Katok Closing Lemma for Non-singular Endomorphisms

Abstract

This work is an adaptation of the Katok Closing Lemma for non-singular, endomorphisms on a compact Riemannian manifold. In special, it appears the idea of the proof of the hyperbolicity as well as the admissibility of local stable-unstable manifolds of a proper periodic point in adapted Katok Closing Lemma.

Appendix

In the last step of the proof of Katok Closing Lemma for $C^{2}$ endomorphisms (local diffeomorphisms), it is necessary to show that z is a hyperbolic periodic point. For $0<\gamma <1$, from Proposition 3, consider the following two cones,

$$\begin{aligned} C_{\gamma }^{u}=\{(w_1,w_2)\in {\mathbb {R}}^{k}\times {\mathbb {R}}^{s-k}:\Vert w_1\Vert \le \gamma \Vert w_2\Vert \};\\ C_{\gamma }^{s}=\{(w_1,w_2)\in {\mathbb {R}}^{k}\times {\mathbb {R}}^{s-k}:\Vert w_2\Vert \le \gamma \Vert w_1\Vert \}. \end{aligned}$$

For the sake of simplicity, let us take $m=1$($F^{m}(z)=z$). For ${\tilde{x}}\in {\tilde{{\varDelta }}}_{l}^{k},(u,v)\in B_{\eta _{l}}\times B_{\eta _{l}}$ and $\gamma = \frac{1-\lambda }{20}$, let $w=(w_1,w_2)\in C^{u}_{\gamma }$. One needs to show the following three items,

• There exist expanding and contracting sub-spaces $E^{u}({\bar{z}}),E^{s}({\bar{z}})$ that $T_{z}M=E^{u}({\bar{z}})\oplus E^{s}({\bar{z}})$;

•  $(dF_{{\bar{z}}}){\tilde{C}}_{\gamma }^{u}\subset {\tilde{C}}_{{{\hat{\gamma }}}}^{u},\,\,\,\,\,\,\,\,\,\,\, \& \,\,\,\,\,\,\,\,\,\, (dF_{{\bar{z}}}^{-1}){\tilde{C}}_{\gamma }^{s}\subset {\tilde{C}}_{{{\hat{\gamma }}}}^{s};$

• $\Vert (dF_{{\bar{z}}}(w)\Vert _{{\tilde{x}}}^{\prime }>{{\hat{C}}}\Vert w \Vert _{{\tilde{x}}}^{\prime },\,\,\,\,\, \& \,\,\,\,\,\,\Vert (dF_{{\bar{z}}}^{-1}) (w)\Vert _{{\tilde{x}}}^{\prime }>({{\hat{C}}})\Vert w \Vert$.

Where ${\tilde{C}}^{u}_{\gamma }=({\varPhi }_{{\tilde{x}}})_{(u,v)}(C^{u}_{\gamma }), \,{\tilde{C}}^{s}_{\gamma }=({\varPhi }_{{\tilde{x}}})_{(u,v)}(C^{s}_{\gamma })$ and $\gamma>{{\hat{\gamma }}}>0, {{\hat{C}}}>1$. First we check item 3 and 2. Then we see item 1. By definition, for $w=(w_1,w_2)\in C^{u}_{\gamma }$, the $\Vert w\Vert \le (1-\gamma )^{-1}\Vert w_2\Vert$, and by Proposition 2,

$$\begin{aligned} \Vert (dF_{{\tilde{x}}})_{(u,v)}(w_1,w_2)\Vert&=\Vert (A_{{\tilde{x}}} w_1+B_{{\tilde{x}}}w_2)+(dh_{{\tilde{x}}})_{(u,v)}(w_1,w_2)\Vert \nonumber \\&\ge \Vert B_{{\tilde{x}}} w_2\Vert -\Vert A_{{\tilde{x}}}\Vert - \Vert (dh_{{\tilde{x}}})_{(u,v)}\Vert \Vert w \Vert \nonumber \\&\ge \lambda ^{-1}\Vert w_2\Vert -\lambda \Vert w_1\Vert - \frac{(1-\lambda )^{2}}{100}\Vert w \Vert \nonumber \\&\ge \lambda ^{-1}(1-\gamma )\Vert w \Vert -\lambda \,\gamma \Vert w\Vert - \frac{(1-\lambda )^{2}}{100}\Vert w\Vert \nonumber \\&=\left( \lambda ^{-1}-\left( \frac{\lambda ^{-1}(1-\lambda )}{20}+ \frac{\lambda (1-\lambda )}{20}+ \frac{(1-\lambda )^{2}}{100} \right) \right) \Vert w \Vert \nonumber \\&>\left( \frac{1}{2}+\frac{1}{2\lambda }\right) \Vert w \Vert . \end{aligned}$$

(7.1)

By choosing lambda as in (3.10), it is possible to obtain the validity of above inequality. Similarly,

$$\begin{aligned} \Vert (dF_{{\tilde{x}}}^{-1})_{(u,v)}(w_1,w_2)\Vert > \left( \frac{1}{2}+\frac{1}{2\lambda }\right) \Vert w \Vert . \end{aligned}$$

(7.2)

From the proof of the first item of Proposition 3 referred to Katok’s paper [9], we see that, for $\lambda$ as (3.10)

$$\begin{aligned} (dF_{{\tilde{x}}})_{(u,v)}C_{\gamma }^{u}\subset C_{\lambda \,\gamma }^{u},\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,(dF_{{\tilde{x}}}^{-1})_{(u,v)}C_{\gamma }^{s}\subset C_{\lambda \,\gamma }^{s}. \end{aligned}$$

(7.3)

Now let z be a periodic point (which is assumed to be a fixed point). considering $z={\varPhi }_{{\tilde{x}}}(u,v)$, then from Proposition 2,

$$\begin{aligned} dF_{{\bar{z}}}&=(d{\varPhi }_{{\tilde{f}}({\tilde{x}})})_{(u,v)}\circ (d F_{{\tilde{x}}})_{(u,v)}\circ (d{\varPhi }_{{\tilde{x}}}^{-1})_{z} \end{aligned}$$

(7.4)

$$\begin{aligned}&=(d{\varPhi }_{{\tilde{f}}({\tilde{x}})})_{(u,v)}\circ (d{\varPhi }_{{\tilde{x}}}^{-1})_{z}\circ (d{\varPhi }_{{\tilde{x}}})_{(u,v)}\circ (d F_{{\tilde{x}}})_{(u,v)}\circ (d{\varPhi }_{{\tilde{x}}}^{-1})_{z}. \end{aligned}$$

(7.5)

Observe that in the Equation 4, $(d{\varPhi }_{{\tilde{x}}})_{(u,v)}\circ (d F_{{\tilde{x}}})_{(u,v)}\circ (d{\varPhi }_{{\tilde{x}}}^{-1})_{z}$ is an operator from $T_{z}M\rightarrow T_{z}M$ and $(d{\varPhi }_{{\tilde{f}}({\tilde{x}})})_{(u,v)}\circ (d{\varPhi }_{{\tilde{x}}}^{-1})_{z}$ transforms the $\Vert .\Vert _{{\tilde{x}}}^{\prime }$ norm into the $\Vert .\Vert _{{\tilde{f}}({\tilde{x}})}^{\prime }$.

Since $d{\varPhi }_{{\tilde{x}}}$ is transforming the Euclidean norm from ${\mathbb {R}}^{2}$ to the norm $\Vert .\Vert _{{\tilde{x}}}^{\prime }$, so the properties (7.3) are applied to ${\tilde{C}}^{u}_{\gamma }$ and ${\tilde{C}}^{s}_{\gamma }$. Then,

$$\begin{aligned} (dF_{{\bar{z}}}){\tilde{C}}_{\gamma }^{u}\subset {\tilde{C}}_{{\hat{\lambda }}\,\gamma }^{u}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (dF_{{\bar{z}}}^{-1}){\tilde{C}}_{\gamma }^{s}\subset {\tilde{C}}_{{\hat{\lambda }}\,\gamma }^{s}, \end{aligned}$$

(7.6)

for some ${\hat{\lambda }}<1$, that obtains by choosing a suitable small enough $\varrho$ in (1.1).

From (7.1) and (7.2), and taking $\tau$ in (5.1) as

$$\begin{aligned} \tau =\dfrac{1+\frac{1-\lambda }{2}}{1+\frac{1-\lambda }{2\lambda }}, \end{aligned}$$

(7.7)

then, we see that, for $w\in {\tilde{C}}^{u}_{\gamma }$,

$$\begin{aligned} \begin{aligned} \Vert (dF_{{\bar{z}}})(w)\Vert _{{\tilde{f}}({\tilde{x}})}^{\prime }&> \left( 1+\frac{1}{2} +\frac{1}{2\lambda }\right) \Vert w \Vert _{{\tilde{x}}}^{\prime } \Rightarrow ^{7.7} \Vert (dF_{{\bar{z}}})(w) \Vert _{{\tilde{x}}}^{\prime } \\&>\left( 1+\frac{1-\lambda }{2}\right) \Vert w \Vert _{{\tilde{x}}}^{\prime }. \end{aligned} \end{aligned}$$

(7.8)

and, similarly, for $w\in \tilde{C^{s}_{\gamma }}$,

$$\begin{aligned} \Vert (dF_{{\bar{z}}}^{-1})(w)\Vert _{{\tilde{x}}}^{\prime }> \left( 1+\frac{1-\lambda }{2} \right) \Vert w \Vert _{{\tilde{x}}}^{\prime }. \end{aligned}$$

We can define, respectively, the following expanding and contracting sub-spaces,

$$ \begin{aligned} E^{u}({\bar{z}}):=\bigcap _{k=0}^{\infty }dF_{{\bar{z}}}^{k}({\tilde{C}}^{u}_{\gamma }),\,\,\,\,\, \& \,\,\,\,\,\,E^{s}({\bar{z}}):=\bigcap _{k=0}^{\infty }dF_{{\bar{z}}}^{-k}({\tilde{C}}^{s}_{\gamma }), \end{aligned}$$

(7.9)

that $E^{s}({\bar{z}})\oplus E^{u}({\bar{z}})=\{0\}$, and so $T_{z}M=E^{s}({\bar{z}})\oplus E^{u}({\bar{z}})$.