discusses the dynamics of continuum-wise hyperbolic surface homeomorphisms. We prove that $c{w}_F$-hyperbolic surface homeomorphisms containing only a finite set of spines are $c{w}_2$-hyperbolic. In the case of $c{w}_3$-hyperbolic homeomorphisms we prove the finiteness of spines and, hence, that $c{w}_3$-hyperbolicity implies $c{w}_2$-hyperbolicity. In the proof, we adapt techniques of Hiraide [11] and Lewowicz [15] in the case of expansive surface homeomorphisms to prove that local stable/unstable continua of $c{w}_F$-hyperbolic homeomorphisms are continuous arcs. We also adapt techniques of Artigue, Pacífico and Vieitez [6] in the case of N-expansive surface homeomorphisms to prove that the existence of spines is strongly related to the existence of bi-asymptotic sectors and conclude that spines are necessarily isolated from other spines.