Research 

A dynamical system consists of a set of  states and a map that describes the evolution of the system over time. Such systems arise naturally in a wide range of scientific contexts, including physics, biology, economics, and engineering, and are often used to understand long-term behavior from simple local interactions. A system is said to be chaotic when it exhibits sensitive dependence on initial conditions, namely, when the paths of points (orbits) beginning arbitrarily close to each other diverge exponentially over time, while still being deterministic. A common real-life example of chaotic behavior is the weather: even when governed by well-understood physical laws, a very small change in initial atmospheric conditions can lead to vastly different outcomes, which is why accurate long-term forecasting is notoriously challenging.

Ergodic theory provides tools to study the long-term statistical behavior of dynamical systems. A central concept is that of an invariant measures, which can be understood as a probability distribution that remains unchanged as the system evolves. Instead of tracking individual trajectories, we study how portions of the space are distributed over time. Invariant measures help us quantify the proportion of time an orbit spends in different regions, allowing us to describe complex systems in terms of average behavior rather than tracking every step. This provides insight into systems where individual trajectories may be unpredictable, yet overall statistical behavior remains well-defined.

Thermodynamic formalism extends these ideas by drawing an analogy with statistical mechanics, allowing us to study dynamical systems using concepts such as energy, entropy, and pressure. In this framework, we study particular invariant measures called equilibrium states that balance competing influences, such as maximizing entropy while favoring regions of lower energy (where energy is given by some potential function). Instead of focusing solely on how often trajectories visit different areas, equilibrium states allow us to weight these visits according to additional factors that reflect the dynamics. This enables us to identify the statistically dominant behavior of the system and understand how complex patterns emerge from simple iterative rules. I study when these equilibrium states exist, when they are unique, and how they depend on the underlying dynamics. I use two main methods to study these systems: Symbolic coding and the Climenhaga-Thompson machinery.