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Invited Talks

Generalized u-Gibbs Measures for $ C^{\infty} $ Diffeomorphism
Snir Ben Ovadia (Hebrew University of Jerusalem)
We show that for every $ C^{\infty} $ diffeomorphism of a closed Riemannian manifold, if there exists a positive volume set of points with a positive Lyapunov exponent (in a weak sense) then there exists an invariant probability measure with a disintegration by absolutely continuous conditionals on disks subordinated to unstable leaves. As an application, we solve the strong Viana conjecture in any dimension. This is joint work with D. Burguet.
Regularity and Lyapunov Rigidity of Stationary Measures
Aaron Brown (Northwestern University)
For non-linear random walks on the 2-torus, we study the regularity of stationary measures. Perturbing the generators of certain (Zariski dense) affine random walks, we show the only (ergodic) non-atomic stationary measure is absolutely continuous. In particular, there are no stationary measures of fractal dimension. One can study the top Lyapunov exponent relative to the absolutely continuous stationary measure. Entropy considerations give an inequality between the top Lyapunov exponent for the non-linear and affine random walks. For random walks satisfying a cone condition, equality holds precisely when non-linear random walk is smoothly conjugate to an affine random walk. This is joint with Homin Lee, Davi Obata, and Yuping Ruan and with Yi Shi.
Heat Equation from a Deterministic Dynamics
Giovanni Canestrari (University of Toronto)
I will describe a derivation of the heat equation in the thermodynamic limit, with a diffusive scaling, from a purely deterministic dynamics satisfying Newton's equations under a time-dependent external field. (Work in collaboration with C. Liverani and S. Olla).
Kinetic localization via Poincar´e-type inequalities and applications to the condensation of Bose gases
Jacky Chong (Beijing Institute of Technology)
This talk will review recent mathematical advances in the theory of Bose gases and condensation. We will present a simplified localization method, based on a Poincar´e-type inequality, that provides a new derivation of Bose–Einstein condensation for dilute Bose gases beyond the Gross–Pitaevskii scaling regime. This work is joint with Hao Liang and Phan Th`anh Nam (arXiv:2510.20493).
On the Stability and Instability of Elliptic Equilibria and Quasi-Periodic Invariant Tori of Real Analytic Hamiltonians
Bassam Fayad (University of Maryland)
Any real analytic Hamiltonian with five or more degrees of freedom which possesses an elliptic equilibrium with frequencies not all of the same sign (or a quasi-periodic torus) that is locally integrable can be perturbed in the real analytic category (within fixed bands) so that the equilibrium becomes unstable in the sense of Lyapunov. This is a joint work with Jaime Paradela Diaz, Maria Saprykina, and Tere Sera.
Ergodicity of Asymmetric Lemon Billiards
Boris Hasselblatt (Tufts University)
Lemon billiards consist of the intersection of two circles of which one contains the centers of both. These do not satisfy the usual criteria for hyperbolicity but have been known to be hyperbolic. Wentao Fan established that the hyperbolicity (of the return to a suitable section) is uniform if the larger radius is large enough, and that known conditions for (first local, then global) ergodicity and indeed mixing properties are satisfied. This includes control of singularity curves which in this case do not seem to fit the usual requirements: Fan obtains alignment, regularity, contraction control and the Sinai-Chernov Ansatz conditions.
Derivatives of the Volume Dimension for Polynomial Skew Products
Yan Mary He (University of Oklahoma)
Around 1980, David Ruelle considered an asymptotic expansion of the Hausdorff dimension of Julia sets of the quadratic family $ f_c(z) = z^2+c $ near $ c=0 $. In particular, Ruelle’s results states that $ H.dimJ_c = 1 + |c|^2/(2\log 2) + O(|c|^3) $ as $|c| \to 0 $. In this talk, we consider an analogue of this question for the family $ f_t(z,w)=(z^d,w^d+t(b_2(x)w^{d-2} + … + b_d(x))) $ of polynomial skew products of $CP^2$ near $t=0$. As holomorphic dynamical systems in higher dimensions are non-conformal, we replace Hausdorff dimension by volume dimension (that we introduced earlier) and consider its asymptotic expansion. This is joint work with Fabrizio Bianchi.
Spectral properties of Schrödinger operators with quasi-periodic potential and Aubry-Mather theory
Konstantin Khanin (BIMSA/University of Toronto)
Schrödinger operators with quasi-periodic potentials were intensively studied in the last few decades. Their spectral properties depend on the value of the coefficient in front of the potential, so called. coupling constant. For small values of the coupling constant the spectrum is absolutely continuous, while for large coupling constants the spectrum is pure point. Natural families of Schrödinger operators with quasi-periodic potentials appear in the context of the Aubry-Mather theory. In this setting there are no coupling constants. Instead operators depend on the nonlinearity parameter for related area-preserving maps. We shall discuss the transition from the absolutely continuous to the pure point spectrum for such families of Schrödinger operators. The talk is based on a joint work with Artur Avila and Martin Leguil.
Rapid Mixing for Random Walks on Nilmanifolds.
Minsung Kim (KTH)
In chaotic systems, the mixing property is known for the fast decay of correlation. It is called rapid mixing if the correlation function decays super-polynomially. The mixing mechanism for hyperbolic systems and its compact group extensions were studied by Dolgopyat in a series of his papers in the late 90s'. In this talk, we prove rapid mixing for almost all random walks generated by $m\geq 2$ translations on an arbitrary nilmanifold. For several classical classes of nilmanifolds, we show m = 2 suffices. This provides a partial answer to the question raised in Dolgopyat ('02) about the prevalence of rapid mixing for random walks on homogeneous spaces. This is joint work with Dmitry Dolgopyat and Spencer Durham.
Counting, Equidistribution, and Geometry of Periodic Diagonal Orbits
Jialun Li (Fudan University)
I will begin with the classical setting of the modular surface: the unit tangent bundle of the modular curve $ T^1(SL(2,Z)\backslash H)=SL(2,Z)\backslash SL(2,R) $, where periodic geodesics correspond to periodic orbits of the diagonal subgroup. Margulis and Bowen established dynamical frameworks for studying these orbits, while Selberg and Sarnak developed spectral and number-theoretic approaches. We then turn to the higher-rank generalization, periodic diagonal orbits on $SL(3,Z)\backslash SL(3,R)$. In this case, each periodic diagonal orbit is a two-dimensional flat torus embedded in the space. Different counting problems emerge, depending on how one orders these tori--whether by dynamical ordering, geometric ordering or arithmetic ordering. The dynamical and geometric ordering are closely connected through the shapes of the flat tori. Finally, I will discuss my recent joint work with Thi Dang and Nihar Gargava on the density of shapes of periodic tori.
Ergodic Solutions for 1D Stochastic Burgers Equation with Space-time Homogeneous Noise
Liying Li (SUSTech)
We study the stochastic Burgers equation on the real line, forced by a white-in-time, color-in-space homogeneous noise, with either positive or zero viscosity. Under mild assumptions on the forcing, we establish that for every fixed average velocity, a unique ergodic solution exists and the One Force --- One Solution principle holds. Moreover, in the inviscid limit the positive viscosity ergodic solutions will converge to the zero-viscosity ones. The analysis utilizes the connection between the Burgers equation and the associated variational problems and direct polymers models, where the ergodic solution problem can be reformulated in terms of the existence of the infinite-volume and zero-temperature limits of the polymer measures. Finally, we will discuss the generalizations and challenges in higher dimensions and for general non-quadratic Hamiltonians, as well as the connection to KPZ universality.
A weak entropy rigidity condition on surfaces
Jana Rodriguez Hertz (SUSTech)
Let $f$ be a $C^{\infty}$ diffeomorphism of a surface $S$ preserving $m$. Let $g$ be a pseudo-Anosov diffeomorphism on $S$ preserving $m$ such that $(g,m)$ is a factor of $(f,m)$ via a continuous surjective map $\pi$. If $h_m(f)=h_m(g), $ then $\pi$ is injective $m$-almost everywhere. In particular, it is non-uniformly hyperbolic and Bernoulli. Moreover, $\pi$ is differentiable in the Whitney sense $m$-almost everywhere on $S$.
Anderson localization for the discrete nonlinear Schrodinger equation
Yunfeng Shi (Sichuan University)
In this talk, we will introduce recent progress on Anderson localization for the discrete nonlinear Schrodinger equation with random and quasi-periodic potentials.
On the Universality Conjecture for Hamiltonian Systems with 2 Degrees of Freedom.
Dmitry Turaev (Imperial College London)
We discuss the conjecture claiming that every Hamiltonian system, which is not integrable, not uniformly partially hyperbolic, nor energy homogeneous, is universal — it accurately represents all symplectic dynamics with the corresponding number of degrees of freedom.
Large Entropy of Non-hyperbolic Measures
Raul Ures (SUSTech)
For a partially hyperbolic diffeomorphism with 1-dimensional center, we introduce a counting function, which counts the disjoint preimages of unit disks on every unit unstable disk where the dynamics in the center direction are expanding or contracting. This function is semi-lower continuous with respect to the diffeomorphism, and is a lower bound of the supremum of the metric entropy of non-hyperbolic measures $ I(f) $. As an application, we can estimate a lower bound of the function $ I(f) $ for a $ C^{1} $ open and dense subset of $C^2$ volume preserving partially hyperbolic diffeomorphisms with 1-dimensional center, and we are going to show that for a large family of $ C^{1} $ partially hyperbolic diffeomorphisms, this lower bound is indeed sharp by considering the iterations. Several partially hyperbolic diffeomorphisms are studied: partially hyperbolic diffeomorphism with circle bundle; perturbation of the time-one map of hyperbolic geodesic flows; some kind of derived from Anosov diffeomorphisms. A similar argument also shows that for a family of $SL(2, \mathbb{R})$ random matrices $\begin{pmatrix} 2 & 0 \\ 0 & 1/2 \end{pmatrix}$ and $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$, this estimation is sharp for $ \theta \ne 0 $ sufficiently small, and in particular, $ I\ge 4\theta \log 2/\pi $. This talk is based on joint work with Lorenzo Díaz, Xiaodong Wang, and Jiagang Yang.
Kac's Program for the Landau Equation
Zhenfu Wang (Peking University)
We study the derivation of the spatially homogeneous Landau equation from the mean-field limit of a conservative N-particle system, obtained by passing to the grazing limit on Kac's walk in his program for the Boltzmann equation. Our result covers the full range of interaction potentials, including the physically important Coulomb case. This provides the first resolution of propagation of chaos for a many-particle system approximating the Landau equation with Coulomb interactions, and the first extension of Kac's program to the Landau equation in the soft potential regime. The convergence is established in weak, Wasserstein, and entropic senses,together with strong L1 convergence. To handle the singularity of soft potentials, we extend the duality approach of Bresch-Duerinckx-Jabin and establish key functional inequalities, including an extended commutator estimate and a new second-order Fisher information estimate. Based on a joint work with Xuanrui Feng (PKU).
Marked length spectrum rigidity for generic analytic Birkhoff billiards
Ke Zhang (University of Toronto)
In either billiards for Riemannian manifolds, the marked length spectrum rigidity asks whether the lengths of all periodic orbits marked by their homotopy type is enough to determine the geometry. We will show that for generic analytic billiards, having the same marked length spectrum implies the billiard maps are symplectically analytically conjugate on the neighborhood of a Cantor hyperbolic set. In the special case that the billiard are $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetric, then the billiards are isometric. This is based on a joint work with V. Kaloshin and M. Leguil.
Perturbation of the Time-1 Map of a Generic Volume-preserving 3-dimensional Anosov Flow
Zhiyuan Zhang (Imperial College London)
In a work-in-progress with Masato Tsujii, we show that any $C^r$ diffeomorphism that is sufficiently close to the time-1 map of a $C^r$ generic volume-preserving 3D Anosov flow (for some large $r$) is topologically mixing and has a unique Gibbs state.

Short Talks

The chaotic billiards beyond defocusing mechanism, uniform hyperbolicity, ergodic and mixing criterions
Wentao Fan (Tufts University)
In 2016, Bunimovich, Zhang, Zhang pioneered the study of Asymmetric Lemon Billiards' hyperbolicity. The current talk will come across the earlier results by Bunimovich, Hong-kun Zhang, Xin Jin and Pengfei Zhang. In their found/proved Wojtkowski invariant cone and conclusions, with the tools by Chernov and Markarian, it will be shown there is a uniform expansion for the Lemon Billiard section return map. And given the uniform expansion and new conclusions, the return map singularity curves satisfy the conditions of local ergodicity. Based on the previous local ergodic theorem by Sinai-Chernov, Liverani-Wojtkowski and Del Magno-Markarian, we further obtain the ergodicity of the lemon billiard map. This is a joint work with Boris Hasselblatt.
Linear response of Bernoulli Convolution
Jianning Fu (University of Maryland)
Let $\mu_{\lambda}$ be the Bernoulli convolution measure with parameter $\lambda\in(0,1)$. We prove that $h=h_{\phi}:\lambda\mapsto \int_{\mathbb{R}}\phi(x)\,d\mu_{\lambda}(x)$ is differentiable almost every $\lambda \in (2^{-\frac{1}{3}}+\epsilon,1)$ and for H$\ddot{o}$lder continuous observable $\phi$ with sufficiently large H$\ddot{o}$lder exponent. In contrast, we also show if $\dim \mu_{\lambda}<1$ then there exists H$\ddot{o}$lder observable with H$\ddot{o}$lder exponent arbitrarily close to $1$, so that $h_\phi$ is not differentiable at $\lambda$.
Elliptic islands and zero measure escaping orbit in a class of outer billiards
Zaicun Li (University of Maryland)
We study outer billiard systems around a class of circular sectors. For semi-discs, we prove the existence of elliptic islands occupying a positive proportion of the plane. Combined with known results, this shows the coexistence of stability and diffusion for this system. On the other hand, we show that there exists a countable family of circular sectors for which the outer billiard system has zero measure of escaping orbits.
On $r$-Neutralized Entropy
Qiujie Qiao (Nankai University)
In this talk, we will discuss $r$-neutralized local entropy and $r$-neutralized entropy. We first review some background on measures of maximal entropy, measures of maximal Hausdorff dimension, and neutralized entropy. We obtain formulas for the $r$neutralized local entropy in smooth systems. In particular, for certain hyperbolic systems, we prove the existence of ergodic measures that maximize the $r$-neutralized entropy. This is based on joint work with Changguang Dong.
Ledrappier-Young entropy formula for $C^1$ diffeomorphisms with dominated splitting
Yao Tong (Peking University)
We partially extend the Ledrappier-Young entropy formula to invariant measures of $C^1$ diffeomorphisms with dominated splitting. We prove that for any ergodic measure with dominated splitting, if the i-th Lyapunov exponent has multiplicity one, then the i-th transverse entropy is equal to the product of the i-th Lyapunov exponent and transverse measure dimension. Moreover, if we further assume that every intermediate non-negative Lyapunov exponents have multiplicity one, we can establish the full Ledrappier-Young entropy formula. This relaxes the classical $C^2$ regularity assumption to $C^1$ in the remarkable work by Ledrappier and Young under these one-dimensional assumptions. As a consequence, the Avila–Viana invariance principle holds when the center is one-dimensional. We also derive $C^1$ versions of numerous fundamental results in measure dimension theory, including the famous works by Ledrappier-Young, Barreira-Pesin- Schmeling, and Ledrappier-Xie. This is joint work with Shaobo Gan and Jiagang Yang.