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26 April 2024 |
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Chacon's Type Ergodic Transformations with Unbounded Arithmetic Spacers | | V.V. Ryzhikov
; | | Date: |
18 Nov 2013 | | Abstract: | The following generalizations of the Chacon map are proposed: instead of
classical constant spacer sequence $(0,1,0)$ let a sequence $(0,s_j,0)$ be one
with unbounded $s_j$. (We mention also an analogue of the historical Chacon map
with spacer sequences in the form $(0,s_j)$.) This narrow class of rank-one
transformations may be abundant and inexhaustible source of open questions. All
such constructions have partial rigidity, but some other properties could be
different. For root sequence, $ s_j= [sqrt{j}]$, (or $ s_j= [ln{j}]$) the
corresponding action is rigid, moreover it possesses all polynomials in its
weak closure. In the linear case $s_j={j}$ we get (as well as for the classical
Chacon transformation) the property of minimal self-joinings (MSJ).
We present some observations about MSJ, mild mixing, partial mixing,
$ae$-mixing, absence of factors, triviality of centralizer and spectral
primality and state several problems, providing a place for the subsequent
generalizations and the imagination. | | Source: | arXiv, 1311.4524 | | Services: | Forum | Review | PDF | Favorites | |
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