Consider a dynamical system $Z_{t+1}=f(Z_{t})$ where $Z$ is an $n\times 1$ dimensional vector and $f$ is continuously differentiable and I wish to approximate its dynamics about a fixed point $z=0$ by linearization. Suppose further that its Jacobian $A$ happens to be singular- so its rank is $0 \leq r < n$ and it has a zero eigenvalue. Suppose further $f$ is locally non-constant so I can find an open set where $A$ is full rank at every point apart from $0$. By the inverse function this represents the dimensionality of the local non-linear system. It appears to me that the approximate linear system is topologically inconjugate to the true non-linear model because they have different dimensionalities. However, whenever I have seen the Grobman-Hartman theorem applied to discrete dynamical systems the only eigenvalues where conjugacy is lost are those on the unit circle unlike in the continuous time case. Where have I gone wrong?
Zero Eigenvalue Discrete Time Dynamical System
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differential-topology
dynamical-systems