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Dynamics of multicritical circle maps

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Abstract

This paper presents a survey of recent and not so recent results concerning the study of smooth homeomorphisms of the circle with a finite number of non-flat critical points, an important topic in the area of One-dimensional Dynamics. We focus on the analysis of the fine geometric structure of orbits of such dynamical systems, as well as on certain ergodic-theoretic and complex-analytic aspects of the subject. Finally, we review some conjectures and open questions in this field.

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Notes

  1. The fact that the Schwarzian derivative vanishes at Möbius transformations is a straightforward computation. On the other hand, given an increasing \(C^3\) diffeomorphism f, consider \(g=(Df)^{-1/2}\) and note that \(Sf=-2\,D^2g/g\). If f has zero Schwarzian derivative then g is affine, which implies at once that f is a Möbius transformation.

  2. Here, as usual, the mesh of a partition is the maximum length of its atoms.

  3. The set \({\mathscr {D}}_0\) is precisely the set of numbers of bounded type, as previously defined.

  4. As a first step, the real bounds (Theorem 5.2) can be used to establish \(C^r\) bounds for return maps, see for instance [35, Appendix A]. A standard Arzelà-Ascoli argument gives then pre-compactness of renormalization orbits.

  5. Let \(\Omega \subset {\mathbb {C}}\) be a domain and let \(K \ge 1\). An orientation-preserving homeomorphism \(f:\Omega \rightarrow f(\Omega )\) is K-quasiconformal if it is absolutely continuous on lines and satisfies

    $$\begin{aligned} \left| {\overline{\partial }}f(z)\right| \le \left( \frac{K-1}{K+1}\right) \big |\partial f(z)\big |\quad \text{ for } \text{ Lebesgue } \text{ a.e. } z\in \Omega \,. \end{aligned}$$

    The Beltrami coefficient of such homeomorphism f is the measurable function \(\mu _f:\Omega \rightarrow {\mathbb {D}}\) given by

    $$\begin{aligned} \mu _f(z)=\frac{{\overline{\partial }}f(z)}{\partial f(z)}\quad \text{ for } \text{ Lebesgue } \text{ a.e. } z\in \Omega \,. \end{aligned}$$

  6. However, we warn the reader that the renormalization “operator” is not a complex-analytic operator.

  7. A quasi-circle, we recall, is the image of a round disk under a quasiconformal homeomorphism of the plane.

  8. It is easy to see that \([f^{q_{m+1}}(c), f^{q_m-q_{m+1}}(c)]\supset I_m\cup I_{m+1}\).

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  1. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo, SP, Brasil

    Edson de Faria

  2. Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Prof. Marcos Waldemar de Freitas Reis, S/N, 24.210-201, Bloco H, Campus do Gragoatá, Niterói, Rio de Janeiro, RJ, Brasil

    Pablo Guarino

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Edson de Faria has been supported by “Projeto Temático Dinâmica em Baixas Dimensões” FAPESP Grant 2016/25053-8, while the second author has been supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES) grant 23038.009189/2013-05.

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de Faria, E., Guarino, P. Dynamics of multicritical circle maps. São Paulo J. Math. Sci. 16, 340–395 (2022). https://doi.org/10.1007/s40863-021-00236-1

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