Scarring in classical chaotic dynamics with noise
Domenico Lippolis, Akira Shudo, Kensuke Yoshida, Hajime Yoshino
SScarring in Classical Chaotic Dynamics with Noise
Domenico Lippolis , Akira Shudo , Kensuke Yoshida , and Hajime Yoshino Institute for Applied Systems Analysis, Jiangsu University, Zhenjiang 212013, China Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji 192-0397, Japan (Dated: January 22, 2021)We report the numerical observation of scarring, that is enhancement of probability density aroundunstable periodic orbits of a chaotic system, in the eigenfunctions of the classical Perron-Frobeniusoperator of noisy Anosov (‘cat’) maps, as well as in the noisy Bunimovich stadium. A parallel isdrawn between classical and quantum scars, based on the unitarity or non-unitarity of the respec-tive propagators. We provide a mechanistic explanation for the classical phase-space localizationdetected, based on the distribution of finite-time Lyapunov exponents, and the interplay of noisewith deterministic dynamics. Classical scarring can be measured by studying autocorrelation func-tions and their power spectra.
Introduction.
In the realm of classical- and quantumchaos, phase-space densities tend to mix, due to thestretching and folding action of the dynamics. As a re-sult, every form of localization is an anomaly to the ex-pected ‘random’ behavior of Hamiltonians, propagators,wavefunctions, and various observables.Examples in quantum mechanics include dynamical lo-calization for a kicked rotor [1], which can be related tothe Anderson localization of a tight-binding model [2],opening-induced phase-space localization [3], and proba-bility density enhancement around unstable periodic or-bits of the underlying classical system. The latter isknown as scarring [4], and it has drawn a fair amount ofattention since its first discovery in the quantum Buni-movich stadium billiard. Scars, the regions of enhancedprobability density, have been ascribed to constructiveinterference around periodic orbits [5]. Dismissed for awhile as transients of no effect on the long-term prop-erties of a closed chaotic system subject to thermaliza-tion [6], scars were brought back into the spotlight byrecent numerical evidence of ergodicity breaking in many-body systems [7–12]. Further theoretical and experimen-tal work has extended the notion of scarring to regulardynamics[13–15], to integrable systems with disorder [16–18], of interest in cold atoms and condensed matter, aswell as to relativistic Dirac billiards [19, 20].In the present paper, we report scarring in the eigen-functions of the classical
Perron-Frobenius evolution op-erator [21] with background noise, for two paradigmaticmodels of chaos. The observations presented here suggestthat quantum localization in chaos does not exclusivelyarise from interference, but is also a classical effect.The noiseless Perron-Frobenius operator L t ρ ( x ) = (cid:90) dx δ (cid:0) x − f t ( x ) (cid:1) ρ ( x ) . (1)transports an initial phase-space density of trajectories ρ ( x ) through the flow f t ( x ), that is the solution of theequations of motion, to a new density. The Perron-Frobenius operator is linear, and its spectral proper-ties depend in general on the space of functions it acts upon [22–24]. It is a formal solution to the Liouville equa-tion ∂ t ρ + ∇ · ( ρ v ) = 0, where ˙ x = v ( x ) is the dynamicalsystem in exam. In chaotic Hamiltonian systems with noescape, the Perron-Frobenius spectrum has an isolated,unitary eigenvalue, whose eigenfunction is uniform in thephase space, and it is called natural measure or invariantdensity [25]. The natural measure is the weight to everyphase space average, and, as such, its successful deter-mination enables us to evaluate any long-term averagedobservable under the ergodicity assumption, thus solv-ing the problem of statistical mechanics. Here, instead,we focus on the second or higher eigenfunctions of thePerron-Frobenius spectrum, whose eigenvalues yield thedecay rate of any initial density to the natural measure.In fact, we can expand the evolution of a density as L t ρ ( x ) = (cid:88) n a n e − γ n t φ n ( x ) + (cid:88) m a m ( t ) ψ m ( x ) (2)where φ ( x ) = 1, γ = 0, while the summation over n is an expansion over the eigenfunctions, all decayingwith rates γ n , and the (cid:80) m represents Jordan blocs,since the Perron-Frobenius operator is in general non-diagonalizable [its spectrum also has a continuous part,neglected in (2)] .In reality, every physical system experiences noise insome form, which is modeled as a random variable ξ ( t )in the equations of motion, ˙ x = v ( x ) + ξ ( t ) . If the noiseis assumed as Gaussian-distributed and uncorrelated, theLiouville equation above acquires a diffusion term, say D ∇ ρ ( D is the noise amplitude or variance of ξ ), and isknown as Fokker-Planck equation. Its formal solution isa path (‘Wiener’) integral [26], whose kernel can be re-garded as an evolution operator analogous to the Perron-Frobenius in Eq. (1), but with a finite-width distributioninstead of the delta function. Noise smears out densities,and thus it balances contractions from the deterministic,chaotic dynamics. Distributions of trajectories, as wellas eigenfunctions of the noisy Perron-Frobenius opera-tor, are then expected to be smooth. Classical scars.
Figure 1 illustrates localization of thesubleading eigenfunctions of the noisy Perron-Frobenius a r X i v : . [ n li n . C D ] J a n FIG. 1. Magnitude of the second eigenfunction of the noisyPerron-Frobenius operator, numerically evaluated for: (a)Bunimovich quarter stadium billiard (in the inset with thebouncing-ball orbit) with noise of amplitude D = 10 − , usingthe Ulam matrix (3) with N = 2 ; (b) perturbed cat map(periodic point at the origin) with (cid:15) = 0 . ν = 1, using thescheme (4) with diffusivity ∆ = 5 × − and M = 100. operator near classical periodic orbits, for the Buni-movich quarter stadium billiard [27], as well as thecat map perturbed with a nonlinear shear, ( x (cid:48) , y (cid:48) ) = (cid:0) x + y − εν sin(2 νπy ) , x + 2 y − εν sin(2 νπy ) (cid:1) mod 1 . In analogy with the corresponding enhancement of probabil-ity density of quantum eigenstates, we dub the observedphenomenon classical scarring , with the caveat that, topresent knowledge, while some mechanisms behind theformation of scars are common in classical and in quan-tum mechanics, others may differ between the two. Thedynamics of the cat map is everywhere unstable (‘hyper-bolic’) [28], and the slowly-decaying eigenfunctions of thePerron-Frobenius spectrum are striated along the unsta-ble manifold [29–32]. On the other hand, the stadiumbilliard is chaotic, ergodic [33], and has infinitely manyunstable periodic orbits, but it also possesses a family ofmarginally stable (‘bouncing ball’) orbits that give riseto the so-called superscars in the eigenfunctions of thequantized system [13, 34]. A newly found example oftheir classical counterpart is shown in Fig. 1(a). As forlocalization arising from unstable orbits in the stadium,results in Fig. 2 suggest that the same scars recur in dis-tinct second- or higher-order eigenfunctions, and, con-versely, that a single eigenfunction often displays severalscars. FIG. 2. Classical scars of short periodic orbits: (a)-(c) Bunimovich quarter stadium billiard. Here the Ulam matrix (3) has N = 2 , and D = 10 − . (a) Bowtie orbit (corresponding scar pointed to by an arrow); (b) rectangular orbit; (c) triangularorbit; (d) scar of a period-3 orbit [marked by ( × )] of the perturbed cat map ( (cid:15) = 0 . , ν = 2) on the unit torus, obtained as thesecond eigenfunction of the transfer matrix (4), and M = 50, ∆ = 10 − . The fixed point at the origin (+) is ‘anti-scarred’. Methodology.
The noisy Perron-Frobenius operator isprojected onto a finite-dimensional vector space, and thusimplemented as a finite matrix. Previous literature warnsus that the choice of the discretization is crucial and maydeeply affect the eigenspectrum beyond the leading eigen-value in the linear map [35]. It has been established, onthe other hand, that both nonlinear perturbations to lin-ear maps on a torus, and background noise, increase therobustness of the numerically evaluated spectrum undercertain conditions [23].The simplest discretization scheme is Ulam’smethod [36], that amounts to subdividing the phasespace into N intervals M i of equal area. The evolutionoperator is thus approximated with a N × N transfermatrix whose entries L ij are the transition probabilitiesfrom M i to M j L ij = µ ( M i (cid:84) f ξ ( M j )) µ ( M i ) (3) in one time step, where f ξ ( x ) = f ( x ) + ξ is the noisymapping. We use a known Monte Carlo method [37]to estimate the non-symmetric transfer matrix L ij , a(weighted) directed network [38], in today’s parlance. Athorough study of stability and convergence of discretiza-tion algorithms has been reported elsewhere by the au-thors [39].We implement Ulam’s scheme for the Bunimovichquarter stadium, where more sophisticated discretiza-tions (e.g. Markov partitions) appear impractical. Onthe other hand, the perturbed cat map also allows for analternative realization of the transfer operator, by whosemeans we rule out the possibility that the detected scar-ring be just a numerical artifact. Since the dynamics ofthe cat map lives on the unit torus, a basis of smooth,periodic functions is suitable for the evolution operator,that can be defined in Fourier space as [41] L ∆ ρ ( x ) = (cid:90) M (cid:88) k x ,k y e πi k · ( f − ( x ) − x ) − ∆ k ρ ( x ) d x . (4)Here the diffusivity ∆ is equivalent to the variance D ofthe random variable ξ defined above in the Langevin pic-ture. The spectrum of the operator in this basis is robustunder perturbations [39] (e.g. dimension of the transfermatrix, noise amplitude, nonlinearity of the perturbedcat map), and classical scarring is consistently detectedin the second eigenfunction of the spectrum: the one inFig. 1(b) is computed with the Fourier basis of Eq. (4),while, for the same map, the second eigenfunction of theUlam matrix (3) is displayed in Fig. 3(b). Analogy with quantum scars.
The route to understandclassical scarring begins by comparing it directly withits quantum counterpart. However, the propagator U t ,that regulates the evolution of quantum cat maps, isunitary [40], unlike our realizations of the noisy Perron-Frobenius operator L t , and that constitutes a clear asym-metry in our quest for classical-to-quantum correspon-dence. Breaking the unitarity of U t by coupling the FIG. 3. The perturbed cat map: magnitudes of the (a) left-and (b) right second eigenfunctions of the Perron-Frobeniusoperator, realized through the Ulam matrix (3), with N =10 , D = 10 − ; Husimi distributions of (c) right- and (d)left eigenfunctions of the spectrum of the subunitary quan-tum propagator coupled to a single-channel opening (dashedcircle). Details of the quantization in ref. [42]. quantized cat map to an opening is the simplest way torestore the symmetry between classical and quantum evo-lution. The result is exemplified in Fig. 3, that featuresscars around the periodic orbit at the origin of the phasespace, in both noisy classical and quantum cat maps.In the classical setting, the areas of enhanced probabil-ity density are striated along the stable- (left eigenfunc- tion of L ) or unstable (right eigenfunction of L ) man-ifold, that emanates from the periodic orbit located atthe origin. The correspondence left/right eigenfunction-stable/unstable manifold is less straightforward for theopen quantum map [42], but still one-to-one.Conversely, we may assimilate the classical scars to theoriginal quantum scars of closed chaotic systems, wherethe propagation is unitary. Quantum scars of a unitarypropagator are typically concentrated around a periodicorbit with no elongations on the manifolds, as a resultof the unitary evolution. Using the known technique ofeigenfunctions unwrapping [43], we map a right eigen-function of the noisy L backward in time by means of theadjoint (‘Koopman’) evolution operator, whose noiselessdefinition reads (cid:2) L t (cid:3) † ρ ( x ) = (cid:90) dx δ (cid:0) x − f t ( x ) (cid:1) ρ ( x ) = ρ (cid:0) f t ( x ) (cid:1) . (5)When noise is included and the Fourier representation (4)is used for L ∆ , the adjoint evolution operator is L † ∆ ρ ( x ) = M (cid:88) k x ,k y e πi k · f ( x ) − ∆ k ˆ ρ k , (6)where the ˆ ρ k ’s are the Fourier coefficients of the density ρ ( x ). Repeated adjoint mapping rids the eigenfunction ofthe striation along the unstable manifold, and regularizesit. As a result, the scar shown in Fig. 4(a) is almostinvariant under both forward and backward iterations,and it now closely resembles its quantum analog obtainedfrom unitary propagation [Fig. 4(b)]. FIG. 4. (a) Unwrapping: the operator (6) is applied for t = 3iterations to the second eigenfunction of the noisy Perron-Frobenius operator for the perturbed cat map [parameters asin Fig. 5(a)]; (b) Husimi distribution of a scarred eigenfunc-tion of the unitary quantum propagator of the same map. Origin of classical scars.
We now examine the localaction of the classical Perron-Frobenius operator (1) onphase-space densities, in order to gain some insight onthe mechanism behind classical scarring: L t ρ = (cid:90) dx δ (cid:0) x − f t ( x ) (cid:1) ρ ( x ) = (cid:88) y = f − t ( x ) ρ ( y ) | det J t ( y ) | , (7)where J tij ( y ) = ∂ i f t ( y ) ∂y j is the Jacobian of the flow. Nowrestrict the analysis to the unstable manifold, where den-sities are stretched and squished. If the map is 2D, theunstable manifold is a 1D curve characterized by an arclength s ( y ), and densities are mapped like L tu ρ u ( s ) ∝ t − (cid:89) k =0 (cid:12)(cid:12)(cid:12) [ f ku ( s )] (cid:48) (cid:12)(cid:12)(cid:12) ρ u ( s ) ≡ e − Λ u ( y ,t ) t ρ u ( s ) . (8)Here we call Λ u ( y , t ) finite-time Lyapunov exponent ofthe map f t ( x ), that expresses the rate of exponential di-vergence of nearby trajectories within the time t . On theother hand, assume that the relaxation of ρ ( y ) towardsequilibrium is well described by the expansion (2), re-tain its first two terms, and assume the decaying part isalmost entirely supported on the unstable manifold: L t ρ ( y ) (cid:39) a + a e − γ t φ ( y ) ∼ a + e − Λ u ( y ,t ) t ρ u ( s ) . (9)Since the evolution (8) of a density on the unstable man-ifold does not depend on the initial condition ρ u ( s ), weinfer that Λ u ( y , t ) > Λ u ( y , t ) ⇒ | φ ( y ) | < | φ ( y ) | ,and thus in general the probability density of the secondeigenfunction of the spectrum along the unstable mani-fold is ruled by the finite-time Lyapunov exponent: thelesser instability, the higher the (magnitude of the) den-sity [44–46]. As it can be inferred from Eq. (9), the Lya-punov exponent is to be evaluated over a time t (cid:39) γ − ,thus of the order of the decay time of the eigenfunction φ ( x ). Densities stretch out along the unstable mani-fold, while they are contracted along the stable manifold.Asymptotically, the compression makes them infinitesi-mally thin, but noise counters that effect, and fattensdensities along the unstable manifold. As a result, scar-ring also becomes apparent by visual inspection. FIG. 5. (a) the second eigenfunction of the noisy Perron-Frobenius operator (4), numerically evaluated for a perturbedcat map with ∆ = 10 − and cutoff M = 100; (b) the phase-space distribution of the finite-time Lyapunov exponents ( t =5, sampling: 10 initial points) for the same map. Figure 5 supports this hypothesis: the second eigen-function of the noisy Perron-Frobenius operator is shownfor the perturbed cat map, and it is localized along theunstable manifold that emanates from the origin. On the other hand, the numerically computed phase-space dis-tribution of the finite-time Lyapunov exponent displaysa suppression pattern that nearly overlaps with the scar.
Power spectra.
Scarring can be quantified by studyingthe power spectrum [47] S ( ω ) = 1 √ T T (cid:88) t =0 C ( t )e πiωt/T (10)of a Gaussian density (the classical analog of a wavepacket), that gradually decays into a uniform phase-spacedistribution, as the evolution operator is applied. Here C ( t ) is the autocorrelation function of the density [definedin Eq. (11)], while T is the length of the time series. Fig- FIG. 6. (a) An initial Gaussian density centered at the fixedpoint of the perturbed cat map is mapped by the Ulam ma-trix (3); (b) t=1 iterations; (c) t=5; (d) t=20. (e) Magnitudeof the power spectrum of the autocorrelation function minusthe steady state: numerics for (dots) the density in (a)-(d),(diamonds) the same initial density centered at random in thephase space, (solid line) prediction (12). ure 6 shows the outcome of the numerical experiment. Afast decay of C ( t ) occurs if we place the initial density atrandom in the phase space, resulting in a flat power spec-trum. Instead, centering the initial distribution aroundthe scarred fixed point [Fig. 6(a)] produces a slower de-cay in the autocorrelation function, and a peaked powerspectrum. The quantum analog of | S ( ω ) | is the localdensity of states, whose energy-dependent, peaked en-velope is a well-known signature of scarring [48]. Thepower spectrum is related to the second eigenfunction ofthe transfer operator in the following way. Truncatingthe expansion (2) at the first order, the autocorrelationfunction is estimated as C ( t ) = ρ T [ L t ρ ] ρ T ρ ≈ c + c e − γ t , (11)for some c , c . The discrete Fourier transform (10) ofEq. (11) then yields a delta function from the asymp-totic overlap c of the evolved density with the naturalmeasure, plus the actual power spectrum of the exponen-tial, δS ( ω ) ∝ − e iωT/ π − γ t . (12)This approximation [solid line in Fig. (6)], with γ deter-mined from the diagonalization of the transfer matrix (3),shows close agreement with the direct numerical compu-tation. Using the results of Eq. (9) and Fig. 5, one couldreplace γ in Eq. (12) with the minimum finite-time Lya-punov exponent evaluated at t (cid:39) γ − . Conclusion.
We have reported the observation of clas-sical scars, that is enhancement of probability densitynear periodic orbits, in the eigenfunctions of a noisy evo-lution operator of chaotic systems. We have detectedscars in two model systems: perturbed cat maps, andthe Bunimovich stadium billiard, both with backgroundnoise. The observed localization is ascribed to the inho-mogeneity of instabilities near the periodic orbit of in-terest, on a time scale consistent with the decay rate ofthe second eigenfunction of the evolution operator, thatis also the rate of correlation decay. In support of thisexplanation for the formation of classical scarring, wehave compared the second eigenfunction with the distri-bution of finite-time Lyapunov exponents of the dynam-ical system in exam. The inevitable presence of noisedoes not alter this mechanism, but merely thickens thescars by its smearing action along the unstable mani-folds [cf. Figs. 1(b) and 5(a)]. Although we do not claima one-to-one correspondence between classical and quan-tum scarring, we have pointed out apparent similaritiesin their phase-space patterns (provided the same symme-tries) and power spectra, with the important differenceof the time scales associated to the relevant instabilities.It is then natural to suppose that the mechanism at theorigin of classical scars must also play a role in the for-mation of their quantum counterparts, in the same spiritas classical dynamical localization [49], or phase-spacelocalization in open systems [3, 50].
Acknowledgments-
The authors are supported by NSFChina (Grant No. 11750110416-1601190090) and JSPSKAKENHI (Grants No. 15H03701 and No. 17K05583).DL thanks S. Pascazio and R. Bellotti for hospitality atINFN Bari, and ReCaS Bari for computational resources. [1] G. Casati, B. V. Chirikov, F. M. Izraelev, and J. Ford, in
Stochastic Behavior in Classical and Quantum Hamilto-nian Systems , edited by G. Casati and J. Ford, LectureNotes in Physics Vol. 93 (Springer, Berlin, 1979).[2] D. R. Grempel, S. Fishman, and R. E. Prange, Localiza-tion in an Incommensurate Potential: An Exactly Solv-able Model, Phys. Rev. Lett. , 833 (1982)[3] K. Clauß, M. J. K¨orber, A. B¨acker, and R. Ketzmerick,Resonance eigenfunction hypothesis for chaotic systems,Phys. Rev. Lett. , 074101 (2018).[4] E. J. Heller, Bound-State Eigenfunctions of ClassicallyChaotic Hamiltonian Systems: Scars of Periodic Orbits,Phys. Rev. Lett. , 1515 (1984).[5] E. Bogomolny, Smoothed wave functions of chaotic quan-tum systems, Physica D , 169 (1988) [6] M. Srednicki, Chaos and quantum thermalization, Phys.Rev. E , 888 (1994)[7] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn,and Z. Papi´c, Weak ergodicity breaking from quantummany-body scars, Nature Phys. , 745 (2018)[8] W. W. Ho, S. Choi, H. Pichler, and M. D. Lukin, PeriodicOrbits, Entanglement, and Quantum Many-Body Scarsin Constrained Models: Matrix Product State Approach,Phys. Rev. Lett. , 040603 (2019)[9] S. Choi, C. J. Turner, H. Pichler, W. W. Ho,A. A. Michailidis, Z. Papi´c, M. Serbyn, M. D. Lukin,and D. A. Abanin, Emergent SU(2) Dynamics and Per-fect Quantum Many-Body Scars, Phys. Rev. Lett. ,220603 (2019)[10] C.-J. Lin and O. I. Motrunich, Exact Quantum Many-Body Scar States in the Rydberg-Blockaded Atom Chain,Phys. Rev. Lett. , 173401 (2019)[11] T. Iadecola and M. Schecter, Weak Ergodicity Breakingand Quantum Many-Body Scars in Spin-1 XY Magnets,Phys. Rev. Lett. , 147201 (2019)[12] S. Pai and M. Pretko, Dynamical Scar States in DrivenFracton Systems , Phys. Rev. Lett. , 136401 (2019)[13] E. Bogomolny and C. Schmit, Structure of Wave Func-tions of Pseudointegrable Billiards, Phys. Rev. Lett. ,244102 (2004).[14] M. Lebental, N. Djellali, C. Arnaud, J.-S. Lauret, J. Zyss,R. Dubertrand, C. Schmit, and E. Bogomolny, Inferringperiodic orbits from spectra of simply shaped microlasers,Phys. Rev. A , 023830 (2007).[15] B. Dietz, T. Friedrich, M. Miski-Oglu, A. Richter, andF. Sch¨afer, Properties of nodal domains in a pseudoin-tegrable barrier billiard, Phys. Rev. E , 045201(R)(2008)[16] P. J. J. Luukko, B. Drury, A. Klales, L. Kaplan,E. J. Heller, and E. R¨as¨anen, Strong quantum scars bylocal impurities, Sci. Rep. , 37656 (2016).[17] J. Keski-Rahkonen, P. J. J. Luukko, L. Kaplan, E. J.Heller, E. R¨as¨anen, Controllable quantum scars in semi-conductor quantum dots, Phys. Rev. B , 094204 (2017)[18] J. Keski-Rahkonen, A. Ruhanen, E. J. Heller,E. R¨as¨anen, Quantum Lissajous Scars, Phys. Rev. Lett. , 214101 (2019)[19] L. Huang, Y.-C. Lai, D. K. Ferry, S. M. Goodnick, andR. Akis, Relativistic Quantum Scars, Phys. Rev. Lett. , 054101 (2009)[20] M.-Y. Song, Z.-Y. Li, H.-Y. Xu, L. Huang, and Y.-C. Lai,Quantization of massive Dirac billiards and unificationof nonrelativistic and relativistic chiral quantum scars,Phys. Rev. Res. , 033008 (2019).[21] P. Cvitanovi´c, R. Artuso, R. Mainieri, G. Tanner and G.Vattay, Chaos: Classical and Quantum , ChaosBook.org (Niels Bohr Institute, Copenhagen 2016)[22] D. Dolgopyat, On decay of correlations in Anosov flows,Ann. Math. (2) , 357 (1998)[23] M. Blank, G. Keller, and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity ,1905 (2002).[24] C. Liverani, On contact Anosov flows, Ann. Math. (2) , 1275 (2004)[25] P. Gaspard, Chaos, Scattering, and Statistical Mechanics (Cambridge University Press, Cambridge, 1999)[26] H. Risken,
The Fokker-Planck Equation (Springer,Berlin, 1996)[27] L. A. Bunimovich, Funct. Anal. Appl. , 254 (1974) [28] V. I. Arnold and A. Avez, Ergodic Problems of ClassicalMechanics , (Benjamin, New York, 1968)[29] J. Weber, F. Haake, and P. ˘Seba, Frobenius-Perron Res-onances for Maps with a Mixed Phase Space, Phys. Rev.Lett. , 3620 (2000).[30] G. Blum and O. Agam, Leading Ruelle resonances ofchaotic maps, Phys. Rev. E , 1977 (2000).[31] J. Weber, F. Haake, P. A. Braun, C. Manderfeld andP. ˘Seba, Resonances of the Frobenius-Perron operator fora Hamiltonian map with a mixed phase space, J. Phys.A: Math. Gen. , 7195 (2001)[32] C. Manderfeld, Classical resonances and quantum scar-ring, J. Phys. A: Math. Gen. , 6379 (2003)[33] L. A. Bunimovich, On the ergodic properties of nowheredispersing billiards, Commun. Math. Phys. , 295[34] A. Hassell, Ergodic billiards that are not quantum uniqueergodic, Ann. Math. , 605 (2010)[35] F. Brini, S. Siboni, G. Turchetti, and S. Vaienti, Decay ofcorrelations for the automorphism of the torus T , Non-linearity , 1257 (1997).[36] S. M. Ulam, A Collection of Mathematical Problems (In-terscience, New York, 1960).[37] L. Ermann and D. L. Shepelyansky, The Arnold cat map,the Ulam method, and time reversal, Physica D , 514(2012).[38] A.-L. Barab´asi,
Network Science (Cambridge UniversityPress, 2016).[39] K. Yoshida, H. Yoshino, A. Shudo, and D. Lippolis,Eigenfunctions of the Perron-Frobenius operator and the finite-time Lyapunov exponents in uniformly hyperbolicarea-preserving maps, in preparation.[40] S. C. Creagh, Quantum zeta function for perturbed catmaps, Chaos , 477 (1995)[41] J. L. Thiffeault and S. Childress, Chaotic mixing in atorus map, Chaos , 502 (2003).[42] D. Lippolis, J. W. Ryu, S. Y. Lee, and S. W. Kim, On-manifold localization in open quantum maps, Phys. RevE , 066213 (2012).[43] G. Froyland, Unwrapping eigenfunctions to discover thegeometry of almost-invariant sets in hyperbolic maps,Physica D , 840 (2008).[44] P. H. Haynes and J. Vanneste, What controls the decay ofpassive scalars in smooth flows?, Phys. Fluids , 097103(2005).[45] J. -L. Thiffeault, Scalar decay in chaotic mixing, Lect.Notes Phys. , 3 (2008)[46] G. Haller, Lagrangian coherent structures, Annu. Rev.Fluid Mech. , 137 (2015).[47] T. M. Antonsen, Z. Fan, E. Ott, and E. Garcia-Lopez,The role of chaotic orbits in the determination of powerspectra of passive scalars, Phys. Fluids , 3094 (1996)[48] L. Kaplan and E. J. Heller, Linear and Nonlinear Theoryof Eigenfunction Scars, Ann. Phys. , 171 (1998).[49] I. Guarneri, G. Casati, and V. Karle, Classical DynamicalLocalization, Phys. Rev. Lett. , 174101 (2014).[50] K. Clauß, E. G. Altmann, A. B¨acker, and R. Ketzmerick,Structure of resonance eigenfunctions for chaotic systemswith partial escape, Phys. Rev. E100