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Register a new userComputation of Topological Entropy via $phi$-expansion, an Inverse Problem for the Dynamical Systems $beta x+alpha mod 1$
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We give an algorithm, based on the $phi$-expansion of Parry, in order to compute the topological entropy of a class of shift spaces. The idea is the solve an inverse problem for the dynamical systems $beta x+alpha mod1$.The first part is an exposition of the $phi$-expansion applied to piecewise monotone dynamical systems. We formulate for the validity of the $phi$-expansion, necessary and sufficient conditions, which are different from those in Parrys paper.
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We consider the map $T_{alpha,beta}(x):= beta x + alpha mod 1$, which admits a unique probability measure of maximal entropy $mu_{alpha,beta}$. For $x in [0,1]$, we show that the orbit of $x$ is $mu_{alpha,beta}$-normal for almost all $(alpha,beta)in[0,1)times(1,infty)$ (Lebesgue measure). Nevertheless we construct analytic curves in $[0,1)times(1,infty)$ along them the orbit of $x=0$ is at most at one point $mu_{alpha,beta}$-normal. These curves are disjoint and they fill the set $[0,1)times(1,infty)$. We also study the generalized $beta$-maps (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 $beta$.
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