Critical circle maps near bifurcation

1991/04/27 by Jacek Graczyk, Graczyk, Jacek, Grzegorz Świa̧tek +1
Mathematics · Physics and Astronomy · #54H20 #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.math/9201283

openalex publication_date 1991/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We estimate harmonic scalings in the parameter space of a one-parameter family of critical circle maps. These estimates lead to the conclusion that the Hausdorff dimension of the complement of the frequency-locking set is less than 1 but not less than 1/3. Moreover, the rotation number is a Hölder continuous function of the parameter.

Related

• The rigidity problem for analytic critical circle maps
• Rigidity of critical circle mappings, I
• Scaling of the rotation number for perturbations of rational rotations
• Asymptotic rigidity of scaling ratios for critical circle mappings
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