The use of biomass as an energy source has advanced in recent decades, given the scientific evidence that it is a solution to the environmental problems faced globally. In this context, biofuels derived from biomass have a prominent role. Among the countries where this alternative is the most promising, Brazil stands out, just behind the USA. It is, therefore, necessary to assess whether such a replacement is economically viable. For such an assessment, the behavior of the relative price of bioethanol/gasoline is crucial. In the present work, the degree of temporal persistence of relative prices, considering the existence of shocks to which they are exposed, is evaluated, considering 15 important Brazilian capitals, via the detrended fluctuation analysis (DFA). The degree of correlation is also evaluated through the detrended cross-correlation analysis (DCCA) between fuel prices in São Paulo, the capital of the most populous state and main producer of bioethanol, with the capitals of the 14 states selected for the analysis. The period of analysis takes place between 2004 and 2020. The use of DCCA with sliding windows was recently proposed and we also evaluate DFA dynamically in this way, and this, together with an extended sample in the context of Brazilian fuel prices, represents the main innovations of the present work. We found that the degree of persistence varies significantly depending on the capitals analyzed, which means that price variations are localized and demand regional stimulus policies. Furthermore, it was found that the correlation with São Paulo is less intense in the most geographically distant capitals. Such evidence is important and complementary to infer how integrated the national bioethanol market is, in order to support public policies aimed at its consolidation.