bio | website | home.exetel.com.au/mansfield |
---|---|---|
location | Sydney | |
age | 32 | |
visits | member for | 4 years, 10 months |
seen | Apr 7 at 0:00 | |
stats | profile views | 469 |
PhD student at UNSW
Apr 7 |
comment |
System with invariant measure, but no ergodic measure.
There will always be a $T$-ergodic measure $\mu^\prime$. By this construction $\mu^\prime \ll \mu$ however $\mu \ll \mu^\prime$ provided $T$ is conservative. Take any set $A$ of positive measure and define the $T$-invariant set $X^\prime = \cup_{n=-\infty}^\infty T^nA$. Then $$\mu^\prime = \frac{1}{\mu(X^\prime)} \mu\circ 1_{X^\prime}$$ is an ergodic, measure preserving, $T$-invariant measure. |
Apr 4 |
comment |
System with invariant measure, but no ergodic measure.
I believe this is true provided $T$ is conservative. The $\mu$-invariance means that for any $T$-invariant set $A$ of positive measure $$1= \mu(\cup_{n=1}^\infty T^nA) = \mu(A) $$. |
Jul 2 |
awarded | Curious |
May 31 |
awarded | Yearling |
May 20 |
comment |
Is an odometer action on a product space always conjugate to its inverse by an involution?
For Q2 I mean a Bratteli diagram with more than one vertex at each level. I wish to distinguish between reversing the ordering and the map $\phi$ defined in the question. 1) reversing the order of the B-diagram. Let $\psi : (V,E,\leq) \mapsto (V,E,\geq), \phi(x) = x$. In this case the action of $T$ and $T^{-1}$ are the same and $\frac{dT^{-1}\mu}{d\mu} = \frac{dT\mu}{d\mu}$ 2) The map $\phi : (V,E,\leq) \mapsto (V,E,\leq)$ does not change the order, and the action of $T$ and $T^{-1}$ are different, the derivatives are not necessarily the same. |
May 15 |
asked | Is an odometer action on a product space always conjugate to its inverse by an involution? |
May 8 |
accepted | Is an non-singualr invertable ergodic transformation on a measure space isomorphic to its inverse? |
May 3 |
asked | Is an non-singualr invertable ergodic transformation on a measure space isomorphic to its inverse? |
Apr 18 |
awarded | Nice Question |
Nov 6 |
accepted | ITPFI factors with restricted growth |
Nov 2 |
asked | ITPFI factors with restricted growth |
Jan 31 |
comment |
Permutations that preserve Cesaro mean
Permutations preserving Cesaro mean for any sequence would be the Levy group. See theorem 2 of M. Blümlinger; N. Obata "Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences" (1991) |
Jan 31 |
revised |
Permutations that preserve Cesaro mean
deleted 2 characters in body |
Jan 31 |
comment |
Permutations that preserve Cesaro mean
I do mean the set $S$ of permutations preserving a given Cesaro mean. |
Jan 27 |
asked | Permutations that preserve Cesaro mean |
Jan 13 |
comment |
Question about entropy
I like this answer because it answers what the question should have been. |
Jan 13 |
revised |
Question about entropy
added 9 characters in body |
Jan 13 |
revised |
Question about entropy
added 18 characters in body |
Jan 13 |
revised |
Question about entropy
added 9 characters in body |
Jan 13 |
comment |
Question about entropy
Thank you Anthony, $TQ$ is not necessarily an element of the partition. I have modified the answer in response to your comment. |