1. GAO Rui gave me the following example, which is featured as a nonuniform hyperbolic map with uniform hyperbolicity on periodic points.

Let . The Julia set of is .

Let be a parametrization of the Julia set. Then inductively we see and . In particular we have

(1). every point with parameter is an periodic point of with its orbit ;

(2). the multiplier of at (counting to periodic) is .

In other words, the Lyapunov exponent of every periodic point is . Hence such a system is uniformly hyperbolic on periodic points but not uniformly hyperbolic: it even contains the critical point .

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2. Let be a flow generated by a vector field on a compact manifold . In general a small perturbation of the time-1 map may not be related to some perturbation of the vector field .

If there is a global cross-section of the flow , then there is a roof function and the first return map with such that is isomorphic to the suspension .

In this case a small perturbation of still have as a global section and can be viewed as a reparemetrization of a flow where is close to and is close to , that is, there is a continuous map on such that .

In particular a small perturbation of the time-1 map is a time-varied map of the perturbed flow.

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3. This part is from

http://mathoverflow.net/questions/62916/trivial-map-on-sigma-algebra-mod0-is-trivial

Suppose is an invertible measure-preserving system. If for all measurable subsets , when could we say ?

A sufficient condition is that there exists a countable basis , such that for all distinct points , there exists some such that and . For example is a compact metric space and contains its Borel algebra.

Then if , then there exists some such that and , or equivalentlly for some . So and hence .

Or equally let . For each containing , we have . So are indistinguishable by the structure on .

4. Let be a fixed point of on with . Then is hyperbolic if . If is orientation-preserving, then and have the same sign, and with absolute value greater than 2. In fact has been used as the definition of hyperbolicity of fixed points.

However, if , then it may happens that the absolute value of is less than 1. For example . Note that .

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