satisfying $\|M_j\|<1/2$ ([@JordanPollicottSimon2007; @Jordan2011; @Rossi2014]); however, the result does not apply to any concrete case.

Theorem \[thm-1.0\] finally gives a full confirmative answer to the problem of the existence of local dimensions in the context of affine IFSs. It completes the aforementioned previous works on the problem.

Exact dimensionality and Ledrappier-Young type dimension formula play a significant role in many of the recent advances in dimension theory of self-affine sets and measures (see e.g. [@Barany2015; @BaranyHochmanRapaport2017; @BaranyKaenmaki2015; @BKK2018; @BaranyRams2018; @BaranyRamsSimon2016; @BRS2018; @BarralFeng2013; @DasSimmons2017; @FalconerKempton2018; @HochmanSolomyak2017; @MorrisShmerkin2018; @PrzytyckiUrbanski1989; @Rapaport2015]). In the remaining part of this section, we will present some applications of Theorem \[thm-1.0\] along the line of this development. The proof of Theorem \[thm-1.0\] is based on some ideas from the work of Ledrappier and Young [@LedrappierYoung1985]. It also adopts and extends some ideas used in [@FengHu2009; @BaranyKaenmaki2015; @QianXie2008] for the construction of measurable partitions and the density estimates of associated conditional measures. Since our construction of measurable partitions is much different from these works (see Remark \[rem-tt\]), and the IFSs in consideration may be non-invertible and non-contractive, many estimates of conditional measures need to be rebuilt or re-justified. A key part of our arguments is on the estimation of 