Smooth proper self-maps on Rn

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In summary, a smooth proper self-map on Rn is a continuous and differentiable function that maps a point in n-dimensional Euclidean space onto itself, satisfying certain conditions for well-behaved and desirable properties. These conditions include continuity, differentiability, finite limit at infinity, mapping the boundary of the space onto itself, and being proper. This type of self-map differs from a general self-map in its proven properties and has applications in mathematics, physics, and engineering. However, it is limited to continuous and differentiable functions and may not accurately model systems with discontinuities or non-differentiable behavior. Additionally, its use may be limited in higher-dimensional spaces.
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Let ##f : \mathbb{R}^n \to \mathbb{R}^n## be a smooth proper map that is not surjective. If ##\omega## is a generator of ##H^n_c(\mathbb{R}^n)## (the ##n##th de Rham cohomology of ##\mathbb{R}^n## with compact supports), show that $$\int_{\mathbb{R}^n} f^*\omega = 0$$
 
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I haven't learned (or remember?) compactly supported cohomology well, so please let me know if there are errors here.

If we consider ##\mathbb{R}^n## as ##S^n-\{N\}## we can view ##\omega## as a top form on ##S^n## that vanishes in a neighborhood of the north pole and ##f## as a smooth map ##S^n\to S^n## that fixes the north pole. Since ##f:S^n\to S^n## misses a point it is homotopic to a constant map as ##S^n-\{\text{point}\}## is contractible. So, an application of Stokes' theorem gives
##\int_{\mathbb{R}^n} f^*\omega=\int_{S^n} f^*\omega=\int_{S^n}(\text{constant map})^*\omega=0.##

Stokes' theorem is used to say that integrating pullbacks of a closed form by homotopic maps on a closed manifold give the same answer. This is seen by considering a homotopy ##H(x,t)## and applying Stokes' theorem to the integral ##\int_{S^n\times [0,1]} d(H^*\omega).##
 
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1. What is a smooth proper self-map on Rn?

A smooth proper self-map on Rn is a function that maps a vector in n-dimensional Euclidean space to another vector in the same space, and is both smooth (meaning it has continuous derivatives of all orders) and proper (meaning it maps compact sets to compact sets).

2. What is the significance of smooth proper self-maps on Rn?

Smooth proper self-maps on Rn have many applications in mathematics and physics, particularly in the study of dynamical systems and differential equations. They also have important implications in the study of manifolds and topology.

3. How are smooth proper self-maps on Rn different from general self-maps on Rn?

Smooth proper self-maps on Rn have the additional properties of being both smooth and proper, while general self-maps on Rn may not have these properties. This means that smooth proper self-maps have more restrictions and can be studied in a more precise and rigorous manner.

4. Can smooth proper self-maps on Rn be composed?

Yes, smooth proper self-maps on Rn can be composed. This means that if we have two smooth proper self-maps, we can apply one after the other to get a new smooth proper self-map. This composition operation is associative, meaning the order in which we compose the maps does not matter.

5. Are there any known examples of smooth proper self-maps on Rn?

Yes, there are many known examples of smooth proper self-maps on Rn. One simple example is the identity map, which maps each vector to itself. Another example is the rotation map, which rotates a vector by a fixed angle around a fixed point. There are also more complex examples, such as the Hénon map and the Lorenz system, which have important applications in chaos theory.

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