Georgia Institute of Technology College of Sciences

We give a comprehensive parameter study of the three-dimensional quadratic diffeomorphism to understand its attracting and chaotic dynamics. For large parameter values, we use a concept introduced 30 years ago for the Frenkel--Kontorova model of condensed matter physics: the anti-integrable (AI) limit. At the traditional AI limit, orbits of a map degenerate to sequences of symbols and the dynamics is reduced to the shift operator, a pure form of chaos. For the 3D quadratic map, the AI limit that we study becomes a pair of one-dimensional maps, introducing symbolic dynamics on two symbols. Using contraction arguments, we find parameter domains such that each symbol sequence corresponds to a unique AI state. In some of these domains, sufficient conditions are then found for each such AI state to continue away from the limit becoming an orbit of the original 3D map. Numerical continuation methods extend these results, allowing computation of bifurcations, and allowing us to obtain orbits with horseshoe-like structures and intriguing self-similarity.

For small parameter values, we focus on the dissipative, orientation preserving case to study the codimension-one and two bifurcations. Periodic orbits, born at resonant, Neimark-Sacker bifurcations, give rise to Arnold tongues in parameter space. Aperiodic attractors include invariant circles and chaotic orbits; these are distinguished by rotation number and Lyapunov exponents. Chaotic orbits include Hénon-like and Lorenz-like attractors, which can arise from period-doubling cascades, and those born from the destruction of invariant circles. The latter lie on paraboloids near the local unstable manifold of a fixed point.

Lastly, we present a generalized proof for the existence of AI states using similar contraction arguments to find larger parameter domains for the one-to-one correspondence of symbol sequences and AI states. We apply numerical continuation to these results to determine the persistence of low-period and heteroclinic AI states to the full, deterministic 3D map for a volume-contracting case. We find the corresponding AI state of a chaotic attractor and continue this state towards the full map. The numerical results show that the AI states continue to resonant and chaotic attractors along a 3D folded horseshoe that is similar to the classical 2D Hénon attractor.

We say that a Riemannian manifold has good higher expansion if every rationally null-homologous i-cycle bounds an i+1 chain of comparatively small volume. The interactions between expansion, spectral geometry, and topology have long been studied in the settings of graphs and surfaces. In this talk, I will explain how to construct rational homology 3-spheres which are good higher expanders. On the other hand, I will show that such higher expanders must be rather topologically complicated; in particular, we will demonstrate a super-polynomial-in-volume lower bound on their torsion homology.

In the 1990s, Arkadi Nemirovski asked the question: 

How hard is it to estimate a solution to an unknown homogeneous linear difference equation with constant coefficients of order S, observed in the Gaussian noise on [0,N]?

The class of all such solutions, or "signals," is parametric -- described by 2S complex parameters -- but extremely rich: it contains all exponential polynomials over C with total degree S, including harmonic oscillations with S arbitrary frequencies. Geometrically, this class corresponds to the projection onto C^n of the union of all shift-invariant subspaces of C^Z of dimension S. We show that the statistical complexity of this class, quantified by the squared minimax radius of the (1-P)-confidence Euclidean norm ball, is nearly the same as for the class of S-sparse signals, namely (S log(N) + log(1/P)) log^2(S) log(N/S) up to a constant factor. Moreover, the corresponding near-minimax estimator is tractable, and it can be used to build a test statistic with a near-minimax detection threshold in the associated detection problem. These statistical results rest upon an approximation-theoretic one: we show that finite-dimensional shift-invariant subspaces admit compactly supported reproducing kernels whose Fourier spectra have nearly the smallest possible p-norms, for all p ≥ 1 at once. 

The talk is based on the recent preprint https://arxiv.org/pdf/2411.03383.

Dynamical systems exhibiting some degree of hyperbolicity often admit “fractal" invariant objects.  However, extra symmetries or “randomness” in the system often preclude the existence of such fractal objects.

I will give some concrete examples of the above and then discuss problems and results related to random dynamics and group actions on surfaces.  I will especially focus on questions related to absolute continuity of stationary measures.