I tried to prove following statement and use some techniques but I couldn't get result :

Question: If Non wandering Set is whole space then Recurent set is dense??

when $T:X \to X$ is hemeomorphism on compact metric space

Thanks for any hint My attempt: we want to show that for all $x$ and any neighborhood $U_x$ about $x$ intersect $R(T)$ $$R(T)\cap U_x\neq\emptyset$$

or equivalently aim is to find element $y$ in nbhd which and there exist subseqquence$\{n_i\}$ such that $T^{n_i}(y)\to y$ as $n_i\to \infty$

Since $x$ is in Non wandering set every nbhd comes back in some natural number

$$T^{m}(U_x)\cap U_x\neq\emptyset\to \exists y,z\in U_x \quad,\quad T^mz=y $$

to make sub sequence I assume this nhbhd $B(x,\frac{1}{n})$ that is goes to $x$ as $n \to\infty$ also $$T^{m_n}(B(x,\frac{1}{n}))\cap B(x,\frac{1}{n})\neq\emptyset\to \exists y_n,z_n\in B(x,\frac{1}{n}) \quad,\quad T^{m_n}z_n=y_n $$

I tried to prove following statement and use some techniques but I couldn't get result :

Question: If Non wandering Set is whole space then Recurent set is dense??

when $T:X \to X$ is hemeomorphism on compact metric space

Thanks for any hint

$ \text{My attempt}:$ we want to show that for all $x$ and any neighborhood $U_x$ about $x$ intersect $R(T)$ $$R(T)\cap U_x\neq\emptyset$$

or equivalently aim is to find element $y$ in nbhd which and there exist subseqquence$\{n_i\}$ such that $T^{n_i}(y)\to y$ as $n_i\to \infty$

Since $x$ is in Non wandering set every nbhd comes back in some natural number

$$T^{m}(U_x)\cap U_x\neq\emptyset\to \exists y,z\in U_x \quad,\quad T^mz=y $$

to make sub sequence I assume this nhbhd $B(x,\frac{1}{n})$ that is goes to $x$ as $n \to\infty$ also $$T^{m_n}(B(x,\frac{1}{n}))\cap B(x,\frac{1}{n})\neq\emptyset\to \exists y_n,z_n\in B(x,\frac{1}{n}) \quad,\quad T^{m_n}z_n=y_n $$

I tried to prove following statement and use some techniques but I couldn't get result :

Question: If Non wandering Set is whole space then Recurent set is dense??

when $T:X \to X$ is hemeomorphism on compact metric space

Thanks for any hint My attempt: we want to show that for all $x$ and any neighborhood $U_x$ about $x$ intersect $R(T)$ $$R(T)\cap U_x\neq\emptyset$$

or equivalently aim is to find element $y$ in nbhd which and there exist subseqquence$\{n_i\}$ such that $T^{n_i}(y)\to y$ as $n_i\to \infty$

Since $x$ is in Non wandering set every nbhd comes back in some natural number

$$T^{m}(U_x)\cap U_x\neq\emptyset\to \exists y,z\in U_x \quad,\quad T^mz=y $$

to make sub sequence I assume this nhbhd $B(x,\frac{1}{n})$ that is goes to $x$ as $n \to\infty$ also $$T^{m_n}(B(x,\frac{1}{n}))\cap B(x,\frac{1}{n})\neq\emptyset\to \exists y_n,z_n\in B(x,\frac{1}{n}) \quad,\quad T^{m_n}z_n=y_n $$
Claim: there exist $k\in\mathbb N$ such that $\{B(x,\frac{1}{n})\}_{n=0}^{\infty}$ stop in $k$ and $$T^{m_n}(B(x,\frac{1}{k}))\cap B(x,\frac{1}{k})=y_k $$

I tried to prove following statement and use some techniques but I couldn't get result :

Question: If Non wandering Set is whole space then Recurent set is dense??

when $T:X \to X$ is hemeomorphism on compact metric space

Thanks for any hint My attempt: we want to show that for all $x$ and any neighborhood $U_x$ about $x$ intersect $R(T)$ $$R(T)\cap U_x\neq\emptyset$$

or equivalently aim is to find element $y$ in nbhd which and there exist subseqquence$\{n_i\}$ such that $T^{n_i}(y)\to y$ as $n_i\to \infty$

Since $x$ is in Non wandering set every nbhd comes back in some natural number

$$T^{m}(U_x)\cap U_x\neq\emptyset\to \exists y,z\in U_x \quad,\quad T^mz=y $$

to make sub sequence I assume this nhbhd $B(x,\frac{1}{n})$ that is goes to $x$ as $n \to\infty$ also $$T^{m_n}(B(x,\frac{1}{n}))\cap B(x,\frac{1}{n})\neq\emptyset\to \exists y_n,z_n\in B(x,\frac{1}{n}) \quad,\quad T^{m_n}z_n=y_n $$