## Formalizm termodynamiczny i własności metryczne funkcji meromorficznych

### Agnieszka Badeńska

#### Abstract

The dissertation is divided into two parts. First we consider transcendental meromorphic functions f : C → C of finite order in the hyperbolic case, satisfying a rapid (balanced) derivative growth condition. For this class of functions Mayer and Urbański have constructed probability conformal measures m and invariant measures µ, equivalent to m. Assuming that the Hausdorff dimension of the Julia set is greater than one, we prove the real analyticity of Jacobian of the invariant measure µ. We use this result to prove a measure rigidity theorem for hyperbolic functions of the form f = R ◦ exp, where R is a non-constant rational function, in the case when ∞ is an asymptotic value and f does not belong to an exceptional class. Finally, again for the class of hyperbolic meromorphic functions satisfying a rapid derivative growth condition, we derive some stochastic properties of the measure µ under the condition that on a subset of the Julia set the dynamics of f is conjugated to the shift map over a countable alphabet. We prove that µ is exponentially fast mixing on cylinders and for an appropriate choice of a potential ψ: J(f) → C, the sequence {ψ ◦ f n} of random variables satisfies the almost sure invariance principle. In the second part we investigate a set of Misiurewicz parameters for some classes of transcendetal functions. We prove that this set has the Lebesgue measure zero in C for the exponential family fλ = λ exp, λ ∈ C \ {0}, as well as for two families Wt and Ws of the Weierstrass elliptic functions, generated by triangle and square lattices.Diploma type | Doctor of Philosophy | ||||

Author |
Agnieszka Badeńska (FMIS / DODE)
Agnieszka Badeńska
| ||||

Title in Polish | Formalizm termodynamiczny i własności metryczne funkcji meromorficznych | ||||

Language | pl polski | ||||

Certifying Unit | Faculty of Mathematics and Information Science (FMIS) | ||||

Discipline | mathematics / (mathematics domain) / (physical sciences) | ||||

Defense Date | 16-06-2010 | ||||

End date | 24-06-2010 | ||||

Supervisor |
Janina Kotus (FMIS / DFE)
Janina Kotus
| ||||

Internal reviewers |
Grzegorz Świątek (FMIS / DFE)
Grzegorz Świątek
| ||||

External reviewers |
Krzystzof Frączek
Krzystzof Frączek
| ||||

Pages | 122 | ||||

Keywords in English | xxx | ||||

Abstract in English | The dissertation is divided into two parts. First we consider transcendental meromorphic functions f : C → C of finite order in the hyperbolic case, satisfying a rapid (balanced) derivative growth condition. For this class of functions Mayer and Urbański have constructed probability conformal measures m and invariant measures µ, equivalent to m. Assuming that the Hausdorff dimension of the Julia set is greater than one, we prove the real analyticity of Jacobian of the invariant measure µ. We use this result to prove a measure rigidity theorem for hyperbolic functions of the form f = R ◦ exp, where R is a non-constant rational function, in the case when ∞ is an asymptotic value and f does not belong to an exceptional class. Finally, again for the class of hyperbolic meromorphic functions satisfying a rapid derivative growth condition, we derive some stochastic properties of the measure µ under the condition that on a subset of the Julia set the dynamics of f is conjugated to the shift map over a countable alphabet. We prove that µ is exponentially fast mixing on cylinders and for an appropriate choice of a potential ψ: J(f) → C, the sequence {ψ ◦ f n} of random variables satisfies the almost sure invariance principle. In the second part we investigate a set of Misiurewicz parameters for some classes of transcendetal functions. We prove that this set has the Lebesgue measure zero in C for the exponential family fλ = λ exp, λ ∈ C \ {0}, as well as for two families Wt and Ws of the Weierstrass elliptic functions, generated by triangle and square lattices. | ||||

Thesis file |
| ||||

Citation count* | 5 (2020-09-18) |

Back