Nonlinear map having conserved quantity I am reading the following discussion:　　　
　　　　　　　　　　　　　　　　　　　　　　
Does this simple 2D dynamical system have a conserved quantity?　　　
Does this dynamical system have another conserved quantity?
Conserved Quantities in Dynamical Systems
And also consider the counterpart for a nonlinear map $$x_{n+1}=g(x_n).$$ 
I think the conserved quantity $V(x_n)$ for a nonlinear map should also be constant along thajectory, i.e., 
$$V(g(x_{k}))=V(x_k),$$ compared to $\dot{V}(x) = 0$ for the vector field, $\dot{x} = f(x)$.
Is there any simple example of a nonlinear map having conserved quantity? 
 A: Maybe this is cheating, but here goes.
Take $g(x,y)=(\frac{x}{2}, 2y)$ : this is a linear map with the conserved quantity $V(x,y)=xy$. It is also a hyperbolic diffeomorphism of $\mathbb R^2$, so by structural stability, any small enough $C^1$ perturbation $g_\epsilon$ of $g$ is topologically conjugated to $g$.
This means that if you take for instance $g_\epsilon(x,y):=g(x,y)+\epsilon (\cos(x), \sin(y))$, then for small enough $\epsilon>0$ there exists a homeomorphism $h_\epsilon : \mathbb R^2 \to \mathbb R^2$ such that $g_\epsilon \circ h_\epsilon = h_\epsilon \circ g$. 
Then $V_\epsilon:=V \circ h_\epsilon^{-1}$ is a conserved quantity for the non-linear map $g_\epsilon$.
Note that having a non-constant conserved quantity is a rather strong condition. In particular it means you can't have a dense orbit, so you can't get transitive dynamics. I don't know of any setting in discrete dynamics where you find a conserved quantity without first proving that your map is conjugated to a simpler (often linear) normal form, as in this example.

Addendum
Another silly example: take your favorite ODE $\dot x= X(x)$ with a conservated quantity $V$ (meaning $V$ is constant along the solutions of $\dot x(t) = X(x(t))$). Then $V$ is a conservated quantity for the time 1 map of the flow of $X$.
