A natural (and perhaps surprising at first sight) source of counter-examples to IFT are the Fréchet-Lie groups of diffeomorphisms of smooth closed manifolds, and the exponential map from vector fields to diffeomorphisms (both smooth).

Even in the simplest case of the circle, the image of the exponential is not a neighbourhood of identity (whereas its differential at $0$ is -- by definition -- the identity).

The reason for this is "dynamical" : if a member $\phi_t=\exp(tX)$ of a one parameter subgroup of $\mathrm{Diff}(S^1)$ has no fixed point, it is conjugated to a rotation by a diffeomorphism sending the -- necessarily -- nowhere vanishing $X$ to a "constant" vector field.

If such a $\phi_t$ has an $n$-periodic point, it is globally $n$-periodic, but any neighbourhood of the identity contains (for large enough $n$) a diffeomorphism whith isolated $n$-periodic points.

A natural (and perhaps surprising at first sight) source of counter-examples to IFT are the Fréchet-Lie groups of diffeomorphisms of smooth closed manifolds, and the exponential map from vector fields to diffeomorphisms (both smooth).

Even in the simplest case of the circle, the image of the exponential is not a neighbourhood of identity (whereas its differential at $0$ is -- by definition -- the identity).

The reason for this is "dynamical" : if a member $\phi_t=\exp(tX)$ of a one parameter subgroup of $\mathrm{Diff}(S^1)$ has no fixed point, it is conjugated to a rotation by a diffeomorphism sending the -- necessarily -- nowhere vanishing $X$ to a "constant" vector field.

If such a $\phi_t$ has one $n$-periodic point, then all points are $n$-periodic. But any neighbourhood of the identity contains (for large enough $n$) a fixed point free diffeomorphism whith isolated $n$-periodic points.