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1. CJM 2013 (vol 66 pp. 57)
Perfect Orderings on Finite Rank Bratteli Diagrams Given a Bratteli diagram $B$, we study the set $\mathcal O_B$ of all
possible orderings on $B$ and its subset
$\mathcal P_B$ consisting of perfect orderings that produce
Bratteli-Vershik topological dynamical systems (Vershik maps). We
give necessary and sufficient conditions for the ordering $\omega$ to be
perfect. On the other hand, a
wide class of non-simple Bratteli diagrams that do not admit Vershik
maps is explicitly described. In the case of finite rank Bratteli
diagrams, we show that the existence of perfect orderings with a prescribed
number of extreme paths constrains significantly the values of the entries of
the incidence matrices and the structure of the diagram $B$. Our
proofs are based on the new notions of skeletons and
associated graphs, defined and studied in the paper. For a Bratteli
diagram $B$ of rank $k$, we endow the set $\mathcal O_B$ with product
measure $\mu$ and prove that there is some $1 \leq j\leq k$ such that
$\mu$-almost all orderings on $B$ have $j$ maximal and $j$ minimal
paths. If $j$ is strictly greater than the number of minimal
components that $B$ has, then $\mu$-almost all orderings are imperfect.
Keywords:Bratteli diagrams, Vershik maps Categories:37B10, 37A20 |
2. CJM 2011 (vol 64 pp. 1341)
Bowen Measure From Heteroclinic Points We present a new construction of the entropy-maximizing, invariant
probability measure on a Smale space (the Bowen measure). Our
construction is based on points that are unstably equivalent to one
given point, and stably equivalent to another: heteroclinic points.
The spirit of the construction is similar to Bowen's construction from
periodic points, though the techniques are very different. We also
prove results about the growth rate of certain sets of heteroclinic
points, and about the stable and unstable components of the Bowen
measure. The approach we take is to prove results through direct
computation for the case of a Shift of Finite type, and then use
resolving factor maps to extend the results to more general Smale
spaces.
Keywords:hyperbolic dynamics, Smale space Categories:37D20, 37B10 |
3. CJM 2001 (vol 53 pp. 382)
Building a Stationary Stochastic Process From a Finite-Dimensional Marginal If $\mathfrak{A}$ is a finite alphabet, $\sU \subset\mathbb{Z}^D$, and
$\mu_\sU$ is a probability measure on $\mathfrak{A}^\sU$ that ``looks like''
the marginal projection of a stationary stochastic process on
$\mathfrak{A}^{\mathbb{Z}^D}$, then can we ``extend''
$\mu_\sU$ to such a process? Under what conditions can we make this
extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying
classical work on this problem when $D=1$, we provide some sufficient
conditions and some necessary conditions for $\mu_\sU$ to be extendible
for $D>1$, and show that, in general, the problem is not formally decidable.
Categories:37A50, 60G10, 37B10 |