9,469 reputation
12153
bio website math.uvic.ca/faculty/aquas
location University of Victoria
age 47
visits member for 4 years, 4 months
seen 10 hours ago
The pic is the phase portrait of a simple "piecewise isometry". Define a map by sliding the two halves of the plane: the top half to the right; the lower half to the left and then rotate. Around each periodic point of the map there's a "periodic island". These are what are in the image...

Mar
19
comment Maximization of Funcional
Where does this question come from?
Mar
18
comment A weak Perron-Frobenius property for sets of positive matrices
So I guess you could get something by showing the family of matrices uniformly contract the positive orthant in the Hilbert metric. Probably not the kind of answer you had in mind though.
Mar
17
awarded  Enlightened
Mar
17
awarded  Nice Answer
Mar
17
comment Which kind of subsets of primes one needs to generate a positive ratio of the natural numbers?
My guess is that so crude that it's unlikely there is really fine information hidden there. I'm not at all an analytic number theorist so can't really comment further.
Mar
17
comment Which kind of subsets of primes one needs to generate a positive ratio of the natural numbers?
I'd say this is probably reasonably well known. Ideas very much like this show up in Mertens' Theorem. Also some morally similar ideas occur in my paper math.uvic.ca/faculty/aquas/papers/paper38.pdf
Mar
17
answered Which kind of subsets of primes one needs to generate a positive ratio of the natural numbers?
Mar
17
comment Which kind of subsets of primes one needs to generate a positive ratio of the natural numbers?
I think you're right. The condition should be the sum of the reciprocals of $\mathbb P\setminus S$ should be finite. I strongly suspect this is necessary and sufficient.
Mar
17
answered Monotonicity of a ratio of conditional expectation operator
Mar
11
comment Density of periodic points and density of periodic measures
The north-south map solves 2.
Mar
10
answered Lyapunov exponents of dual / adjoint / transpose random dynamical system (RDS)
Mar
8
comment When are (Abelian) Cayley graphs also expanders?
Here's a proof. Suppose there are $d$ generators. Then you are interested in counting the number of distinct group elements you can make by composing $n$ generators. I will give an upper bound on the size of the $n$-ball by only taking account of coincidences that occur by rearrangement; but not any other coincidences. Hence we're asking how many ways are there to pick $n$ items with replacement but without order from $d$. It's a well known combinatorial fact ("stars and bars") that there are $\binom{n+d-1}{d-1}\sim n^{d-1}/(d-1)!$ ways to do this.
Mar
8
comment Upper bound for the number of integral points in a convex set
What makes you think this is correct?
Mar
8
comment When are (Abelian) Cayley graphs also expanders?
Abelian Cayley graphs have polynomial neighbourhood size, and so cannot be expanders. To be an expander, you have to be able to get from any vertex to any other by following a short path. The difficulty with Abelian groups is that many different paths take you to the same place (following $g$ then $h$ is the same as following $h$ then $g$). To have expansion, you need most of the $N^l$ paths of length $l$ to take you to different places (so the number of places you can get to grows exponentially). In an Abelian gp, you can only get to $l^N$ places, roughly.
Mar
5
comment Volume of randomly changing sphere follows beta distribution
So if the $(X_k)$ were placed at positions of a discrete distribution, then the volume would take on discrete values, and so could not possibly be beta-distributed. Now if you change the discrete distribution by spreading it out just slightly, you don't really alter the distribution of the volume of the sphere - it's still very close to a discrete distribution. This cannot be close to a beta distribution.
Mar
4
comment Volume of randomly changing sphere follows beta distribution
Sounds much too good to be true. Is $p(X)$ meant to be the absolutely continuous density function of the distribution of the $X$'s?
Mar
1
comment Classification of ergodic measures for circle expanding maps
To be more specific about producing enormous numbers of measures all satisfying your positive entropy condition, given any measure on $\{0,1\ldots,d-1\}^{\mathbb Z}$, take its product with a product measure on $\{0,1\}^{\mathbb Z}$ (where 1 has measure $\epsilon$ and 0 has measure $1-\epsilon$). You can think of the new measure as a measure with $2d$ symbols $\{(0,0),\ldots,(1,d-1)\}$. All of these measures satisfy your concentration condition. As explained before, these measures have any factor with entropy $<\log d$. The randomization part adds $|\epsilon\log\epsilon|$ entropy.
Feb
28
comment Classification of ergodic measures for circle expanding maps
So you can reformulate Furstenberg's conjecture in terms of shift spaces. Indeed this was what Dan Rudolph did when he proved his positive result in the presence of positive entropy. The usefulness of the shift is that it makes it very easy to construct lots of measures using what Kalikow calls the 'monkey method' (see his book with McCutcheon). So for example, it would be easy using this book to construct lots of very bizarre measures that still satisfy your non-concentration condition.
Feb
28
answered Classification of ergodic measures for circle expanding maps
Feb
27
comment When an integral with respect to a Poisson point process is finite?
So I think I may have been misled by wikipedia, where it says that any left continuous process is predictable. Maybe it should have said any left continuous adapted process is predictable.