Thermodynamic formalism for dispersing billiards

For any finite horizon Sinai billiard map \begin{document}$T$\end{document} on the two-torus, we find \begin{document}$t_*>1$\end{document} such that for each \begin{document}$t\in (0,t_*)$\end{document} there exists a unique equilibrium state \begin{document}$\mu_t$\end{document} for \begin{document}$- t\log J^uT$\end{document}, and \begin{document}$\mu_t$\end{document} is \begin{document}$T$\end{document}-adapted. (In particular, the SRB measure is the unique equilibrium state for \begin{document}$- \log J^uT$\end{document}.) We show that \begin{document}$\mu_t$\end{document} is exponentially mixing for Hölder observables, and the pressure function \begin{document}$P(t) = \sup_\mu \{h_\mu -\int t\log J^uT d \mu\}$\end{document} is analytic on \begin{document}$(0,t_*)$\end{document}. In addition, \begin{document}$P(t)$\end{document} is strictly convex if and only if \begin{document}$\log J^uT$\end{document} is not \begin{document}$\mu_t$\end{document}-a.e. cohomologous to a constant, while, if there exist \begin{document}$t_a\ne t_b$\end{document} with \begin{document}$\mu_{t_a} = \mu_{t_b}$\end{document}, then \begin{document}$P(t)$\end{document} is affine on \begin{document}$(0,t_*)$\end{document}. An additional sparse recurrence condition gives \begin{document}$\lim_{t\downarrow 0} P(t) = P(0)$\end{document}.

more » « less
Award ID(s):
NSF-PAR ID:
10382743
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Journal of Modern Dynamics
Volume:
18
Issue:
0
ISSN:
1930-5311
Page Range / eLocation ID:
415
Format(s):
Medium: X
National Science Foundation
##### More Like this
1. Abstract

Let$$(h_I)$$$\left({h}_{I}\right)$denote the standard Haar system on [0, 1], indexed by$$I\in \mathcal {D}$$$I\in D$, the set of dyadic intervals and$$h_I\otimes h_J$$${h}_{I}\otimes {h}_{J}$denote the tensor product$$(s,t)\mapsto h_I(s) h_J(t)$$$\left(s,t\right)↦{h}_{I}\left(s\right){h}_{J}\left(t\right)$,$$I,J\in \mathcal {D}$$$I,J\in D$. We consider a class of two-parameter function spaces which are completions of the linear span$$\mathcal {V}(\delta ^2)$$$V\left({\delta }^{2}\right)$of$$h_I\otimes h_J$$${h}_{I}\otimes {h}_{J}$,$$I,J\in \mathcal {D}$$$I,J\in D$. This class contains all the spaces of the formX(Y), whereXandYare either the Lebesgue spaces$$L^p[0,1]$$${L}^{p}\left[0,1\right]$or the Hardy spaces$$H^p[0,1]$$${H}^{p}\left[0,1\right]$,$$1\le p < \infty$$$1\le p<\infty$. We say that$$D:X(Y)\rightarrow X(Y)$$$D:X\left(Y\right)\to X\left(Y\right)$is a Haar multiplier if$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$$D\left({h}_{I}\otimes {h}_{J}\right)={d}_{I,J}{h}_{I}\otimes {h}_{J}$, where$$d_{I,J}\in \mathbb {R}$$${d}_{I,J}\in R$, and ask which more elementary operators factor throughD. A decisive role is played by theCapon projection$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$$C:V\left({\delta }^{2}\right)\to V\left({\delta }^{2}\right)$given by$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$$C{h}_{I}\otimes {h}_{J}={h}_{I}\otimes {h}_{J}$if$$|I|\le |J|$$$|I|\le |J|$, and$$\mathcal {C} h_I\otimes h_J = 0$$$C{h}_{I}\otimes {h}_{J}=0$if$$|I| > |J|$$$|I|>|J|$, as our main result highlights: Given any bounded Haar multiplier$$D:X(Y)\rightarrow X(Y)$$$D:X\left(Y\right)\to X\left(Y\right)$, there exist$$\lambda ,\mu \in \mathbb {R}$$$\lambda ,\mu \in R$such that\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$\begin{array}{c}\lambda C+\mu \left(\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}-C\right)\phantom{\rule{0ex}{0ex}}\text{approximately 1-projectionally factors through}\phantom{\rule{0ex}{0ex}}D,\end{array}$i.e., for all$$\eta > 0$$$\eta >0$, there exist bounded operatorsABso thatABis the identity operator$${{\,\textrm{Id}\,}}$$$\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}$,$$\Vert A\Vert \cdot \Vert B\Vert = 1$$$‖A‖·‖B‖=1$and$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta$$$‖\lambda C+\mu \left(\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}-C\right)-ADB‖<\eta$. Additionally, if$$\mathcal {C}$$$C$is unbounded onX(Y), then$$\lambda = \mu$$$\lambda =\mu$and then$${{\,\textrm{Id}\,}}$$$\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}$either factors throughDor$${{\,\textrm{Id}\,}}-D$$$\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}-D$.

more » « less
2. We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control \begin{document}$g$\end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when \begin{document}$g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma)$\end{document}, which, moreover, in the canonical case \begin{document}$\gamma = 0$\end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether \begin{document}$\gamma = 0$\end{document} or \begin{document}$0 \neq \gamma \in L^{\infty}(\Omega)$\end{document}, since \begin{document}$\gamma \neq 0$\end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with \begin{document}$g$\end{document} "smoother" than \begin{document}$L^2(\Sigma)$\end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than \begin{document}$L^2(0, T;L^2(\Gamma))$\end{document}, and [44] for control less regular in space than \begin{document}$L^2(\Gamma)$\end{document}. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [24,Section 9.8.2].

more » « less
3. Consider the linear transport equation in 1D under an external confining potential \begin{document}$\Phi$\end{document}:

For \begin{document}$\Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2$\end{document} (with \begin{document}$\varepsilon >0$\end{document} small), we prove phase mixing and quantitative decay estimates for \begin{document}${\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v$\end{document}, with an inverse polynomial decay rate \begin{document}$O({\langle} t{\rangle}^{-2})$\end{document}. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in \begin{document}$1$\end{document}D under the external potential \begin{document}$\Phi$\end{document}.

more » « less
4. This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar \begin{document}$\theta$\end{document} on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity \begin{document}$u$\end{document} is of lower singularity, i.e., \begin{document}$u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta$\end{document}, where \begin{document}$p$\end{document} is a logarithmic smoothing operator and \begin{document}$\beta \in [0, 1]$\end{document}. We complete this study by considering the more singular regime \begin{document}$\beta\in(1, 2)$\end{document}. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.

more » « less
5. Stochastic differential games have been used extensively to model agents' competitions in finance, for instance, in P2P lending platforms from the Fintech industry, the banking system for systemic risk, and insurance markets. The recently proposed machine learning algorithm, deep fictitious play, provides a novel and efficient tool for finding Markovian Nash equilibrium of large \begin{document}$N$\end{document}-player asymmetric stochastic differential games [J. Han and R. Hu, Mathematical and Scientific Machine Learning Conference, pages 221-245, PMLR, 2020]. By incorporating the idea of fictitious play, the algorithm decouples the game into \begin{document}$N$\end{document} sub-optimization problems, and identifies each player's optimal strategy with the deep backward stochastic differential equation (BSDE) method parallelly and repeatedly. In this paper, we prove the convergence of deep fictitious play (DFP) to the true Nash equilibrium. We can also show that the strategy based on DFP forms an \begin{document}$\epsilon$\end{document}-Nash equilibrium. We generalize the algorithm by proposing a new approach to decouple the games, and present numerical results of large population games showing the empirical convergence of the algorithm beyond the technical assumptions in the theorems.

more » « less