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As the questioner notes in a comment, the answer is Yes for n<3. One way to create counterexamples for larger n is to use the work on the Seifert Conjecture. Start with a vector field pointing inwar …

This is a good question. For some reason, terminology in dynamical systems is not standardized at all--and it's interesting to disentangle various definitions. A good book to look at is Differential e …

Your intuition is on target, since you mention the Banach fixed point theorem together with the Hartman-Grobman theorem. The proofs I've seen of Hartman-Grobman find the topological conjugacy as the f …

For the specific case you mention of an irrational rotation of the circle, it depends on the rotation number $r$. You can get slow asymptotic convergence for something like $$r=\sum{10^{-n!}}$$ (and y …

You can't always approximate by ergodic flows, because ergodic flows might not even exist. For example, on $S^2$ the Poincare-Bendixson theorem rules out ergodic flows, but there are many measure-pres …

Yes, there is. Conceptually, imagine a flow on $\mathbb{R}^3$ with a single closed orbit on the unit circle in the plane $z=0$, while every other trajectory has an increasing z-coordinate. It's then e …

Unless I'm misinterpreting the question, the SRB measure is just Lebesgue measure... the cat map is hyperbolic, preserves area, and is topologically transitive. See Theorem 3.10 and the following rema …

For $n = 2$, the answer is Yes. Topological considerations (as in the proof of the Poincare-Bendixson theorem) mean that the first-return map for a transversal to a point on a periodic orbit will be t …

You can construct a smooth counterexample in $\mathbb{R}^2$: a function $f$ whose gradient flow has a unique equilibrium, which is also stable, but whose basin of attraction is not the entire plane. I …

You can visualize a solenoid as part of a simple 3D dynamical system. The Wikipedia page on solenoids has excellent illustrations. If you want something even more concrete, and have some "Silly Putty" …

The reason that the matrices are alike is that, up to a multiplicative factor, each is approximately encoding $Area(T(R_i) \cap R_j))$. This is because the rational points with denominator $q$ are clo …

(edited to include Willie Wong's idea for $C^0$ case.) This kind of flow can't exist in any dimension. Let $S$ be the unit sphere and $B$ be the open unit ball. If the origin is a global attractor for …

Let $\mu$ be Lebesgue measure on $S^1$, and $\delta_P$ be a point-mass at a point $P \in S^1$. Then there is no flow on $S^1$ whose time averages lead to $\frac{1}{2}(\mu + \delta_P)$. (Consider the o …

No, because it has a unique leaf that is a torus. That can't be a stable or unstable manifold; one reason is that it can't get either bigger or smaller under the action of the hyperbolic dynamical sys …

The Lorenz equations are quadratic, and already have an infinite number of distinct knotted and linked orbits. An answer to one of your other questions has good references on Lorenz orbits, but I also …