IN ONE-DIMENSIONAL DYNAMICS

Organizers: Feliks Przytycki and Juan Rivera-Letelier

Coffee | |

10:00 | G. Levin |

11:10 | P. Haissinsky I |

12:20 | M. Urbanski I |

9:15 | J. Rivera-Letelier |

10:10 | Coffee break |

10:40 | P. Haissinsky II |

11:50 | M. Urbanski II |

16:00 | V. Mayer I |

17:10 | M. Zinsmeister |

9:15 | J. Kotus |

10:10 | Coffee break |

10:40 | P. Haissinsky III |

11:50 | M. Urbanski III |

16:00 | V. Mayer II |

17:10 | G. Havard |

9:15 | M. Todd |

10:10 | Coffee break |

10:40 | H. Bruin |

11:50 | M. Holland |

16:00 | V. Mayer III |

17:10 | G. Iommi |

9:15 | D. Meyer |

10:10 | Coffee break |

10:40 | M. Denker |

11:50 | ...... |

MINICOURSES:

- Peter Haissinsky,
*Coarse conformal dynamics*

**Abstract.**The aim is to introduce geometric methods coming from the theory of negatively curved metric spaces to the iteration of rational maps, and more generally to the dynamics of finite branched coverings on metric spaces. - Volker Mayer,
*Thermodynamical formalism for meromorphic functions of finite order*

**Abstract.**One of the main tools of the geometric and ergodic study of holomorphic dynamically systems is the thermodynamical formalism. We present joint work with Mariusz Urbanski in which we make this theory available for a very general class of meromorphic functions and deduce several applications. We first present this class of so called balanced meromorphic functions. Then we pass to the thermodynamical formalism itself. The key point here is that, given a balanced function, we can associate to it a Riemannian metric with respect to which the transfer operator is well defined and can be controlled thanks to Nevanlinna Theory. - Mariusz Urbanski,
*Conformal Iterated Function Systems, generalizations and applications*

**Abstract.**The class of conformal graph directed Markov systems (with countably many edges and finitely many vertices) will be defined. The pressure function and conformal measures will be introduced and the classification into regular, strongly regular, cofinitely regular and irregular systems systems will be provided. An appropriate version of Bowen's formula for the Hausdorff dimension of the limit set will be discussed and the formula for the dimension of the closure of the limit set will be provided. The θ parameter will be discussed. Dynamical and geometrical rigidities will be mentioned. The space of Iterated function systems (the set of vertices is a singleton) acting on a phase space`X`will be topologized and regularity properties of the Hausdorff dimension function will be discussed. They include continuity and real-analyticity. This theory will be applied to elliptic functions to get lower bounds for the Hausdorff dimension of the Julia set. It will be also applied for generalized polynomial-like maps to demonstrate real-analyticity of pressure and real-analytic dependence of the Hausdorff dimension on parameter if an appropriate family of such maps is given. Applications to diophantine approximations and statistics of return times will be mentioned. The case when the number of vertices is also infinite will be discussed and applications to infinitely generated Kleinian groups of Schottky type will be provided.

- Henk Bruin,
*Equilibrium states and Young tower constructions in one-dimensional dynamics.*(Joint work with Mike Todd)

**Abstract:**Young towers are being used for an increasing number of applications; recently for the construction of equilibrium states for specific potential functions.

Equilibrium states are invariant measures that maximize a certain functional involving the potential function and entropy. However, when using a Young tower construction, the resulting equilibrium state may only be unique and maximal within the class of measures that can be lifted to this specific Young tower. A priori, a different Young tower may result in different equilibrium states.

In this talk I want to restrict attention to smooth maps on the interval and discuss a canonical way of constructing Young towers using the Hofbauer tower. I will show that, within the class of invariant measures with positive Lyapunov exponent, equilibrium states are indeed independent of the choice of the Young tower. - Manfred Denker,
*A Local Limit Theorem for Beta-Transformations*

**Abstract.**This is joint work with Aaronson, Sarig and Zweimueller. We establish conditions for aperiodicity of cocycles (in the sense of Guivarc'h and Hardy, obtaining, via a study of perturbations of transfer operators, conditional local limit theorems. Our results apply to a large class of Markov and non-Markov interval maps, including beta-transformations. - Guillame Havard,
*Non-finite equilibrium measures vs conformal measures in iteration of a rational map on its Julia set*

**Abstract.**When studying equilibrium measure for a given potential φ, which means to find an invariant probability measure which maximizes the pressure`P`(φ)$ of φ, one may first look for a measure ω with Jacobian equal to exp(`P`(φ)-φ). Then one may try to find an invariant probability measure μ equivalent with ω. This will be our equilibrium measure. It appears that sometimes this strategy leads to a non-finite invariant measure. This could be the case if for instance there are some parabolic points. In this talk I will introduce a way to picture those invariant measures as equilibrium measures. Then I will discuss unicity and existence of such generalized equilibrium measures. The case of the iteration of a parabolic rational map on its Julia set is the perfect field to illustrate those notions. - Mark Holland,
*Statistical Properties of Lorenz-like maps and flows*

**Abstract.**Using methods of inducing we prove exponential decay of correlations for a class of one-dimensional maps with singularities and critical points. The motivation for studying such maps arises naturally from the Lorenz equations. Furthermore, we can use inducing to analyse the statistical properties of the actual Lorenz attractor, where we prove that a Central Limit Theorem and Almost-Sure-Invariance-Principal (approx by Brownian motion) holds. - Godofredo Iommi,
*Suspension flows over countable Markov shifts*

**Abstract.**In this joint work with Luis Barreira, we introduce a notion of topological pressure for suspension flows over countable Markov shifts. We develop the associated thermodynamic formalism. In particular, we establish a variational principle and find conditions that guarantee the existence of equilibrium measures. As an application we present a multifractal analysis for the entropy spectrum of Birkhoff averages. - Janina Kotus,
*Invariant Measures for Meromorphic Misiurewicz Maps*

**Abstract.**We study the existence of finite absolutely continuous invariant measures for meromorphic nonrecurrent maps whose Julia set is the whole sphere. In the rational context, these hypotheses imply that such a measure must exist. We show that it is not so for meromorphic maps unless an additional condition on the behavior of the map, which can be stated in terms of its Nevanlinna characteristic, is satisfied.

(joint work with G. Swiatek) - Genadi Levin,
*Cantor attractors for critical circle covers*

**Abstract.**I'll describe some tools in the proof of the existence of such attractors (joint with G. Swiatek). - Daniel Meyer,
*Thurston Maps, subdivisions, and conformal dimension*

**Abstract.**We consider postcritically finite branched coverings of the sphere, that are expanding. There exists an invariant curve, so there is a natural Markov partition for the dynamics. This eqips the sphere in a natural way with a metric`d`. A theorem of Thurston gives a condition when such a map is equivalent to a rational one. In this case the sphere with metric`d`is quasisymmetric to the standard one. We prove a generalization of Thurstons theorem which gives a precise bound for the dimension that can be attained under a qs map. This is joint work with Mario Bonk. - Juan Rivera-Letelier,
*Statistical properties of Topological Collet-Eckmann maps*

**Abstract.**I will show that every Topological Collet-Eckmann (non-uniformly hyperbolic) rational map on the Riemann sphere possesses a unique conformal probability measure of minimal exponent and that this measure is non-atomic ergodic and its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Moreover there is a unique invariant probability measure absolutely continuous with respect to this conformal measure; it has exponential decay of correlations and satisfies Central Limit Theorem (on Hölder functions). The existence of such a measure characterizes the Topological Collet-Eckmann condition.

(joint work with F. Przytycki) - Michael Todd,
*Return time statistics in dynamical systems*

**Abstract.**This is joint work with Henk Bruin. One way of finding the return time statistics is to induce. A result of Bruin, Saussol, Troubetzkoy and Vaienti shows that return time statistics to balls/cylinders are often the same as those for a first return map. This is useful since the statistics of the first return map is often, though not always, well understood. Our method extends this idea to inducing schemes. One result of this is that smooth interval maps (I,f) such that almost every point has positive Lyapunov Exponent have exponential return time statistics to balls with respect to an acip. - Michel Zinsmeister,
*Rectifiability and Teichmüller theory*