Uniform hyperbolicity on periodic points

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

Let f:\mathbb{C}\to\mathbb{C},z\mapsto 2-z^2. The Julia set of f is J=[-2,2].

Let z_t=-2\cos t be a parametrization of the Julia set. Then inductively we see f^n(-2\cos t)=-2\cos 2^n t and (f^n)'(-2\cos t)=\frac{\sin 2^nt}{\sin t}\cdot 2^n. In particular we have

(1). every point z_n with parameter t_n=\frac{2\pi}{2^n-1} is an n-periodic point of f with its orbit \mathcal{O}(z_n)=[z_t:t=2^k\cdot t_n,0\le k\le n-1];

(2). the multiplier of f at z_n (counting to periodic) is 2^n.

In other words, the Lyapunov exponent of every periodic point is 2. Hence such a system is uniformly hyperbolic on periodic points but not uniformly hyperbolic: it even contains the critical point z=0.


2. Let \phi_t:M\to M be a flow generated by a vector field X on a compact manifold M. In general a small perturbation f of the time-1 map \phi_1 may not be related to some perturbation of the vector field X.

If there is a global cross-section N of the flow \phi_t, then there is a roof function r:N\to[c,C] and the first return map g:N\to N with \bigcup_{x\in N}\phi(x,[0,r(x)])=M such that \phi_t is isomorphic to the suspension (g,r).

In this case a small perturbation f of \phi_t\sim(g,r) still have N as a global section and can be viewed as a reparemetrization of a flow \psi=(h,s) where h is close to g and s is close to r, that is, there is a continuous map t(x) on M such that f(x)=\psi(x,t(x)).

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


3. This part is from


Suppose (X,\mathcal{X},\mu,T) is an invertible measure-preserving system. If \mu(A\backslash T^{-1}A)=0 for all measurable subsets A\in\mathcal{X}, when could we say \mu(\{x:Tx\neq x\})=0?

A sufficient condition is that there exists a countable basis A_n\in\mathcal{X}, such that for all distinct points x,y\in X, there exists some A_n such that x\in A_n and y\notin A_n. For example X is a compact metric space and \mathcal{X} contains its Borel algebra.

Then if Tx\neq x, then there exists some A_n such that x\in A_n and Tx\notin A_n, or equivalentlly x\in A_n\backslash T^{-1}A_n for some n\ge1. So \{x:Tx\neq x\}\subset\bigcup_{n\ge1}A_n\backslash T^{-1}A_n and hence \mu(\{x:Tx\neq x\})=0.

Or equally let x\notin\bigcup_{n\ge1}A_n\backslash T^{-1}A_n. For each A_n containing x, we have Tx\in A_n. So x,Tx are indistinguishable by the structure on X.

4. Let p be a fixed point of f on \mathbb{R}^2 with \mathrm{Jac}(f,p)=1. Then p is hyperbolic if 0<|\lambda_1|<1<|\lambda_2|. If f is orientation-preserving, then \lambda_1 and \lambda_2 have the same sign, and \mathrm{Tr}(D_pf)=\lambda_1+\lambda_2=\lambda_1+1/\lambda_1 with absolute value greater than 2. In fact |\mathrm{Tr}(D_pf)|>2 has been used as the definition of hyperbolicity of fixed points.

However, if \mathrm{Jac}(f,p)=-1, then it may happens that the  absolute value of \mathrm{Tr}(D_pf) is less than 1. For example A=\begin{pmatrix}1 & 1\\ 1 & 0\end{pmatrix}. Note that A^2=\begin{pmatrix}2 & 1\\ 1 & 1\end{pmatrix}.

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