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The answer is "yes". Consider $\mathbb S^1$ as the quotient $\mathbb R/\mathbb Z$. Your homeomorphism $f$ lifts to a homeomorphism $\phi : \mathbb R \to \mathbb R$ such that $\phi(x+1)=\phi(x)+1$. For …

This question relates to this one and that one. Some background In the setting of discrete holomorphic dynamics (say, Julia sets) an irrational $\lambda$ is said to be well approximated by rational nu …

If I understand correctly the setting (say $f$ is a germ at $0\in\mathbb C$ of a biholomorphic function), the answer is no in general, but yes generically. The diffeomorphismn $f$ can always be writte …

The question admits a positive answer when one looks at it for the ring of formal power series $\mathbb R[[x,y]]$. For instance the vector field $x^k\partial_x+y\partial_y$ has a formal cokernel of di …

Consider a foliation of $\mathbb{R}^2$, say coming from the trajectories of a vector field $X$. Its orbit space (the quotient of $\mathbb{R}^2$ by the relation "lying on the same trajectory") is seldo …

First, this case is totally uninteresting regarding Hilbert XVI. Indeed, there are no limit cycles in such systems. The $\alpha / \omega$-limit of a trajectory is either a point or a non-isolated cycl …

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 sim …

Unfortunately you'll have to face a «no» answer, as there is no general/generic way to handle your problem… First of all, your definition of «non-hyperbolic» is not clear to me when $n>1$. As far as …

Such an example is impossible. We can always assume $W$ is a polydisc, and part of its boundary $\partial W$ is included in the $3$-space $T=\{(x,y) : |y|=r\}$. Take $p\in T\cap\bar W$. If the foliati …

This is not a complete answer, but it should give some elements towards it (especially regarding the link with PDEs). First, the Liénard system has a non-trivial centralizer in the case $F''=0$, i.e. …

I don't know much about the general setting you discuss, but a reasonable notion of homotopy between singular vector fields in the local holomorphic setting has been given, and studied, by J.-F. Matte …

Here is a couple of references regarding the holomorphic world, although they do not represent an answer to your question per se (but I'm afraid this is too long a comment, and might anyhow interest p …

The answer is "no". In fact, it is still "no" for germs of curves : generically, a germ of an analytic curve $\gamma : (\mathbb R,0)\rightarrow (\mathbb R^2,0)$ is not tangent to any polynomial vector …

The period of the compact, non-singular orbit $\gamma$ is given by $$T(\gamma)=\oint_\gamma \tau$$ where $\tau$ is any differential $1$-form such that $\tau(F)=1$, e.g. $\tau:=\frac{\mathrm{d}y_2}{\mu …

For your first question: $dy=y^2$ can be integrated by quadratures. The solutions are homographies. For your second question: no it is not true. By the uniformization theorem, the universal covering …