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

Table of Contents

1 Dynamics in a Sinaï billard (with finite horizon)

At each bounce there is “dispersion of trajectories”:
image: 62_home_faure_enseignement_Systemes_dynamiques_M1_Images_chap_intro_rebond_billard_disque.jpg
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):
image: 63_home_faure_articles_2016_Geodesic_flow_avec____e_billard_sinai_rapport_1_Animation_1_bille.gif
Trajectories starting from closed initial conditions ( Δ y=1 0 -4 ), have very different behaviour after few bounces only (they become decorrelated).
image: 64_home_faure_enseignement_Systemes_dynamiques_M1_Images_chap_intro_billard_t200_2.png
Observe through a smooth observable v , the evolution of a smooth distribution of probability u , i.e. v observable , u φ -t evolutionof u L 2 := v( x ) u( φ -t x ) x : "Correlation function"
The distribution u is represented here by N=1 0 6 independent points with closed initial conditions Δ y=1 0 -4 .
Colors of directions: image: 65_home_faure_articles_2016_Geodesic_flow_avec_Tsujii_expose_billard_sinai_rapport_colors.png
We observe a predictable but irreversible behavior:
image: 66_home_faure_articles_2016_Geodesic_flow_avec_____animation_10e6_billes_billard_et_diffusion.gif
Statistical laws emerge (diffusion):
image: 67_home_faure_articles_2016_Geodesic_flow_avec____i_rapport_3_Animation_10e4_billes_diffusion.gif
Information escapes to microscopic scales escapes to high Fourier modes:
irreversibility at macroscopic scales, the entropy increases.
Emergent effective dynamics of fluctuations that we want to characterize.
image: 68_home_faure_articles_2016_Geodesic_flow_avec_Tsujii_expose_fuite.svg image: 69_home_faure_articles_2016_Geodesic_flow_avec_Tsujii_expose_fuite_1.svg
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:( x,y ) S 1 × R ( 2x  mod  1,y+ sin 2 π x ) Transfer operator: L :u C ( S 1 × R ) ( L u ) :=uf
Reduced transfer operator to Fourier mode e i ν y : ( L ν u ˜ ) ( x ) := e i ν sin ( 2 π x ) u ˜ ( 2x )
image: 70_home_faure_articles_2016_Geodesic_flow_avec____pose_04_Animation_resonances_nu_echelle_log.gif image: