Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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DYNAMICAL PROPERTIES OF A FAMILY OF SKEW PRODUCTS WITH THREE PARAMETERS

  • Ahn, Young-Ho (Department of Mathematics, Mokpo National University)
  • Received : 2009.08.25
  • Accepted : 2009.11.10
  • Published : 2009.12.25
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Abstract

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].

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References

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