User alejandro - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:03:04Z http://mathoverflow.net/feeds/user/889 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104080/given-a-vector-field-all-of-whose-integral-curves-are-closed-is-the-period-a-smo/105873#105873 Answer by Alejandro for Given a vector field all of whose integral curves are closed, is the period a smooth function? Alejandro 2012-08-29T19:48:33Z 2012-08-29T19:48:33Z <p>Sam is completely right. In general, the period function $\tau\colon M\to\mathbb{R}$ is not even continuous. </p> <p>A very nice reference for a (counter-)example to Giuseppe's question is the paper <a href="http://www.numdam.org/item?id=PMIHES_1976__46__5_0" rel="nofollow">A counterexample to the periodic orbit conjecture</a>, by Dennis Sullivan. In the paper, Sullivan constructs a singularity-free flow on a compact 5-manifold such that all its orbits are periodic and function $\tau$ is unbounded!</p> http://mathoverflow.net/questions/89116/twist-maps-of-the-annulus/89138#89138 Answer by Alejandro for Twist maps of the annulus Alejandro 2012-02-21T21:17:44Z 2012-02-21T21:17:44Z <p>I assume that the expression "that preserves the measure" means that preserves area (a.k.a. the Lebesgue, or Haar measure) on $\mathbb{R}/\mathbb{Z}\times\mathbb{R}$.</p> <p>In this case the answer is no. The most simple example of a twist map which is not topologically conjugate to an area-preserving one is given by a dissipative twist diffeomorphism (i.e. $0&lt;|\mathrm{det} f_x|&lt;1$, for every $x\in A$) exhibiting a non-trivial attractor. These attractors were first studied by Birkhoff, so now they are called "Birkhoff attractors" in the literature. </p> http://mathoverflow.net/questions/84749/transitivity-of-a-flow-and-its-time-1-map/88625#88625 Answer by Alejandro for Transitivity of a flow and its time-1 map Alejandro 2012-02-16T13:10:24Z 2012-02-16T13:10:24Z <p>I think the answer for the second question is no. We can construct a counterexample as follows: </p> <p>Let $f\colon M\to M$ be an arbitrary transitive homeomorphism and $u\colon M\to\mathbb (0,1/4)$ be an arbitrary non-constant continuous function. Then, let's define $$c(x):=u(x)-u(f(x))+1,\quad\forall x\in M,$$ and consider the flow $f_t\colon M_c\to M_c$ as above.</p> <p>We claim $f_1$ is not transitive. In fact, given any $x\in M$ and any $t\in (0,1/4)$ it holds: $$f_1(x,u(x)+t)=(x,u(x)+t+1)=(x,c(x)+u(f(x))+t)=(f(x),u(f(x))+t).$$ That means that every compact set $U_t:=\lbrace (x,u(x)+t) : x\in M\rbrace \subset M_c$ is $f_1$-invariant and hence it cannot be transitive. Notice the function $c$ is not constant because $f$ is transitive $u$ is not constant itself.</p> http://mathoverflow.net/questions/87448/minimality-of-time-t-minimal-flows Minimality of time-t minimal flows Alejandro 2012-02-03T16:34:46Z 2012-02-03T16:34:46Z <p>This question is mainly motivated by the question <a href="http://mathoverflow.net/questions/84749/transitivity-of-a-flow-and-its-time-1-map" rel="nofollow">http://mathoverflow.net/questions/84749/transitivity-of-a-flow-and-its-time-1-map</a></p> <p>Let $M$ be a closed smooth manifold and $\Phi\colon\mathbb{R}\times M\to M$ be a smooth minimal flow, i.e. given any $x\in M$ its $\Phi$-orbit $\lbrace\Phi^t(x) : t\in\mathbb{R}\rbrace$ is dense in $M$.</p> <p>Question 1: Is it true that there exists at least one $t\in\mathbb{R}$ such that the time-$t$ diffeomorphism $\Phi^t\colon M\to M$ is minimal?</p> <p>Question 2: Assuming $\Phi$ is not conjugate to a suspension flow (i.e. there is no closed codimension-1 submanifold everywhere transverse to the flow), then is it true that $\Phi^t\colon M\to M$ is a minimal diffeomorphism for any $t\in\mathbb{R}$? </p> http://mathoverflow.net/questions/82844/locally-free-lie-group-action-not-preserving-any-measure locally-free Lie group action not preserving any measure Alejandro 2011-12-07T02:26:01Z 2011-12-08T15:31:22Z <p>I'd like to know if there exists a connected Lie group $G$ and a closed manifold $M$ such that there is a locally-free smooth action $G\times M\to M$ (i.e. the stabilizer of any point of $M$ is a discrete subgroup of $G$) with no invariant (Borel) probability measure.</p> http://mathoverflow.net/questions/1454/is-there-a-co-hahn-mazurkiewicz-theorem-for-line-filling-spaces/1976#1976 Answer by Alejandro for Is there a co-Hahn-Mazurkiewicz theorem for line-filling spaces? Alejandro 2009-10-22T23:27:42Z 2009-10-25T15:05:12Z <p>Konstantin, you're right. But the disjoint union of spaces produces a non-connected space and well, I imagine, Skupers should be interested in characterizing connected spaces.</p> <p>UPDATE: After the second remark of Konstantin, I think we should reformulate the original question of Skupers asking about the characterization of connected "MINIMAL line-filling spaces", i.e. spaces which have no proper line-filling subspace. </p> http://mathoverflow.net/questions/1633/homotopy-type-of-stabilizers Homotopy type of stabilizers Alejandro 2009-10-21T11:41:13Z 2009-10-22T04:30:51Z <p>Let X be a contractible metric space and G a topological group acting transitively on X (i.e. given any two points x,y \in X, there exists g \in G such that gx=y).</p> <p>My question is the following: is it true that given any x \in X its stabilizer Stab(x)={ g \in G : gx=x } and the whole group G have the same homotopy type?</p> <p>If the answer is "no", I'd like to know some "mild" hypothesis that could be add to have an affirmative response.</p> <p>For instance, I know that whenever G is a Lie group and H &lt; G is a closed subgroup such that G/H is contractible, then G and H are homotopically equivalent (in this case H can be seen as the stabilizer of the coset H under the natural G-action on G/H). However, to assume that G is a Lie group seems to be too restrictive. In fact, I'd like to apply this "result" to some groups which are not locally compact.</p> http://mathoverflow.net/questions/89291/topological-and-dynamical-complexity Comment by Alejandro Alejandro 2012-02-23T15:46:58Z 2012-02-23T15:46:58Z Sorry, for me this question is not clear at all. What do you mean by &quot;positive maximum Lyapunov exponent&quot;? I mean, if the system has an unstable (i.e. a repeller) equilibrium point, then the system does have a positive Lyapunov exponent and it can have an arbitrary amount of unstable equilibrium points. Moreover, since you're considering systems in $\mathbb{R}^n$ (which is non-compact), you can even have a system with positive entropy and with n unstable equilibrium points, with $n\in \mathbb{N}\cup \lbrace 0,\infty\rbrace$. So, please, could you reformulate your question? http://mathoverflow.net/questions/84749/transitivity-of-a-flow-and-its-time-1-map Comment by Alejandro Alejandro 2012-02-16T09:52:10Z 2012-02-16T09:52:10Z @Pengfei You're completely right, my example is not an answer to your questions, and that's the reason I just wrote a comment for my example. The only purpose of it was to point out that Zarathustra's remark wasn't right, i.e. the last statement after your second question is not correct. http://mathoverflow.net/questions/84749/transitivity-of-a-flow-and-its-time-1-map Comment by Alejandro Alejandro 2012-02-08T16:33:03Z 2012-02-08T16:33:03Z @Pengfei: Sorry, but i don't understand your doubt. What do you mean when you &quot;it seems that $\Phi_1$ and $\phi_1$ are not related&quot;? What is $\Phi_1$ and what $phi_1$ for you? And why should we consider the suspension of a diffeomorphism not isotopic to the identity? http://mathoverflow.net/questions/84749/transitivity-of-a-flow-and-its-time-1-map Comment by Alejandro Alejandro 2012-02-05T07:13:19Z 2012-02-05T07:13:19Z @Pegnfei: I'm sorry, I changed your notation and that produced some confusion. However, I think my example can be rewritten following your notation: consider the map $f : \mathbb{T}^2\ni (x,y) \mapsto (x+\sqrt{2},y+\qrt{3})\in\mathbb{T}^2$. The suspension flow of $f$ with $c\equiv \sqrt{2}$ is isomorphic to the flow $\Phi_t(x,y,z)=(x+t,y+t\sqrt{3/2},z+\frac{t}{\sqrt{2}})$ and hence, we get $\Phi_1(x,y,z)=(x+1,y+\sqrt{3/2},z+\frac{1}{\sqrt{2}})$, which is not transitive. http://mathoverflow.net/questions/84749/transitivity-of-a-flow-and-its-time-1-map Comment by Alejandro Alejandro 2012-02-03T16:12:18Z 2012-02-03T16:12:18Z I don't think, in general, the time-$\sqrt{2}$ map of a suspension flow of a transitive map is transitive itself. In fact, consider the following example: let $f\colon\mathbb{T}^2\to\mathbb{T}^2$ be given by $f(x,y):=(x+\sqrt{3},y+\sqrt{2}) \mathrm{mod}\mathbb{Z}^2$. Then, $f$ is a minimal diffeomorphism. The suspension flow of $f$ is $\Phi_t\colon\mathbb{T}^3\to\mathbb{T}^3$ given by $\Phi_t(x,y,z)=(x+t\sqrt{3},y+t\sqrt{2},z+t)$. In this case $\Phi_{\sqrt{2}$ is not transitive. For me it is not clear at all that any transitive suspension flow admits a time-t transitive map. http://mathoverflow.net/questions/83085/minimum-distance-to-a-sampled-point-with-given-pdf Comment by Alejandro Alejandro 2011-12-10T17:34:37Z 2011-12-10T17:34:37Z @Jennifer: the expression &quot;$N$ independent samples $X_1,\ldots X_N$ of $f$&quot; is not completely clear for me. Does it mean you take $N$ independent points on $[0,1]^2$ with probability $f dA$? http://mathoverflow.net/questions/82844/locally-free-lie-group-action-not-preserving-any-measure/82974#82974 Comment by Alejandro Alejandro 2011-12-08T21:37:52Z 2011-12-08T21:37:52Z @Alain: if you consider the trivial action on the second factor, I cannot figure it out how the non-existence of $\Gamma$-invariant measures (for the projective action) appears in your argument. http://mathoverflow.net/questions/82844/locally-free-lie-group-action-not-preserving-any-measure/82974#82974 Comment by Alejandro Alejandro 2011-12-08T21:19:42Z 2011-12-08T21:19:42Z For me too, it isn't clear how $G$ acts on $M$, but as Yves mentioned above, Alain's argument seems to work for the diagonal left $G$-action on $(G/\Gamma)\times\mathbb{P}^1(\mathbb{R})$. So I'll consider my question as answered... http://mathoverflow.net/questions/82844/locally-free-lie-group-action-not-preserving-any-measure Comment by Alejandro Alejandro 2011-12-08T12:57:16Z 2011-12-08T12:57:16Z @Igor: By closed I mean a compact manifold without boundary. http://mathoverflow.net/questions/1633/homotopy-type-of-stabilizers/1808#1808 Comment by Alejandro Alejandro 2009-10-23T13:43:52Z 2009-10-23T13:43:52Z I'm particularly interested in the case G &lt; Diff(M) and X is some manifold, in general, different of M. So, the CW-complex-hypotheses seems reasonable, but the existence of slices seems to be too strong. http://mathoverflow.net/questions/1633/homotopy-type-of-stabilizers/1808#1808 Comment by Alejandro Alejandro 2009-10-22T12:18:24Z 2009-10-22T12:18:24Z Tyler, thanks for the ideas. About the counterexample, I didn't say it explicitly (sorry), but I'm interested in the case where G is connected. About the second and third paragraphs, you're completely right. But, is there any chance of getting an affirmative answer imposing extra hypothesis on G and/or on X rather than on the action itself? Thanks again.