On Dynamics Of Infinitely Generated Contractive Mapping Systems On PZ_p And P-adic Continued Fractions

The p-adic dynamical systems have been studies deeply by many mathematicians and many results have been obtained.One of the topics in p-adic dynamical systems is the study of the structure of the limit set in the non-Archimedean setting.The limit sets in European spaces have been studied by mathematicians deeply.Naturally,the study of the limit sets in non-Archimedean spaces has arose the interest of mathematicians.The p-adic continued fractions are also hot topics recently.The first part of this thesis focuses on the limit set of semigroup G which is generated by infinitely many contractive maps on pZp.In 2007,Aihua Fan,Lingmin Liao,Yuefei Wang and Dan Zhou studied p-adic repellers in Qp.In 2014,Weiyuan Qiu,Yuefei Wang,Jinghua Yang and Yongchen Yin studied the limit set of a semigroup which is generated by finitely many contractive analytic functions on the closed unit ball of an algebraically closed field which is complete with respect to the non-trival non-Archimedean absolute value,and they mainly studied the metric properties of it.In 2016,Farrukh Mukhamedov and Obatek Khakimov studied the unconventional limit set of the semigroup finitely generated by contractive functions on the closed unit ball of the non-Archimedean Banach algebra,and they mainly studied the dynamics of the limit sets.We study the limit set of the semigroup G which is generated by infinitely many contractive maps on pZp.It is shown that there exists a shift transformation on the limit set of G and the shift transformation is ergodic with respect to the Haar measure on p2p,if G satisfies the open tiling conditions.The second part is concerned with the p-adic continued fractions.Khinchin studied the real continued fractions and constructed the metric theory of it.In 1970,Ruban defined Ruban's continued fraction and generalized the metric properties of continued fractions to Qp.In 1970,Schneider defined Schneider's p-adic continued fraction.In 2011,Jordan Hirsh and Lawrence C.Washington published an paper about Schneider's p-adic continued fraction and got p-adic Khinchin's theorem.In 2013,Jaroslav Hancl,A.Jassova,Poj Lertchoosakul and Rajit Nair studied the metric properties of p-adic continued fractions and generalize p-adic Khinchin's theorem.In 2017,Hui Hu,Yueli Yu and Yanfen Zhao published a paper concerning the metric properties of an(x)in Schneider's p-adic continued fraction(see the formula 1.2).They proved that an(x)is independent and identically distributed with respect to Haar measure on pZp.Based on results of the first part of the thesis,we generalize p-adic Khinchin's theorem and p-adic Lochs' theorem to any semigroup satisfying the open tiling conditions in the p-adic settings.In Schneider's p-adic continued fraction and Ruban's continued fraction,we get p-adic Lochs theorems.The third part deals with the dynamics of affine map on ×Qp In 1975,the first work on affine dynamics on Zp was given by Oselies and Zieschang.In 2006,Aihua Fan,Mingtian Li,Jiayan Yao and Dan Zhou studied the structure of affine p-adic dynamical system on Zp and they found all ergodic components for any such system.In 2011,Aihua Fan and Youssef Fares described the dynamics of an arbitrary affine dynamical system on a local field.We investigate the dynamics of the affine map TA,b(X)=AX+b on Qp ×Qp.When A is topological conjugate to(?)we get a corresponding minimal decomposition.