## 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 $z=0$.

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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.

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3. This part is from

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

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 $A^2=\begin{pmatrix}2 & 1\\ 1 & 1\end{pmatrix}$.