Multifractal rigidity for piecewise linear Markov maps

We give an answer to the multifractal rigidity problem presented by Barreira, Pesin and Schmeling for the dimension spectra of Markov measures on the repellers of piecewise linear Markov maps with two branches. Thermodynamic formalism provides us with a one-parameter family of measures. Zero-temperature limit measures of this family and the concept of nondegeneracy of spectra play important roles.