User lbdl - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T20:15:50Z http://mathoverflow.net/feeds/user/18334 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56892/entropy-of-nested-compact-invariant-sets/81809#81809 Answer by lbdl for Entropy of nested compact invariant sets lbdl 2011-11-24T15:15:40Z 2011-11-24T15:15:40Z <p>The strategy of Benoît Kloeckner fails for differentiable maps. Indeed if $K$ is a single point and $K_n$ invariant balls around $K$ it implies that $K$ is an attracting fixed point. Therefore the log of the differential of the diffeo $f$ should be close to zero near $K$ and so does the entropy.</p> <p>However conter-examples in any finite smoothness ($C^r$ maps with $1\leq r&lt;+\infty$) were given by Misisurewicz in the early seventies :</p> <p>Diffeomorphism without any measure with maximal entropy, Bull. Acad. Pol. Sci., Ser. sci. math., astr. et phys. 21 (1973), 903--910 </p> http://mathoverflow.net/questions/79800/margulis-ruelle-inequality-for-piecewise-continuous-interval-maps/81807#81807 Answer by lbdl for Margulis-Ruelle inequality for piecewise continuous interval maps lbdl 2011-11-24T14:10:20Z 2011-11-24T14:10:20Z <p>Do you want just a reference or a proof? I didnt know any reference but for a proof the following should work. </p> <p>Let $\mu$ be an ergodic measure of a piecewise $C^1$ interval maps (with finitely many pieces!). Either the boundary of the partition of smoothness (the finite set of singularities) has zero $\mu$ measure or $\mu$ is periodic. In the last case Ruelle's inequality trivialy holds. In the first case there exists neighborhoods of the boundary with arbitrarily small $\mu$-measure and therefore by ergodicity generic $\mu$ points visits this neighborhood very rarely. Then, when you are far from the boundary, you just apply the classical proof of Ruelle's inquality (you compute the entropy at a scale small compared to the size of the neighborhood). The part where you are closed to the boundary is negligible because local multiplicity is trivial (I mean that any singularity meets the boundary of at mosts two elements of the partition). This last fact is false in higher dimension and that's why to estimate from above entropy of piecewise (even affine) map one needs to add a term of multiplicity (see for example <a href="http://www.math.uit.no/seminar/Preprints/05-04-BKMR.pdf" rel="nofollow">http://www.math.uit.no/seminar/Preprints/05-04-BKMR.pdf</a>).</p> http://mathoverflow.net/questions/77342/submonoid-of-a-matrix-monoid-with-a-common-eigenvector submonoid of a matrix monoid with a common eigenvector lbdl 2011-10-06T08:56:58Z 2011-10-06T17:20:51Z <p>Hello, I am considering two real invertible $3\times 3$ matrices $A$ and $B$ and a nonzero vector $v\in\mathbb{R}^3$ and i am wondering if the submonoid $E$ of the monoid $(A,B)$ genererated by $A$ and $B$ with $v$ as an eigenvector </p> <p>$$E:= \left( C\in (A,B), \ v \text{ is an eigenvector of } C \right)$$</p> <p>is finitely generated.</p> <p>Does it even hold for $2\times 2$ matrices? Thank you in advance.</p>