Final answer:
An example of a C¹ diffeomorphism with the desired properties is the function f(x) = x + x²sin(2π/x) for x ≠ 0, and f(0) = 0. This has a non-hyperbolic fixed point at x = 0 and an infinite number of hyperbolic fixed points accumulating at this non-hyperbolic fixed point.
Step-by-step explanation:
The question is asking for an example of a C¹ diffeomorphism that has a non-hyperbolic fixed point which is also an accumulation point of other hyperbolic fixed points. In mathematics, specifically in dynamical systems, a C¹ diffeomorphism is a continuously differentiable bijection with a continuously differentiable inverse.
A simple example of a function that satisfies these conditions is f(x) = x + x²sin(2π/x) for x ≠ 0, and f(0) = 0. At x = 0, we have a non-hyperbolic fixed point because f'(0) = 1, which means the derivative of the function at the fixed point is neither less than 1 nor greater than -1, thus not satisfying the hyperbolicity condition.
However, for points where x is the reciprocal of any integer (1/n for n an integer), the fixed points are hyperbolic since the derivative there is not equal to 1 (it actually oscillates infinitely as x approaches 0), leading to an alternating pattern of stable and unstable fixed points accumulating at x = 0. This provides an interesting dynamic system with a mix of fixed point behaviors.