In the book "Introduction to the Modern Theory of Dynamical Systems" the author uses the term "smooth affines structures". 
What is this structure really?
Proposition $2.1.3:$ Let $I = [-\delta,\delta ]$, $f: I\to I$ be a real analytic contracting map, $f(0) = 0$, and $\mu :=f'(0) \neq 0$. Then there are an interval $J_1 \subset I$ containing $0$ and a real-analytic diffeomorphism $h: J_1 \to J_2 \subset \mathbb{R}$ preserving the origin and conjugating $f$ with the linear map $\Lambda(x)= \mu x$.