# Smooth affine structures

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 $$\Lambda(x)= \mu x$$.

## This question has an open bounty worth +50 reputation from Lucas ending in 2 days.

Looking for an answer drawing from credible and/or official sources.

I have a hard time understanding this discussion that Katok makes after the corollary and I'm not sure what structure he refers to

• What is the assertion of Proposition 2.1.3? – Arnaud Mortier Nov 6 at 22:26
• 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 $\Lambda(x)= \mu x$. – Lucas Nov 6 at 22:33
• The general terminology is that a (flat) affine structure on a manifold is a special atlas where the transition maps are restrictions of affine maps. – Moishe Kohan Nov 14 at 4:02