K-rational approximations in k-Luroth expansions
Ergodic Theory and Dynamical Systems Seminar
16th April 2020, 2:00 pm – 3:00 pm
Online seminar at http://www.bluejeans.com/287667352,
Given an irrational number, one can think about the speed of approximation by rationals. The speed of approximation depends on the expansion that we expand the numbers. Examples of typical expansions are the numerical systems generated by continued fractions, as well as base b expansions. In this talk, we are interested in a one parameter family of numerical systems called k Lurton expansions, each one generated by an interval map L_k. Each k-Lurton expansion allows us to consider k-rationals and irrational numbers in [0,1]. In particular, we are interested in the size of numbers in [0,1] having the same exponential speed of approximation by k-rationals, for different values of k. With this in mind we show that the Hausdorff dimension of these sets varies analytically with the parameter k by using tools from thermodynamic formalism for countable Markov shifts.