Finite and infinite ergodic theory for linear and conformal dynamical systems

Ergodic theory;Linear operators;Conformal mapping

Munday, Sara ; Stratmann, Bernd ; Stratmann, Bernd

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Ergodic theory;    Linear operators;    Conformal mapping;   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/3220/SaraMundayPhDThesis.pdf?sequence=3&isAllowed=y

来源: DR-NTU

【 摘 要 】

The first main topic of this thesis is the thorough analysis of two families of piecewise linearmaps on the unit interval, the α-Lüroth and α-Farey maps. Here, α denotes a countably infinitepartition of the unit interval whose atoms only accumulate at the origin. The basic propertiesof these maps will be developed, including that each α-Lüroth map (denoted Lα) gives rise to aseries expansion of real numbers in [0,1], a certain type of Generalised Lüroth Series. The firstexample of such an expansion was given by Lüroth. The map Lα is the jump transformationof the corresponding α-Farey map Fα. The maps Lα and Fα share the same relationship as theclassical Farey and Gauss maps which give rise to the continued fraction expansion of a realnumber. We also consider the topological properties of Fα and some Diophantine-type sets ofnumbers expressed in terms of the α-Lüroth expansion.Next we investigate certain ergodic-theoretic properties of the maps Lα and Fα. It will turnout that the Lebesgue measure λ is invariant for every map Lα and that there exists a uniqueLebesgue-absolutely continuousinvariant measure for Fα. We will give a precise expression forthe density of this measure. Our main result is that both Lα and Fα are exact, and thus ergodic.The interest in the invariant measure for Fα lies in the fact that under a particular condition onthe underlying partition α, the invariant measure associated to the map Fα is infinite.Then we proceed to introduce and examine the sequence of α-sum-level sets arising fromthe α-Lüroth map, for an arbitrary given partition α. These sets can be written dynamically interms of Fα. The main result concerning the α-sum-level sets is to establish weak and strongrenewal laws. Note that for the Farey map and the Gauss map, the analogue of this result hasbeen obtained by Kesseböhmer and Stratmann. There the results were derived by using advancedinfinite ergodic theory, rather than the strong renewal theorems employed here. This underlinesthe fact that one of the main ingredients of infinite ergodic theory is provided by some delicateestimates in renewal theory.Our final main result concerning the α-Lüroth and α-Farey systems is to provide a fractal-geometricdescription of the Lyapunov spectra associated with each of the maps Lα and Fα.The Lyapunov spectra for the Farey map and the Gauss map have been investigated in detail byKesseböhmer and Stratmann. The Farey map and the Gauss map are non-linear, whereas thesystems we consider are always piecewise linear. However, since our analysis is based on a largefamily of different partitions of U , the class of maps which we consider in this paper allows usto detect a variety of interesting new phenomena, including that of phase transitions.Finally, we come to the conformal systems of the title. These are the limit sets of discretesubgroups of the group of isometries of the hyperbolic plane. For these so-called Fuchsiangroups, our first main result is to establish the Hausdorff dimension of some Diophantine-typesets contained in the limit set that are similar to those considered for the maps Lα. These setsare then used in our second main result to analyse the more geometrically defined strict-Jarníklimit set of a Fuchsian group. Finally, we obtain a “weak multifractal spectrum” for the Pattersonmeasure associated to the Fuchsian group.