A remark on Gibbs distributions associated by critical circle maps
Keywords:
critical circle map, critical point, rotation number, invariant measure, symbolic dynamics, Gibbs measureAbstract
In present work we investigate circle maps $f$ with one critical point and irrational rotation number $\rho_{f}.$ It is well known that each such map $f$ has a unique probability invariant measure $\mu_{f}$ and it is singular w.r.t Lebesque measure on the circle. J. Yoccoz proved that if $f\in C^[2]$ then $f$ is topoligically congugated by linear rotation to angle $\rho_{f}.$ In the space of critical maps with "Golden mean " rotation number $\rho=\frac{\sqt5}-1}{2}$ the renormalization transformation has unique nontrivial fixed point $f_{0}$ . For the map $f_{0}$ can be constructed the thermodynamic formalism. We prove that on the space one sided sequences there exist unique Gibbs measure corresponding to the map $f_{0}$ .
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