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A point is normal for almost all maps βx+α mod 1 or generalized β-transformations

Published online by Cambridge University Press:  03 February 2009

B. FALLER  and
C.-E. PFISTER
B. FALLER
Affiliation:
EPF-L, SB IACS, Station 8, CH-1015 Lausanne, Switzerland (email: bastien.faller@a3.epfl.ch, charles.pfister@epfl.ch)
C.-E. PFISTER
Affiliation:
EPF-L, SB IACS, Station 8, CH-1015 Lausanne, Switzerland (email: bastien.faller@a3.epfl.ch, charles.pfister@epfl.ch)
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Abstract

We consider the map Tα,β(x):=βx+α mod 1, which admits a unique probability measure μα,β of maximal entropy. For x∈[0,1], we show that the orbit of x is μα,β-normal for almost all (α,β)∈[0,1)×(1,∞) (with respect to Lebesgue measure). Nevertheless, we construct analytic curves in [0,1)×(1,∞) along which the orbit of x=0 is μα,β-normal at no more than one point. These curves are disjoint and fill the set [0,1)×(1,∞). We also study the generalized β-transformations (in particular, the tent map). We show that the critical orbit x=1 is normal with respect to the measure of maximal entropy for almost all β.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 5 , October 2009 , pp. 1529 - 1547
Copyright
Copyright © Cambridge University Press 2009

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