100 research outputs found

    Simultaneous dense and nondense orbits for commuting maps

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    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

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    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 PP 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∈Zmn\in\Z^{m} such that EE 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∈Nk\in\N there exists n∈Zmn\in\Z^{m} such that the integers p_{1}(n),\ld,p_{r}(n) are all divisible by kk. 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, Z=⋃i=1kEi\Z=\bigcup_{i=1}^{k}E_{i}, there exist i\in\{1,\ld,k\} and a,n∈Eia,n\in E_{i} such that \{a,a+p_{1}(n),\ld,a+p_{r}(n)\}\sln E_{i}
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