Georgia Institute of Technology College of Sciences

We introduce the notion of TDFOE studying the thermodynamic formalism on topological Markov shifts where the potential is random, depending on a Gibbs random walk on a compact metric space.

We introduce the associated Pressure Out of Equilibrium, and discuss its properties (e.g. variational principle), and applications (Azuma inequality for potentials expanding on average w.r.t. to a Gibbs measure).

We discuss the associated Semi-Ruelle operator which generates the POE, and its properties (e.g. existence of conformal measures, spectral radius). We provide an application by studying Effectively Expanding on Average diffeomorphisms on closed manifolds (this is a $C^1$-open condition, extends previously studied conditions).

We prove quasi-compactness (and later a spectral gap) for the Averaged Semi-Ruelle operator on a Sobolev space.

Notably, the diffeos are allowed to be highly dissipative. Finally, we mention a few applications such as: Dimension bounds on the stationary measure, and CLT and LLT for the (not necessarily invariant) volume measure.

The classical Livsic theorem says that a H\"older cocycle over a transitive Anosov diffeomorphism/flow is a coboundary if and only if it satisfies the periodic obstruction on all periodic orbits. It is natural to ask whether satisfying the periodic obstruction for all closed orbits of period at most $T$ is enough to conclude that the cocycle is, in some quantitative sense, close to being a coboundary. We show that for transitive Anosov flows, this is indeed enough to find an approximate solution to the cohomological equation with error decaying exponentially in $T$, improving on the polynomial rates obtained first by S. Katok for contact flows in dimension 3, and then later Gouëzel and Lefeuvre in higher dimensions. This is joint work with Jonathan DeWitt, Spencer Durham, and James Marshall Reber.

In dynamics, the speed of mixing depends on the dynamical features of the map and the regularity of the observables. Notably, two classical linear models—the Bernoulli doubling map and the CAT map—exhibit double exponential mixing for analytic observables. Are ergodic linear maps the only ones with this property? In dimension one, we provide a full classification for maps from the space of volume-preserving finite Blaschke products acting on the circle (as well as for free semigroup actions generated by a finite collection of such maps). In higher dimensions, we identify a necessary condition for double exponential mixing and present several families of examples and non-examples. Key ideas of the proof involve the Koopman precomposition operator on spaces of hyperfunctions (elements of the dual space of analytic functions), which turns out to be non-self-adjoint, compact, and quasi-nilpotent, with spectrum reduced to zero.

https://gatech.zoom.us/j/97077908574?pwd=bnP9YWZwqKsU5YrgFZR40asqub0GOR.1

Meeting ID: 970 7790 8574

Passcode: 604975

Cislunar space—the region between Earth and the Moon—has reemerged as a critical area for space exploration. From a mathematical perspective, this region is governed by multi-body dynamics that give rise to rich structures, including invariant manifolds, resonant orbits, and homoclinic chaos. This talk will introduce classical and modern tools from celestial mechanics to analyze motion in the Earth–Moon system, with an emphasis on restricted 3- and 4-body problems. We will discuss how perturbative methods (normal forms) and invariant manifold theory (parameterization method) reveal the underlying organization of the phase space. Particular attention will be placed on connecting the perturbative regime, where classical methods apply, with the realistic system, which often lies far outside that regime, using computer-assisted techniques. Our ultimate goal is to establish rigorous results for the real solar system while enhancing the engineering capabilities needed to design and fly missions, highlighting how mathematics contributes both to theory and to the practical challenges of contemporary space exploration.

No prior knowledge is needed; the talk will be self-contained and accessible. Undergraduates are encouraged to attend.