Problem 10

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Statistics plus dynamics of transformations of $[0,1]$ - 'non-linear' $\beta$-expansions like examples.

Tags

FL

cf. [1] and question 111.

References

1. [hofbauer1982ergodic] Hofbauer F, Keller G.  1982.  The Theory of Chaotic Attractors. :120–141.

For the $\beta$

For the $\beta$-transformations themselves (including nonlinear versions with the same coding space), [1] gives unique equilibrium state for every Lipschitz potential; this was extended to Holder potentials in [2].  Of course general piecewise expanding interval maps have a huge literature: in the transitive case, every Holder potential has a unique equilibrium state provided it satisfies the condition $\sup_\mu \int \phi\,d\mu < P(\phi)$, where the supremum is over all invariant measures.  Buzzi conjectured in [3] that this holds for all Holder potentials, and proved it when the map is continuous (not just piecewise continuous).  So far as I know the conjecture is still open even for the examples $x\mapsto \alpha + \beta x \pmod 1$.

References

1. [Walters1978] Walters P.  1978.  Math. Z.. 159:65–88.
2. [CT13] Climenhaga V, Thompson DJ.  2013.  J. Lond. Math. Soc. (2). 87:401–427.
3. [Buzzi2004] Buzzi J.  2004.  Stoch. Dyn.. 4:84–94.