Coarse expanding conformal dynamics.

*(English)*Zbl 1206.37002
Astérisque 325. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-266-2/pbk). viii, 139 p. (2009).

This monograph consists of three main parts. The first introduces two closely related classes of dynamical systems, (1) topologically, and (2) metrically, course expanding conformal dynamical systems. The setting is a continuous map of the form \(f:{\mathcal X}_1\to {\mathcal X}_0\) where \({\mathcal X}_1\) is an open subset of \({\mathcal X}_0\) with compact closure. The main object of interest is the invariant set \({\mathcal X}\) of points whose entire forward orbits remain in \({\mathcal X}_1\). A topological course expanding conformal system is such a map that is a bounded-to-one branched covering with certain irreducibility, expansiveness, and regularity properties that are defined in terms of a finite open cover \({\mathcal U}\) of \({\mathcal X}\) by open, connected subsets of \({\mathcal X}_0\) and the iterated preimagesof elements of \({\mathcal U}\). A motivating example is a subhyperbolic rational map of the Riemann sphere, with the Julia set playing the part of \({\mathcal X}\) and \({\mathcal X}_0\), \({\mathcal X}_1\) appropriate neighborhoods of \({\mathcal X}\). When the spaces involved are metric spaces, the system is a metric course expanding conformal system if in addition certain metric properties concerning the diameters and ‘roundness’ of the sets in the finite open cover and their iterated preimages. After these introductions, the first main part concludes with derivations of basic properties of these two types of systems.

The second part introduces an abstract graph \(\Gamma\) defined in terms of the fundamental open cover and its preimages, a map \(F\) induced by \(f\) on \(\Gamma\) that extends to a compactification of \(\Gamma\), such that the restriction of \(F\) to the boundary of \(\Gamma\) is topologically conjugate to the restriction of \(f\) to \({\mathcal X}\). Metrics on \(\partial \Gamma\) are used to induce metrics \(d_\varepsilon\) on \({\mathcal X}\) with good properties for all small \(\varepsilon\). It is shown that there is a unique invariant quasiconformal measure; it is mixing, the measure of maximal entropy, and describes the distributions of both periodic points and preimages of points.

The third part is devoted applying the preceding theory to various classes of examples, in particular rational functions and uniformly quasiregular systems. It is shown that if \(f:S^2\to S^2\) is an orientation preserving and metrically course expanding conformal system (here \({\mathcal X} = {\mathcal X}_0 = {\mathcal X}_1\)), then \(f\) is quasiconformmally conjugate to a chaotic semi-hyperbolic rational map. Also, if \(M\) is a Riemannian manifold of dimension three or more, then \(f:M\to M\) is an orientation preserving and metrically course expanding conformal system if and only if it is a generalized Lattès map.

The second part introduces an abstract graph \(\Gamma\) defined in terms of the fundamental open cover and its preimages, a map \(F\) induced by \(f\) on \(\Gamma\) that extends to a compactification of \(\Gamma\), such that the restriction of \(F\) to the boundary of \(\Gamma\) is topologically conjugate to the restriction of \(f\) to \({\mathcal X}\). Metrics on \(\partial \Gamma\) are used to induce metrics \(d_\varepsilon\) on \({\mathcal X}\) with good properties for all small \(\varepsilon\). It is shown that there is a unique invariant quasiconformal measure; it is mixing, the measure of maximal entropy, and describes the distributions of both periodic points and preimages of points.

The third part is devoted applying the preceding theory to various classes of examples, in particular rational functions and uniformly quasiregular systems. It is shown that if \(f:S^2\to S^2\) is an orientation preserving and metrically course expanding conformal system (here \({\mathcal X} = {\mathcal X}_0 = {\mathcal X}_1\)), then \(f\) is quasiconformmally conjugate to a chaotic semi-hyperbolic rational map. Also, if \(M\) is a Riemannian manifold of dimension three or more, then \(f:M\to M\) is an orientation preserving and metrically course expanding conformal system if and only if it is a generalized Lattès map.

Reviewer: Mike Hurley (Cleveland)

##### MSC:

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |

37B99 | Topological dynamics |

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |

37F15 | Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems |

37F20 | Combinatorics and topology in relation with holomorphic dynamical systems |

37F30 | Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010) |

54E40 | Special maps on metric spaces |