Minimal polynomials of some beta-numbers and Chebyshev polynomials

The natural extensions of β-transformations which generalize baker's transformations

“…The connection between β-shifts and mechanical words has been studied and has disclosed it's fertility [2,3,8,[11][12][13]. Proposition 2.1 indicates that this interdisciplinary study inevitably involves lexicographic order between mechanical words.…”

“…Transcendence [8] and algebraic properties [11,12] of -values were investigated. And it turned out that the regularity of depends upon the Diophantine property of α.…”