Conjugate maps

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Yuri A. Kuznetsov (2007), Scholarpedia, 2(12):5420. doi:10.4249/scholarpedia.5420 revision #91160 [link to/cite this article]
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Figure 1: Conjugate maps \(f\) and \(g\) where \(U,V \subseteq {\mathbb R}^n\ .\)

Two maps \[ x \mapsto f(x),\ \ f: {\mathbb R}^n \to {\mathbb R}^n, \] and \[ y \mapsto g(y),\ \ g: {\mathbb R}^n \to {\mathbb R}^n, \] are called topologically conjugate if there is a homeomorphism (i.e., continuous invertible map with continuous inverse) \[ y=h(x),\ \ h:{\mathbb R}^n \to {\mathbb R}^n, \] such that \[ h(f(x))=g(h(x)) \] for all \(x \in {\mathbb R}^n\ .\)

This is an equivalence relation and the set of all maps \( {\mathbb R}^n \to {\mathbb R}^n\) is divided into classes of topologically conjugate maps. Using the symbol of the map composition, the last equation can be rewritten as \[ h \circ f = g \circ h~~~~{\rm or}~~ f = h^{-1} \circ g \circ h. \]

If \(f\) and \(g\) are invertible, the conjugating homeomorphism \(h\) maps an orbit \[ \ldots, f^{-2}(x),f^{-1}(x),x,f(x),f^2(x),\ldots \] of \(f\) onto an orbit \[ \ldots, g^{-2}(y),g^{-1}(y),y,g(y),g^2(y),\ldots \] of \(g\ ,\) where \(y=h(x)\) and the order of points is preserved. If the maps are noninvertible, \(h\) maps forward orbits of \(f\) onto the corresponding forward orbits of \(g\ ,\) preserving the order of points.

Two topologically conjugate maps are often merely called conjugate. They have identical topological properties, in particular the same number of fixed points and periodic orbits of the same stability types.

If both \(h\) and \(h^{-1}\) are smooth (e.g. \(C^k\) maps), the maps \(f\) and \(g\) are called smoothly (\(C^k\)) conjugate (or diffeomorphic). In this case, we have just one map written in two coordinate systems.


Contents

Examples

The 1D maps \( x \mapsto \frac{1}{2} x \) and \( y \mapsto \frac{1}{3} y \) are topologically conjugate, while they are not smoothly conjugate and neither of them is topologically conjugate to \( u \mapsto - \frac{1}{2} u \ .\)

Local conjugacy

The basic definition can be localized: Two maps \[ x \mapsto f(x),\ \ f: {\mathbb R}^n \to {\mathbb R}^n, \] and \[ y \mapsto g(y),\ \ g: {\mathbb R}^n \to {\mathbb R}^n, \] are called locally conjugate near respective points \(x_0\) and \(y_0\) if there is a homeomorphism of an open neighborhood \(U\) of \(x_0 \in {\mathbb R}^n\) onto an open neighborhood \(V\) of \(y_0 \in {\mathbb R}^n\) \[ y=h(x),\ \ h:U \to V, \] that satisfies \[ h(f(x))=g(h(x)) \] for all \(x \in U\) such that \( f(x) \in U\ ,\) and \(y_0=h(x_0)\ .\)

Conjugacy of parameter-dependent maps

The conjugacy is also defined for two maps depending on parameters. Consider two \(m\)-parameter families of maps \[ x \mapsto f(x,\alpha),\ \ f: {\mathbb R}^n \times {\mathbb R}^m \to {\mathbb R}^n, \] and \[ y \mapsto g(y,\beta),\ \ g: {\mathbb R}^n \times {\mathbb R}^m\to {\mathbb R}^n. \] Two such families are called conjugate if

  • there is a homeomorphism of the parameter space \(p:{\mathbb R}^m \to {\mathbb R}^m,\ \beta=p(\alpha)\ ;\)
  • there is parameter-dependent homeomorphism \(h_{\alpha}:{\mathbb R}^n \to {\mathbb R}^n,\ \ y=h_{\alpha}(x) \) such that

\[ h_{\alpha}(f(x,\alpha))=g(h_{\alpha}(x),p(\alpha)) \] for all \(x \in {\mathbb R}^n\) and \(\alpha \in {\mathbb R}^m\ .\)

Notice that it is not required above that the map \(h_{\alpha}\) depends continuously on \(\alpha \ .\) Some authors call this conjugacy "weak" or "fiber", reserving the term "conjugate" for the case when the map \((x,\alpha) \mapsto (y,\beta)=(h_{\alpha}(x),p(\alpha))\) is a homeorphism of the direct product \( {\mathbb R}^n \times {\mathbb R}^m \ .\)

This definition can also be localized, so that one can speak about "local conjugacy" of two families, e.g. near the origin of \( {\mathbb R}^n \times {\mathbb R}^m \) assuming \((h_{0}(0),p(0))=(0,0)\ .\)

References

  • Z. Nitecki (1971) Differentiable Dynamics. MIT Press.
  • D.V. Anosov et al. (1988) Smooth dynamical systems. In: "Dynamical Systems I", Encyclopaedia of Mathematical Sciences, v. 1, 149-233.
  • Yu.A. Kuznetsov (2004) Elements of Applied Bifurcation Theory, Springer, 3rd edition.

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