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TBA
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In 1961 Ulam proposed a mathematical simplification extracting the essential accelerating mechanism proposed by Fermi, as to explain the cause of high-energy particles in cosmic rays. In this talk, we shall describe the typical behavior of the very model introduced by Ulam in both the classical original form as well as its quantization. In the classical model, we show that typical orbits are recurrent under resonance assumptions. Meanwhile in the quantum model, the acceleration caused by resonance gets much amplified and we point out a direct relationship between the acceleration behavior of the system and the shape of its quasi-energy spectrum. This is a joint work in progress with Changguang Dong and Disheng Xu.
In this talk, we will discuss recent progress in the theory of smooth star flows that contain singularities and consider their expansiveness, continuity of the topological pressure, and the existence and uniqueness of equilibrium states. We will prove an ergodic version of the Spectral Decomposition Conjecture: $C^1$ open and densely, every singular star flow has only finitely many ergodic measures of maximal entropy, and only finitely many ergodic equilibrium states for Holder continuous potentials satisfying a mild yet optimal condition. Joint with M.J. Pacifico and J. Yang.
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Motivated by question in the dynamics of phase fields, we study the Allen-Cahn equation in dimension 2 with white noise initial datum. We prove the appearance of a universal initial condition for mean curvature flow in a small noise scaling. We also obtain a weak coupling limit when the noise is not tuned down: the effective variance that appears can be described as the solution to an ODE. I will discuss ongoing applications in the perturbative study of other critical SPDEs. Joint works with Simon Gabriel, Martin Hairer, Khoa Lê and Nikos Zygouras.
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Since the mid-to-late 70s, a variety of authors turned their attention to understanding the localization behavior of evolution of discrete ergodic Schr\”odinger operators. This study included the notions of Anderson localization as well as more nuanced properties of the Schr\”odinger semi-group (so-called quantum dynamics). A remarkable result of the work on the latter, due to Y. Last [1996], is that the quantum dynamics is tied to the fractal structure of the operator’s spectral measures. This has been used as a suggestive indicator of certain long-time behavior of the quantum dynamics in the absence of localization.
In the early 2000s, D. Damanik, S. Techeremchantsev, and others linked the long-time behavior of the quantum dynamics to properties of the Green's function of the semi-group generator, which is in turn closely related to the base dynamical system.
In this talk, we will discuss the notion of discrepancy and how it is related to ideal properties of the Green's function. In the process, we will present current and ongoing work establishing novel upper bounds for the discrepancy for skew-shift sequences. As an application of our bounds, we improve the quantum dynamical bounds in Han-Jitomirskaya [2019], Jitomirskaya-Powell [2022], Shamis-Sodin [2023], and Liu [2023] for long-range Schr\”odinger operators with skew-shift base dynamics.
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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 will discuss some surprising rigidity phenomena for Anosov flows in dimension 3. For example, in the context of generic transitive 3-dimensional Anosov flows, any continuous conjugacy is either smooth or reverses the positive and negative SRB measures.
This is joint work with Martin Leguil and Federico Rodriguez Hertz
Margulis inequalities and Margulis functions (a.k.a Foster-Lyapunov stability) have played a major role in modern dynamics, in particular in the fields of homogeneous dynamics and Teichmuller dynamics.
Moreover recent exciting developments in the field of random walks over manifolds give rise to related notions and questions in a much larger geometrical content, largely motivated by upcoming work of Brown-Eskin-Filip-Rodriguez Hertz.
I will explain what are Margulis functions and Margulis inequalities and describe the main lemma due to Eskin-Margulis (“uniform expansion”) that allows one to prove such an inequality. I will also try to sketch some interesting applications.
No prior knowledge is needed, the talk will be self-contained and accessible.
In this talk, we will study random dynamical systems of smooth surface diffeomorphisms. Aaron Brown and Federico Rodriguez Hertz showed that, in this setting, hyperbolic stationary measures have the SRB property, except when certain obstructions occur. Here, the SRB property essentially means that the measure is absolutely continuous along certain “nice” curves (unstable manifolds). In this talk, we want to understand conditions that guarantee that SRB stationary measures are absolutely continuous with respect to the Lebesgue measure of the ambient space. Our approach is inspired on Tsujii’s “transversality” method, which he used to show Palis conjecture for partially hyperbolic endomorphisms. This is a joint work with Aaron Brown, Homin Lee and Yuping Ruan.
The arithmetic quantum unique ergodicity (AQUE) conjecture predicts that the L^2 mass of Hecke-Maass cusp forms on an arithmetic hyperbolic manifold becomes equidistributed as the Laplace eigenvalue grows. If the underlying manifold is non-compact, mass could “escape to infinity”. This possibility was ruled out by Soundararajan for arithmetic surfaces, which when combined with celebrated work of Lindenstrauss completed the proof of AQUE for surfaces.
We establish non-escape of mass for Hecke-Maass cusp forms on a congruence quotient of hyperbolic 4-space. Unlike in the setting of hyperbolic 2- or 3-manifolds (for which AQUE has been proved), the number of terms in the Hecke relations is unbounded, which prevents us from naively applying Cauchy-Schwarz. We instead view the isometry group as a group of quaternionic matrices, and rely on non-commutative unique factorization, along with certain structural features of the Hecke action. Joint work with Zvi Shem-Tov.
