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**Objective:** describe “**emergent behaviors**” in “**complex dynamical systems**”.

## 1 Dynamics in a Sinaï billard (with finite horizon)

At each bounce there is “**dispersion of trajectories**”:

This is the limit case of the **geodesic flow** on a surface with negative curvature:

We observe an **unpredictable, chaotic behavior** of an individual trajectory (but **reversible**):

Trajectories starting from closed initial conditions ($\Delta y=1{0}^{-4}$), have very different behaviour after few bounces only (they become decorrelated).

**Observe** through a smooth observable $v$, the evolution of a smooth **distribution of probability** $u$, i.e.$$\u27e8\underset{observable}{\underset{\u23df}{v}},\underset{evolution\phantom{\rule{6px}{0ex}}of\phantom{\rule{6px}{0ex}}u}{\underset{\u23df}{u\u25cb{\phi}^{-t}}}{\u27e9}_{{L}^{2}}:=\int v\left(x\right)u\left({\phi}^{-t}x\right)dx\phantom{\rule{20px}{0ex}}:"Correlationfunction"$$

The distribution $u$ is represented here by $N=1{0}^{6}$ independent points with closed initial conditions $\Delta y=1{0}^{-4}$.

**Colors of directions**:

We observe a** predictable but irreversible behavior:**

Statistical laws emerge (diffusion):

**Information escapes to microscopic scales** $\leftrightarrow $ escapes to **high Fourier modes**:

$\to $ irreversibility at macroscopic scales, the entropy increases.

**Emergent effective dynamics of fluctuations** that we want to characterize.

**Analogy:** with the escape of quantum waves outside the nucleus (radioactive decay), or photon outside an excited atom (fluorescence). Stationnary decaying states are called meta-stable states, “**resonances**”. The decay of correlation functions and fluctuations is characterized by the **discrete spectrum of resonances** (not in ${L}^{2}$).

Studied in physics and semiclassical analysis (Combes 70', Helffer-Sjöstrand 80')

**Results:** (F.-Tsujii, 2013-2016).

For an **Anosov geodesic flow**, the emergent dynamics (correlation functions) is governed by a **discrete spectrum of Ruelle resonances** that concentrate on a vertical axis, and a wave equation or **emergent Schrödinger equation** that describes the fluctuations of the distribution.

This **spectrum is determined by closed orbits, zeros of a Gutzwiller-Voros zeta function** that generalizes Selberg zeta function to non variable curvature manifolds.

## 2 Partially expanding map

**Model:**$$f:\phantom{\rule{20px}{0ex}}\left(x,y\right)\in {S}^{1}\times R\to \left(2xmod1,y+sin2\pi x\right)$$**Transfer operator:**$$L:u\in {C}^{\infty}\left({S}^{1}\times R\right)\to \left(Lu\right):=u\u25cbf$$

**Reduced transfer operator **to Fourier mode ${e}^{i\nu y}$ :$$\left({L}_{\nu}\tilde{u}\right)\left(x\right):={e}^{i\nu sin\left(2\pi x\right)}\tilde{u}\left(2x\right)$$