Loading...
Please wait, while we are loading the content...
Similar Documents
Weakly mixing rank-one transformations conjugate to their squares
| Content Provider | Semantic Scholar |
|---|---|
| Author | Danilenko, Alexandre I. |
| Copyright Year | 2008 |
| Abstract | Utilizing the cut-and-stack techniques we construct explicitly a weakly mixing rigid rank-one transformation T which is conjugate to T . Moreover, it is proved that for each odd q, there is such a T commuting with a transformation of order q. For any n, we show the existence of a weakly mixing T conjugate to T 2 and whose rank is finite and greater than n. 0. Introduction. Recently there has been progress in studying ergodic transformations isomorphic to their composition squares [Ag2], [Go2], [Go3] (see also [Go1]). In [Ag2], Ageev answered a well known question: he proved the existence of a weakly mixing rank-one transformation T conjugate to T 2. However, his proof given in the Baire category framework is not constructive. Thus, no concrete example of T is known so far. The main purpose of the present paper is to construct such a T via the cutting-and-stacking algorithm with explicitly described spacers. For this, we apply a group action approach suggested first by A. del Junco in [dJ3] to produce a counterexample in the theory of simple actions. The idea is to select an auxiliary countable group H and an element h ∈ H and to construct via (C,F )-techniques a special funny rank-one action V of H in such a way that the transformation Vh has required dynamical properties. In our case, H is the group of 2-adic rationals and h=1. The action V is constructed in §2. For other—sometimes unexpected—applications of the group action approach we refer to [Ma], [Ag1], [Da5], [Da6], [DdJ]. For the basics of the cutting-and-stacking (C,F )techniques we refer to §2 below (see also [Da3], [Da4], [DaS] and a survey [Da7]). A new short category proof of the existence theorem from [Ag2] is given below, in Section 1 (Theorem 1.3). Section 2 contains the main result of the present paper (Theorem 2.2). In Section 3 we discuss “elements” of the general theory of ergodic transformations T conjugate to T 2: generic aspects of 2000 Mathematics Subject Classification: Primary 37A40; Secondary 37A15, 37A20, 37A30. |
| Starting Page | 75 |
| Ending Page | 93 |
| Page Count | 19 |
| File Format | PDF HTM / HTML |
| DOI | 10.4064/sm187-1-4 |
| Volume Number | 187 |
| Alternate Webpage(s) | https://www.impan.pl/shop/publication/transaction/download/product/89832 |
| Alternate Webpage(s) | https://doi.org/10.4064/sm187-1-4 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
