100 research outputs found

### Simultaneous dense and nondense orbits for commuting maps

We show that, for two commuting automorphisms of the torus and for two
elements of the Cartan action on compact higher rank homogeneous spaces, many
points have drastically different orbit structures for the two maps.
Specifically, using measure rigidity, we show that the set of points that have
dense orbit under one map and nondense orbit under the second has full
Hausdorff dimension.Comment: 17 pages. Very minor changes to the exposition. Three additional
papers cite

### Intersective polynomials and polynomial Szemeredi theorem

Let P=\{p_{1},\ld,p_{r}\}\subset\Q[n_{1},\ld,n_{m}] be a family of
polynomials such that p_{i}(\Z^{m})\sle\Z, i=1,\ld,r. We say that the
family $P$ has {\it PSZ property} if for any set E\sle\Z with
d^{*}(E)=\limsup_{N-M\ras\infty}\frac{|E\cap[M,N-1]|}{N-M}>0 there exist
infinitely many $n\in\Z^{m}$ such that $E$ contains a polynomial progression of
the form \hbox{\{a,a+p_{1}(n),\ld,a+p_{r}(n)\}}. We prove that a polynomial
family P=\{p_{1},\ld,p_{r}\} has PSZ property if and only if the polynomials
p_{1},\ld,p_{r} are {\it jointly intersective}, meaning that for any $k\in\N$
there exists $n\in\Z^{m}$ such that the integers p_{1}(n),\ld,p_{r}(n) are
all divisible by $k$. To obtain this result we give a new ergodic proof of the
polynomial Szemer\'{e}di theorem, based on the fact that the key to the
phenomenon of polynomial multiple recurrence lies with the dynamical systems
defined by translations on nilmanifolds. We also obtain, as a corollary, the
following generalization of the polynomial van der Waerden theorem: If
p_{1},\ld,p_{r}\in\Q[n] are jointly intersective integral polynomials, then
for any finite partition of $\Z$, $\Z=\bigcup_{i=1}^{k}E_{i}$, there exist
i\in\{1,\ld,k\} and $a,n\in E_{i}$ such that
\{a,a+p_{1}(n),\ld,a+p_{r}(n)\}\sln E_{i}

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