ABSTRACT:

This lecture will be primarily based on my book "Open Dynamical Systems: Statistics, Geometry, and Thermodynamic Formalism" co-authored with Tushar Das, Giulio Tiozzo, and Anna Zdunik. The starting dynamical system will be given by any countable alphabet finitely primitive subshift of finite type and the invariant Gibbs state of a 1-cylinder summable Holder continuous potential φ. The holes defining open systems will then be formed by collections of open sets with appropriately constrained thin boundaries. They will naturally lead to singularly perturbed transfer operators associated with the potential φ.

The talk will develop conditionally invariant measures and surviving sets. The surviving variational principle, the existence and uniqueness of surviving equilibrium states, the asymptotic of perturbed leading eignevalues, and escape rates when holes approach a given hole will be discussed. The logarithms of leading perturbed eigenvalues will turn out to be equal to topological pressures defined by means of the surviving variational principle. Also, stochastic properties of surviving equilibria such as exponential decay of correlation, the Central Limit Theorem, the Law of Iterated Logarithm, and the Almost Sure Invariance Principle will be discussed. All these objects will be produced and studied in our dynamical setting by means and applications of the Keller-Liverani Perturbation Theorem.

Applications to real systems will also be discussed. It will be shown that in such a context in Euclidean spaces, virtually all Euclidean balls taken as holes give rise to "controlled" open systems. In particular, these holes need not be dynamically defined, such as unions of cylinders of the same length. For conformal systems, the asymptotic of Hausdorff dimension of surviving sets will also be computed.