University of Isfahan Mathematics and Society 2345-6493 Articles in Press 2026 04 22 Dynamical systems with stable statistical behavior Dynamical systems with stable statistical behavior 30366 10.22108/msci.2026.148131.1790 FA Maisam Hedyehloo School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran. Journal Article 2026 02 16 In this paper, a new class of maps on the two-dimensional torus $\mathbb{T}^2$ is introduced that are not uniformly expanding and have no type of dominated splitting; nevertheless, we show that these maps stably possess a finite number of absolutely continuous invariant probability measures (ACIP) with respect to Lebesgue measure. The proof is based on the concept of "virtually expanding", recently introduced by Tsujii. Our method provides a framework for explicitly constructing these examples. To find new examples, we use a geometric interpretation of the virtually expanding property. This geometric perspective allows us to provide a sufficient condition for an endomorphism to be virtually expanding. The construction of this category of examples is based on a precise perturbation and surgery. These perturbations are supported on a small ball. In fact, by using the bump function technique, we change a linear endomorphism near a fixed point and replace its derivative with a suitably chosen matrix. This perturbation is designed in such a way that it preserves the virtual expanding property but eliminates any dominant splitting. In this paper, a new class of maps on the two-dimensional torus $\mathbb{T}^2$ is introduced that are not uniformly expanding and have no type of dominated splitting; nevertheless, we show that these maps stably possess a finite number of absolutely continuous invariant probability measures (ACIP) with respect to Lebesgue measure. The proof is based on the concept of "virtually expanding", recently introduced by Tsujii. Our method provides a framework for explicitly constructing these examples. To find new examples, we use a geometric interpretation of the virtually expanding property. This geometric perspective allows us to provide a sufficient condition for an endomorphism to be virtually expanding. The construction of this category of examples is based on a precise perturbation and surgery. These perturbations are supported on a small ball. In fact, by using the bump function technique, we change a linear endomorphism near a fixed point and replace its derivative with a suitably chosen matrix. This perturbation is designed in such a way that it preserves the virtual expanding property but eliminates any dominant splitting. https://math-sci.ui.ac.ir/article_30366_100c560d16bcd9469765fad6c4c546c7.pdf