Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data"

In this note we correct minor errors in [2, 4] that are due to a mistake in [2, Lemma 1.2] that in turn is based on an uncritical quotation of [3, Theorem 2.6.1] that contains an error in the uniqueness statement. Lemma 1.2 from [2] is incorrect as stated (although it is true for a restriction of the action α to a subgroup of finite index) and should be replaced by Lemma 1 below. All results of [2, 4] are correct with h being a semi-conjugacy between the action α and an affine action α0 with the same homotopy data as α. In [4, Corollary 2.4] “linear models” should be replaced with “affine models”. Proofs are not affected. Let us explain the nature of the error first. Let L be an integer n×n matrix with determinant ±1 and no eigenvalues of absolute value one. It determines an automorphism FL of the torus T n that is an Anosov diffeomorphism. Let f be a diffeomorphism of T homotopic to FL. Then there exists a continuous map h : T → T homotopic to the identity such that