DYNAMICAL PROPERTIES OF A FAMILY OF SKEW PRODUCTS WITH THREE PARAMETERS

• Journal title : Honam Mathematical Journal
• Volume 31, Issue 4,  2009, pp.591-599
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2009.31.4.591
Title & Authors
DYNAMICAL PROPERTIES OF A FAMILY OF SKEW PRODUCTS WITH THREE PARAMETERS
Ahn, Young-Ho;

Abstract
For given $\small{{\alpha},{\omega}\;{\in}\;{\mathbb{R}}}$ and $\small{{\beta}}$ > 1, let $\small{T_{{\beta},{\alpha},{\omega}}}$ be the skew-product transformation on the torus, [0, 1) $\small{{\times}}$ [0, 1) defined by (x, y) $\small{{\longmapsto}\;({\beta}x,y+{\alpha}x+{\omega})}$ (mod 1). In this paper, we give a criterion of ergodicity and weakly mixing for the transformation $\small{T_{{\beta},{\alpha},{\omega}}}$ when the natural extension of the given $\small{{\beta}}$-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].
Keywords
$\small{{\beta}}$-expansions;$\small{{\beta}}$-transformations;skew product;generalized baker's transformations;
Language
English
Cited by
References
1.
F. Blanchard ${\beta}$-expansions and symbolic dynamics, Theoretical Computer Science 65 (1989), 131-141.

2.
G. Brown and Q. Yin, ${\beta}$-transformation, natural extension and invariant measure, Ergod. Theory and Dyn., Syst. 20 (2000), 1271-1285.

3.
Z. Coelho and W. Parry, Shift endomorphisms and compact Lie extensions, Bol. Soc. Brasi. Math. 29(1) (1998), 163-179.

4.
H. Helson and W. Parry, Cocycles and spectra, Ark. Mat. 16 (1978), 195-206.

5.
E. Hewitt and K. Ross, Abstract Harmonic Analysis I, II, Springer-Verlag, 1963,1970.

6.
D. Kwon, The natural extensions of ${\beta}$-transformations which generalize baker's transformations, Nonlinearity. 22 (2009), 301-310.

7.
W. Parry, On the ${\beta}$-expansion of real numbers, Acta Math. Acad. Sci. Hung. 11(1960), 401-416.

8.
A. Renyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung. 8 (1957), 477-493.

9.
W. Rudin, Real and Complex Analysis, McGraw-Hill, 1986.

10.
S. Siboni, Ergodic properties of a class of skew-systems obtained by coupling the transformation of the 1-torus with the endomorphism 2x mod [0, 1[, Nonlinearity. 7 (1994), 1133-1141.

11.
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag New York, 1982.

12.
P. Walters, Equilibrium states for ${\beta}$-transformations and related transformations, Math. Z. 159 (1978), 65-88