I'm trying to prove the Smale's horseshoe set is locally minimal.
More specifically, let $H$ be the horseshoe set described in Section 1.8 in the book "Introduction to Dynamical Systems" by Brin and Stuck. Then, $H$ is said to be locally minimal if there is some open set $U$ containing $H$ such that for every invariant set $Y$ satisfying $H \subset Y \subset U$, we have $H=Y$.
A set $Y$ is said to be invariant if $f^n(Y) \subset Y$, for all $n \in \mathbb{Z}$.
My attempt: Choose $U = \text{int}(R)$. Then, $H \subset U$, $\bar{U}=R$, and $$H = \bigcap_{k \in \mathbb{Z}} f^k(\bar{U}).$$ Let $Y$ be an invariant set such that $H \subset Y \subset U$. Let $n \ge 0$. Since $Y$ is invariant, $$f^n(Y) \subset Y \subset U \subset \bar{U}.$$ Hence, $$Y \subset f^{-n}(f^n(Y)) \subset f^{-n}(\bar{U}).$$ Therefore, $Y \subset f^{-n}(\bar{U})$, for all $n \ge 0$. Now, I would like to show $Y \subset f^{n}(\bar{U})$, for all $n \ge 0$. If I have this, I can conclude $$Y \subset \bigcap_{k \in \mathbb{Z}} f^k(\bar{U}),$$ so $Y \subset H$. But I got stuck here. I can have a similar argument like $$f^{-n}(Y) \subset Y \subset \bar{U}.$$ But then since $f^n(f^{-n}(Y)) \subset Y$, I cannot conclude $Y \subset f^{n}(\bar{U})$.
Any help is highly appreciated.