What is ergodic theory?
Ergodic theory examines the statistical behavior of dynamical systems. Many classical examples of dynamical systems, such as Newton’s Three-Body Problem and the Lorenz model of atmospheric convection, are of such high complexity that their long-term behavior cannot be predicted from knowledge of their initial data. This could be either because small errors in the initial data measurement compound into large errors over time (a phenomenon known as sensitive dependence on initial conditions), or because the equations whose solutions give the trajectories of a dynamical system are too complicated to be solved explicitly. (In particular, both the Three-Body Problem and the Lorenz attractor exhibit both of these phenomena.) However, despite the “chaotic” nature of such dynamical systems, the statistical behavior of different solutions is often highly predictable. Using this framework, ergodic theory examines the statistical behavior of otherwise “unpredictable” dynamical systems.
My work specifically is in smooth ergodic theory, nonuniform hyperbolicity, and thermodynamic formalism. These fields use techniques from statistical mechanics to investigate the ergodic properties of dynamical systems in subsets of smooth manifolds. Often the additional smooth structure of differentiable maps on manifolds leads to new and interesting statistical objects (such as invariant probability measures on hyperbolic attractors, and probability measures that maximize the metric entropy of a system). My research has been in the thermodynamics of nonuniformly and singular hyperbolic dynamical systems, such as almost- and pseudo-Anosov diffeomorphisms and singular hyperbolic attractors.
Smooth ergodic theory and hyperbolic dynamics:
D. Veconi. (2022). SRB measures of singular hyperbolic attractors. Discrete and Continuous Dynamical Systems, 42(7), 3415-3430.
D. Veconi. (2022). Thermodynamics of pseudo-Anosov diffeomorphisms. Ergodic Theory and Dynamical Systems, 42(3), 1284-1326.
D. Veconi. (2020). Equilibrium states of almost Anosov diffeomorphisms. Discrete and Continuous Dynamical Systems, 40(2), 767-780.
K. Ahrendt, L. DeWolf, L. Masurowski, K. Mitchell, T. Rolling, D. Veconi. (2016). Initial and boundary value problems for the Caputo fractional self-adjoint difference equation. Enlightenment of Pure and Applied Mathematics, 2(1).