| Summary: | 博士 === 國立交通大學 === 應用數學系所 === 100 === In this dissertation, we investigate topological dynamics of high-dimensional systems which are perturbed from a continuous map f of the following form F(x,y) = (f(x),g(x,y)). First, we show that if the lower dimensional map f has a snap-back repeller, then the small C^{1} perturbation of f also has a snap-back repeller.
Assume that g is locally trapping and the system is along a one-parameter continuous family {F_{λ}} such that F_{0} = F. We show that if f is a one dimensional map and has positive entropy, or f is a high-dimensional map and has a snap-back repeller then F_{λ} has a positive topological entropy for all small parameter λ. Also, we show that if f is a C^{1} diffeomorphism having a topologically crossing homoclinic point, then F_{λ} has positive topological entropy for all λ close enough to 0.
Moreover, we show that if f has covering relations determined by a transition matrix A, then any small C^{0} perturbed system of F has a compact positively invariant set restricted to which the perturbated system is topologically semi-conjugate to the one-sided subshift of finite type induced by A. In addition, if the covering relations satisfy a strong Liapunov condition and g is a contraction, we show that any small C^{1} perturbed homeomorphism of F has a compact invariant set restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by A.
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