Anthony Quas
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 11h revised Relationship between Multiplicative Ergodic Theorems Add condition 4 as suggested by R.W. 16h comment Relationship between Multiplicative Ergodic Theorems Thanks for this... 19h asked Relationship between Multiplicative Ergodic Theorems 1d comment Why do we care about simplicity of the spectrum in Oseledets' theorem? Certainly this is a big deal if $d=2$ and $A\colon M\to SL(2)$. In this case, the system is (non-uniformly) hyperbolic. 2d comment Necessity of expansiveness for existence absolutely continuous invariant measures for piecewise smooth maps of an interval There certainly are examples of 2-branch (both onto) maps that have a finite absolutely continuous invariant measure and a neutral fixed point at the origin. These have $\inf (\tau^k)'=1$ for each $k$. 2d comment Necessity of expansiveness for existence absolutely continuous invariant measures for piecewise smooth maps of an interval How about the identity map? Apr 28 comment Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology? I believe his last answer contained the reference you're asking for. Apr 28 comment Equivalence classes on an ordered Bratteli diagram PS: I have a recent paper (on ArXiv) with Reem Yassawi and Jeannette Janssen studying the structure of the set of minimal and maximal paths in some Bratteli diagrams. Apr 28 comment Equivalence classes on an ordered Bratteli diagram This is probably true by definition. When you say you have an $S$-invariant measure, this implies that the set of paths where $S$ and $S^{-1}$ are undefined (MIN and MAX, the set of minimal and maximal paths) has measure 0. If MIN has measure 0, then so does $S^{-j}$(MIN), so that the set of $\gamma$ where $k_n(\gamma)\to j$ has measure 0 for each $j$. By countable additivity, the set of $\gamma$ where $k_n(\gamma)$ remains bounded has measure 0. Similarly with $d_n(\gamma)-k_n(\gamma)$. Apr 27 comment Hausdorff dimension = entropy/Lyapunov exponent for the baker's map? I think you can get a lower bound also by taking a measure of maximal entropy, $\mu$, on $X$ and pushing it forward under $\pi$ to a measure on $[0,1]^2$. Now Shannon-MacMillan-Breiman says that for almost every $x$, there is an $n(x)$ such that $\mu([x]_{-k}^k)<\exp(-2k(h-\epsilon))$ for a.e. $x\in X$ and $k\ge n(x)$. Letting $S$ be the collection of $x$'s where $n(x)\le N_0$ for some large $N_0$ (so that $S$ has measure at least $\frac 12$), you should obtain a lower bound on the Haudorff measure of a covering set of $\pi(X)$ Apr 27 comment Hausdorff dimension = entropy/Lyapunov exponent for the baker's map? I think $d_H=2h(X)/\log 2$ would be a better bet: there are roughly $e^{2hn}$ 2-sided $n$-cylinders, mapping to regions of size $2^{-n}$ in the square. Indeed it's not hard to check that $d_H(\pi(X))\le 2h(X)/\log 2$. Apr 26 comment Generalizing Ramanujan's “1729 story” The way I heard the story described was the Ramanujan saw the integers as his "friends". The fact that he immediately commented that 1729 had this property was not something that occurred to him off the top of his head, but a comment about one of his friends. Apr 18 comment Are there good bounds on binomial coefficients? So using Stirling's formula, you get an approximation to within a (small) constant simultaneously valid for all $n$ and $k$. Apr 16 comment Expected number of changes in the sign of a rolling sum of independent normal variables Good observation... Apr 14 comment Generalizing the law of large numbers to multiple sets of samples OK. So you have in mind: $m$ fixed; $n$ large. So here's my best guess so far: a distribution, $\nu$, is attainable with an $m$-step sample if $\nu(A)\in [\lambda(A)^m,1-\lambda(A^c)^m]$ for each subset $A$ of the interval. This is certainly a necessary condition. I'll think about sufficiency... By the way, applying this to a set $A$ of small measure immediately gives that $\nu\ll\lambda$ with density bounded above by $m$. Apr 14 comment Generalizing the law of large numbers to multiple sets of samples I don't think there's very much mileage in this question. If $m$ is large, you should be able to get very close to your favourite distribution by picking suitable points from each set. I think it's really destroying the probabilistic spirit of the LLN by allowing "the opponent" to make all of these choices. Apr 13 answered Expected number of changes in the sign of a rolling sum of independent normal variables Apr 7 comment Integral over the Cantor's set Hausdorff dimension @MarkMcClure: Thanks! My favourite paper on the subject is the one with $3\log 2-\pi^2/8$ in the title. There are still many more unsolved than solved questions in the area. Apr 7 comment Integral over the Cantor's set Hausdorff dimension This is just a routine integration with respect to the natural measure on the fractal. Probably it's not in the literature because people have yet to see importance in the topic. Apr 7 comment Maximum size of minimal sequence of transpositions whose product is a given permutation I see. The term minimal is a bit misleading. As @GerhardPaseman says, this is looking for the longest non-self-intersecting walk on the Cayley graph of $S_n$.