ترغب بنشر مسار تعليمي؟ اضغط هنا

Maximal equicontinuous factors and cohomology for tiling spaces

120   0   0.0 ( 0 )
 نشر من قبل Aurelie Reymond
 تاريخ النشر 2012
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

We study the homomorphism induced on cohomology by the maximal equicontinuous factor map of a tiling space. We will see that this map is injective in degree one and has torsion free cokernel. We show by example, however, that the cohomology of the maximal equicontinuous factor may not be a direct summand of the tiling cohomology.



قيم البحث

اقرأ أيضاً

Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which forces its border. One can then represent the tiling space as an inverse limit of an inflation and substit ution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson-Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.
135 - Marcy Barge 2013
We consider the structure of Pisot substitution tiling spaces, in particular, the structure of those spaces for which the translation action does not have pure discrete spectrum. Such a space is always a measurable m-to-one cover of an action by tran slation on a group called the maximal equicontinuous factor. The integer m is the coincidence rank of the substitution and equals one if and only if translation on the tiling space has pure discrete spectrum. By considering factors intermediate between a tiling space and its maximal equicontinuous factor, we establish a lower bound on the cohomology of a one-dimensional Pisot substitution tillng space with coincidence rank two and dilation of odd norm. The Coincidence Rank Conjecture, for coincidence rank two, is a corollary.
If phi is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Phi on the tiling space T_Phi factors via geometric realization onto a d-dimensional solenoid. Under this realization, the collection of Phi-periodic asympt otic tilings corresponds to a finite set that projects onto the branch locus in a d-torus. We prove that if two such tiling spaces are homeomorphic, then the resulting branch loci are the same up to the action of certain affine maps on the torus.
We show that the group of conformal homeomorphisms of the boundary of a rank one symmetric space (except the hyperbolic plane) of noncompact type acts as a maximal convergence group. Moreover, we show that any family of uniformly quasiconformal homeo morphisms has the convergence property. Our theorems generalize results of Gehring and Martin in the real hyperbolic case for Mobius groups. As a consequence, this shows that the maximal convergence subgroups of the group of self homeomorphisms of the $d$-sphere are not unique up to conjugacy. Finally, we discuss some implications of maximality.
Identity-homotopic self-homeomorphisms of a space of non-periodic 1-dimensional tiling are generalizations of orientation-preserving self-homeomorphisms of circles. We define the analogue of rotation numbers for such maps. In constrast to the classic al situation, additional assumptions are required to make rotation numbers globally well-defined and independent of initial conditions. We prove that these conditions are sufficient, and provide counterexamples when these conditions are not met.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا