Need to Prove Chaos theory in my topology CLass. Can someone show me a nice proo

Question

Need to Prove Chaos theory in my topology CLass. Can someone show me a nice proof. How would I explain this:

Let V be a set. f:V%u2192V is said to be chaotic on V if

1) f has sensitive dependence on initial condition, i.e. there exists %u03B4>0 such that, for any x%u2208Vand any neighborhood N of x, there exists y%u2208N and n%u22650 such that |fn(x)%u2212fn(y)|>%u03B4.

2) f is topologically transitive, i.e. for any pair of open sets J,I%u2282V there exists k>0 such that fk(J)%u2229I%u22600.

3) periodic points are dense in V.

Explanation / Answer

1

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