• Title, Summary, Keyword: skew product transformation

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ERGODICITY AND RANDOM WALKS ON A COMPACT GROUP

  • CHOE, GEON HO
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.5 no.1
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    • pp.25-33
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    • 2001
  • Let G be a finite group with a probability measure. We investigate the random walks on G in terms of ergodicity of the associated skew product transformation.

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DYNAMICAL PROPERTIES OF A FAMILY OF SKEW PRODUCTS WITH THREE PARAMETERS

  • Ahn, Young-Ho
    • Honam Mathematical Journal
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    • v.31 no.4
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    • pp.591-599
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    • 2009
  • For given ${\alpha},{\omega}\;{\in}\;{\mathbb{R}}$ and ${\beta}$ > 1, let $T_{{\beta},{\alpha},{\omega}}$ be the skew-product transformation on the torus, [0, 1) ${\times}$ [0, 1) defined by (x, y) ${\longmapsto}\;({\beta}x,y+{\alpha}x+{\omega})$ (mod 1). In this paper, we give a criterion of ergodicity and weakly mixing for the transformation $T_{{\beta},{\alpha},{\omega}}$ when the natural extension of the given ${\beta}$-transformation can be viewed as a generalized baker's transformation, i.e., they flatten and stretch and then cut and stack a two-dimensional domain. This is a generalization of theorems in [10].

THE PARITIES OF CONTINUED FRACTION

  • Ahn, Young-Ho
    • Honam Mathematical Journal
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    • v.30 no.4
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    • pp.733-741
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    • 2008
  • Let T be Gauss transformation on the unit interval defined by T (x) = ${\frac{1}{x}}$ where {x} is the fractional part of x. Gauss transformation is closely related to the continued fraction expansions of real numbers. We show that almost every x is mod M normal number of Gauss transformation with respect to intervals whose endpoints are rational or quadratic irrational. Its connection to Central Limit Theorem is also shown.