We will discuss Anosov-Katok construction that allows to build smooth realizations of some ergodic systems. Different realizations of this construction will be considered.

We consider three constitutive relations to describe tumor growth: Darcys law, Stokes law, and the combined Darcy-Stokes law. Darcys law is used to describe fluid flow in a porous medium. Stokes law describes the flow of a viscous fluid. In this talk, we will discuss using linear theories to study tumor shape stability (the ability of the tumor to return to being spherical or exhibit protrusions) described by the three physical relations and to evaluate the consistency between theoretical model predictions and experimental data. The motivation behind this work is that shape instabilities (growing protrusions) are associated with local invasiveness, which is often a precursor to tumor metastasis (infiltration of the distant organs). We will discuss the results and further show that it is feasible to extract parameter values from a limited set of data and create a self-consistent modeling framework that can be extended to the multiscale study of cancer. Numerical methods are used to simulate the nonlinear effects of stress on solid tumor growth and invasiveness.