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马尔科夫谱和拉格朗日谱的分形几何学
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演讲者: Carlos Gustavo Moreira(巴西国家理论数学与应用数学研究所)
时间: 2022年9月22日,16:00 - 17:00
地点: 台州楼240A(杰曼诺夫数学中心报告厅)& 腾讯会议ID 859-140-818

Abstract

We will discuss some recent results on the fractal geometry of the Markov and Lagrange spectra, which are classical objects from the theory of Diophantine approximations, and their set difference. In particular, we discuss recent results in collaboration with Erazo, Gutiérrez-Romo and Romaña, which give precise asymptotic estimates for the fractal dimensions of the Markov and Lagrange spectra near 3 (their smaller accumulation point) and other recent collaborations with Jeffreys, Matheus, Pollicott and Vytnova, in which we prove that the Hausdorff dimension of the complement of the Lagrange spectrum in the Markov spectrum has Hausdorff dimension between 0.594179 and 0.796445.

We will relate these results to symbolic dynamics, continued fractions and to the study of the fractal geometry of arithmetic sums of regular Cantor sets, a subject also important for the study of homoclinic bifurcations in Dynamical Systems.

 

 

Biography

Carlos Gustavo T. de A. Moreira is a Brazilian mathematician working on dynamical systems, ergodic theory, number theory and combinatorics. Moreira is currently a researcher at the National Institute of Pure and Applied Mathematics (IMPA, Brazil). He is member of the Brazilian Academy of Science and of the Third World Academy of Sciences. He was awarded with the UMALCA award (2009) and the TWAS Prize (2010). He was invited speaker at the International Congress of Mathematicians of 2014 and plenary speaker in 2018.