I am studying the definition of hyperbolic set: a compact invariant set $\Lambda$ of a diffeomorphism $f$ such that the tangent space in each point of the set $\Lambda$ can be split in two invariant directions (stable and unstable).
However, I am having some trouble trying to figure out examples of such hyperbolic sets.
Here are some standard examples:
Hyperbolic fixed points of a diffeomorphism: Let $f:M\to M$ be a $C^1$ diffeomorphism of a $C^\infty$ manifold and $x\in M$ be such that $f(x)=x$. Then the derivative $T_xf:T_xM\to T_xM$ is an invertible operator acting on the tangent space at $x$. $\Lambda=\{x\}$ is $f$-hyperbolic if $T_xf$ has no eigenvalues of modulus $1$.
In particular, if $X$ is a Banach space and $T:X\to X$ is a hyperbolic invertible operator, then $\Lambda=\{0\}$ is a $T$-hyperbolic set (see Space of linear, continuous, hyperbolic functions is open, dense in the set of invertible functions).
Hyperbolic periodic points of a diffeomorphism: Let $f:M\to M$ be a $C^1$ diffeomorphism of a $C^\infty$ manifold and $x\in M$ be such that $f^p(x)=x$ for some $p\in\mathbb{Z}_{>0}$. Then the derivative $T_xf^p:T_xM\to T_xM$ is an invertible operator acting on the tangent space at $x$. $\Lambda=\{x\}$ is $f$-hyperbolic if $T_xf^p$ has no eigenvalues of modulus $1$.
State space of an Anosov diffeomorphism: Let $f:M\to M$ be a $C^1$ diffeomorphism of a compact $C^\infty$ manifold. $f$ is Anosov iff $\Lambda=M$ is $f$-hyperbolic.
In particular, if $A$ is a $d\times d$ invertible matrix with integer entries whose inverse also is with integer entries, then it defines an diffeomorphism of the torus $\mathbb{T}^d$. If $A$ has no eigenvalues of modulus $1$, then ($A$ is Anosov and) $\Lambda$is $A$-hyperbolic (see my answer in Examples of conjugate-like structures across mathematics, or Lyapunov exponent for 2D map?, or A question on hyperbolic toral automorphism, or What's so hyperbolic about hyperbolic sets?).
There is a diffeomorphism $f$ of a two dimensional closed disk, called Smale's horseshoe map, that has a Cantor set $\Lambda$ as a (compact invariant) hyperbolic set. (see How does Smale's horseshoe map work? or What's the point of a Horseshoe map?).
The Smale-Williams solenoid $\Lambda$ can be obtained as the attractor of a diffeomorphism of a solid torus. This diffeomorphism stretches and makes thinner the donut; see p.2-5 of Milnor's notes available at https://www.math.stonybrook.edu/~jack/DYNOTES/dn2.pdf. $\Lambda$ is a hyperbolic set for this particular diffeomorphism, and along the direction the diffeomorphism makes the donut thinner it looks like a Cantor set. (Also see Solenoid is hyperbolic set)
