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I'm not sure this is a research-level question, but I couldn't find an answer after a bit of searching, so here goes.

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a real-analytic function. Can we always find a Riemann surface $M$ and a real-analytic injective "change of coordinates" $\phi: \mathbb{R}^2 \to M$, such that $\phi \circ f \circ \phi^{-1}$ is holomorphic on $\mathrm{Im}~\phi$? If not, what are necessary and sufficient conditions for this to be the case?

If that is relevant, in my particular case $f$ is also injective, so answers conditional to that are also welcome.

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    $\begingroup$ $(x,y) \mapsto (x,0)$ might be a counterexample for the non-objective case. $\endgroup$
    – Geoffrey Irving
    Commented Jul 28 at 11:58
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    $\begingroup$ What does it mean for the function 𝜙∘f∘𝜙^(-1) to be holomorphic "on Im 𝜙" ? I don't understand what "on Im 𝜙" means in this context. $\endgroup$
    – Daniel Asimov
    Commented Jul 29 at 14:43
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    $\begingroup$ @DanielAsimov the intention was that $\phi$ may not generically be surjective on $M$, so $\phi^{-1}$ may only be defined on some domain (i'm not sure whether assuming surjectiveness would limit generality or not). $\endgroup$
    – Mikhail Tikhomirov
    Commented Jul 29 at 14:58
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    $\begingroup$ Ohhhhhhhhh, I think I see now that the notation "Im 𝜙" is meant to indicate the image of the map 𝜙. I was confused because the notation "Im f" — especially in the context of complex variables — always means "the imaginary part of f". So if f(z) = u(z) + iv(z) with u, v real, then Im(f) = v. $\endgroup$
    – Daniel Asimov
    Commented Jul 29 at 15:10
  • $\begingroup$ May I suggest that you add the tag Dynamical system to your question? $\endgroup$
    – Ali Taghavi
    Commented Jul 31 at 19:59

3 Answers 3

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The answer is negative. For the non-injective case, the reason is that non-constant complex analytic functions are open, discrete maps, while real analytic functions can be neither open nor discrete (see the comment to the question). So you cannot make $f$ complex analytic even by a transformation $\psi\circ f\circ\phi$, where $\phi$ and $\psi$ are distinct.

For the injective case, take $f$ such that $f(0)=0$, and consider the Jacobi matrix $J_f(0)$. Then $J_{\phi\circ f\circ\phi^{-1}}=J_\phi J_f J_\phi^{-1}$ by the chain rule, and Jacobi matrices of complex analytic maps are conformal (that is orthogonal with determinant $1$ times positive scalar, while for general real analytic maps they don't have this property. In particular, two eigenvalues of $J_f$ have equal moduli.

So, for example, the map $(x,y)\mapsto (2x,y)$ cannot be conjugate to a complex analytic function.

It will not help if you allow $\phi$ to be non-differentiable (just a local homeomorphism), since in the example above the map $f$ has a curve of fixed points, while for complex analytic maps (other than identity) fixed points are isolated.

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    $\begingroup$ Thank you, these are good points. I don't quite follow the chain rule argument, shouldn't it be $J_{\phi \circ f \circ \phi^{-1}} = J_{\phi} J_f (J_{\phi})^{-1}$? While it's true that $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ is not a conjugate of an orthogonal matrix, a non-orthogonal $\begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}$ is conjugate to a reflection rotated by $2\pi/3$. $\endgroup$
    – Mikhail Tikhomirov
    Commented Jul 28 at 17:22
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    $\begingroup$ I'm still curious if there are deeper necessary conditions than some good behaviour near fixed points, as well their isolatedness. $\endgroup$
    – Mikhail Tikhomirov
    Commented Jul 28 at 17:24
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    $\begingroup$ @MikhailTikhomirov Yes I agree that the conjugate of a complex matrix is not necessarily a complex matrix. $\endgroup$
    – Ali Taghavi
    Commented Jul 29 at 11:21
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    $\begingroup$ Is there any chance for a topological characterization of holomorphic functions akin to the Thurston's dynamical criterion for holomorphic maps $\mathbb CP^1\to \mathbb CP^1$? $\endgroup$
    – Dmitrii Korshunov
    Commented Jul 30 at 21:43
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    $\begingroup$ @Dmitry Korshunov: yes, people try to generalize Thurston's criterion for functions $\mathbf{C}\to\mathbf{C}$, and this is a substantial area of research. I recall that even for rational functions, Thurston's result applies only to a small subclass of all rational functions. $\endgroup$
    – Alexandre Eremenko
    Commented Jul 31 at 13:46
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To see why the second question cannot have a simple answer, it is sufficient to look at the local context near a fixed-point of a tangent-to-identity mapping, as Alexandre Eremenko suggests. By "a simple answer" I mean that a necessary and sufficient condition solving the problem posed by the OP is unlikely to be a (semi-)algebraic or (semi-)analytic condition expressed on the space of (germs of a) real-analytic diffeo.

On the one hand, an holomorphic parabolic germ $f(z)=z+az^{k+1}+\dotsb$ has its orbits organized along $2k$ petals, alternatively repulsive and attractive (Léau's theorem), and the space of its non-stationary orbits is the (non-Hausdorff) complex curve obtained by gluing $2k$ spheres near $0$ and $\infty$ to form a "necklace" consisting in $2k$ beads (Écalle–Voronin's theorem). In iterative dynamics, the gluing mappings are called "horn maps". Therefore, even at a topological level (i.e. allowing orientation preserving homeos $\phi$), the equivalence classes of orbits spaces of conformal diffeos is stratified by countably many strata (indexed by the number $2k$ of beads).

On the other hand, the dynamics of 2-dimensional tangent-to-identity real-analytic germs $g$ near an isolated fixed-point is much richer than that, in particular because the complexified diffeo $g_\mathbb{C}:(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$ has a curve of fixed-point that might remain unseen from the real plane, but still carries a lot of dynamical constraints. Papers on that topic appeared regularly in the last 30 years (see e.g. Bedford) and still do, and they all tend to show that orbits spaces (and their real slices) of parabolic 2-dimensional holomorphic germs are really complicated objects. For instance, in a recent preprint, Klimeš and Stolovitch study a relatively tame and restrictive family of parabolic germs for which the space of orbits is more like an infinite necklace (i.e. with infinitely many beads and horn maps).

Germs near a fixed-point with irrational multiplier are probably a lot more difficult to fathom. And that's only the local question.

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  • $\begingroup$ Yes local analysis around fixed points and periodic orbits is a complicated problem. but I wonder why OP uses the Riemann surface terminology (if he is not searching for global analysis?) With a global view point existence of two different period for a diffeomorphism simply implies a negative answer . So my question: if there is no any object to work(no periodic orbit) how can one apply your answer to the main question? $\endgroup$
    – Ali Taghavi
    Commented Jul 30 at 13:33
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    $\begingroup$ @Loic Treyssier: which question "does not have a simple answer"? The original question does have a simple (negative) answer. Are you trying to give some non-trivial characterization of germs which are conjugate to complex analytic germs, in terms of dynamics? $\endgroup$
    – Alexandre Eremenko
    Commented Jul 30 at 13:40
  • $\begingroup$ Again with a global point of view, let we have an algebraic foliation of plane whose flow is a complete flow(complete algebraic flow is classified by Chicone and Sotomayor). The flow $\phi_t$ is real analytic . Can one say that it is conjugate to a global linear map?(az+b) $\endgroup$
    – Ali Taghavi
    Commented Jul 30 at 13:40
  • $\begingroup$ @Loic I think both local and global version are exciting problems $\endgroup$
    – Ali Taghavi
    Commented Jul 30 at 13:45
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    $\begingroup$ @AlexandreEremenko: yes I was referring to the non-answered question: " If not, what are necessary and sufficient conditions for this to be the case?" $\endgroup$
    – Loïc Teyssier
    Commented Jul 30 at 21:10
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According to existing informative and interesting answers one gets that the local dynamical behavior around fixed points or around periodic orbits may generates some obstructions for being conjugate to holomorphic function.

But I was thinking to the case that we do not have any fixed point or periodic orbit. I hope the following would be helpful as a partial answer to the OP question:

I restrict the OP question to the case of bijectve maps. We have a real analytic bijective map $f$ on $\mathbb{R}^2$ and we ask that is it conjugate to a biholomorphism of $\mathbb{C}$?

A non trivial hyperbolic invertible dynamics on $\mathbb{R}^2$ is not conjugate to any holomorphic diffeomorphism of the plane

By non trivial hyperbolicity I mean that both stable and unstable space are 1 dimensional space(Non of them is trivial)

Recall that a dynamical system $(M,f)$ is a hyperbolic dynamic if at each point $x$ of $M$ the tangent space $T_x M$ is decomposed to a direct sum $T_x M=T^s_x \oplus T^u_x$ such that both distributions $T_x^s$ and $T_x^u$ are invariant under $Df$. Further more the restriction of $Df$ to the stable bundle $T_x^s$ is a contraction and the restriction to the unstable bundle $T^u$ is an expansion.

As a consequence the hyperbolicity of a dynamic is preserved by smooth diffeomorphisms(smooth change of coordinate)

For more information on hyperbolicity see the following two linkes and reference therein:

Hyperbolic structure

and
Hyperbolic dynamics

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