American Institute of Mathematical Sciences

Advanced
  2018, 12: 55-122. doi: 10.3934/jmd.2018004

Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions

Krzysztof Frączek 1, , Ronggang Shi 2, and  Corinna Ulcigrai 3,

1. 

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
2. 

Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, PR China
3. 

School of Mathematics, University of Bristol, Howard House, Queens Ave, BS8 1SD, Bristol, United Kingdom

Received  June 13, 2016 Revised  November 06, 2017 Published  March 2018

Fund Project: KF: Research partially supported by the National Science Centre (Poland) grant 2014/13/ B/ST1/03153.
RS: Supported by NSFC (11201388), NSFC (11271278), ERC starter grant DLGAPS 279893.
CU: Supported by the ERC, via the Starting Grant ChaParDyn, as well as by the Leverhulme Trust via a Leverhulme Prize and by the Royal Society and the Wolfson Foundation via a Royal Society Wolfson Research Merit Award. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 335989.

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. In the space of affine lattices $ASL_2( \mathbb{R})/ASL_2( \mathbb{Z})$, we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum $\mathscr{H}(1,1)$ of translation surfaces. For these curves we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers is also obtained.

Keywords: Homogeneous dynamics, translation surfaces, Kontsevich-Zorich cocycle, equidistribution, ergodic theorem, Oseledets theorem, Eaton lenses, gap distribution, pseudo-integrable billiards.
Mathematics Subject Classification: Primary: 37A17; Secondary: 37A40, 37J35, 11K38.
Citation: Krzysztof Frączek, Ronggang Shi, Corinna Ulcigrai. Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions. Journal of Modern Dynamics, 2018, 12: 55-122. doi: 10.3934/jmd.2018004
References:
[1]

A. Avila, A. Eskin and M. Möller, Symplectic and isometric SL(2, R)-invariant subbundles of the Hodge bundle, J. Reine Angew. Math., 732 (2017), 1-20.   Google Scholar

[2]

A. Avila, S. Gouëzel and J.-Ch. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211.   Google Scholar

[3]

A. Avila and P. Hubert, Recurrence for the wind-tree model, Ann. Inst. H. Poincaré Anal. Non Linéaire, (2018). inéaire, (2018). doi: 10.1016/j.anihpc.2017.11.006.  Google Scholar

[4]

M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol., 11 (2007), 1887-2073.  doi: 10.2140/gt.2007.11.1887.  Google Scholar

[5]

T. Browning and I. Vinogradov, Effective Ratner theorem for $SL(2, \mathbb R) <\ltimes \mathbb R^2$ and gaps in $\sqrt n$ modulo $1$, J. Lond. Math. oc. (2), 94 (2016), 61-84.  doi: 10.1112/jlms/jdw025.  Google Scholar

[6]

Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (Ⅱ), J. Amer. Math. Soc., 26 (2013), 659-734.  doi: 10.1090/S0894-0347-2013-00760-2.  Google Scholar

[7]

J. Chaika and A. Eskin, Every flat surface is Birkhoff and Oseledets generic in almost every direction, J. Mod. Dyn., 9 (2015), 1-23.  doi: 10.3934/jmd.2015.9.1.  Google Scholar

[8]

V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 1085-1110.  doi: 10.24033/asens.2234.  Google Scholar

[9]

V. Dragović and M. Radnović, Pseudo-integrable billiards and arithmetic dynamics, J. Mod. Dyn., 8 (2014), 109-132.  doi: 10.3934/jmd.2014.8.109.  Google Scholar

[10]

M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010,155–241.  Google Scholar

[11]

N. D. Elkies and C. T. McMullen, Gaps in $\sqrt{n}$ mod $1$ and ergodic theory, Duke Math. J., 123 (2004), 95-139.  doi: 10.1215/S0012-7094-04-12314-0.  Google Scholar

[12]

A. Eskin, S. Filip and A. Wright, The algebraic hull of the Kontsevich-Zorich cocycle, arXiv: 1702.02074. Google Scholar

[13]

A. Eskin, G. A. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141.  doi: 10.2307/120984.  Google Scholar

[14]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL_2(\mathbb{R})$ action on moduli space, arXiv: 1302.3320. Google Scholar

[15]

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution and orbit closures for the $SL(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673-721.   Google Scholar

[16]

S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, Invent. Math., 205 (2016), 617-670.  doi: 10.1007/s00222-015-0643-3.  Google Scholar

[17]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.  doi: 10.2307/3062150.  Google Scholar

[18]

R. Fox and R. Kershner, Concerning the transitive properties of geodesics on a rational polyhedron, Duke Math. J., 2 (1936), 147-150.  doi: 10.1215/S0012-7094-36-00213-2.  Google Scholar

[19]

K. Frączek and P. Hubert, Recurrence and non-ergodicity in generalized wind-tree models, Math. Nachr., to appear, arXiv: 1506.05884. Google Scholar

[20]

K. Frączek and M. Schmoll, Directional localization of light rays in a periodic array of retro-reflector lenses, Nonlinearity, 27 (2014), 1689-1707.  doi: 10.1088/0951-7715/27/7/1689.  Google Scholar

[21]

K. Frączek and C. Ulcigrai, Non-ergodic $\mathbb{Z}$-periodic billiards and infinite translation surfaces, Invent. Math., 197 (2014), 241-298.  doi: 10.1007/s00222-013-0482-z.  Google Scholar

[22]

I. Ya. Goldsheid and G. A. Margulis, Lyapunov exponents of a product of random matrices, Russian Math. Surveys, 44 (1989), 11-71.   Google Scholar

[23]

Y. Guivarc'h and A. N. Starkov, Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms, Ergodic Theory Dynam. Systems, 24 (2004), 767-802.  doi: 10.1017/S0143385703000440.  Google Scholar

[24]

J. H. Hannay and T. M. Haeusserab, Retroreflection by refraction, J. Mod. Opt., 40 (1993), 1437-1442.  doi: 10.1080/09500349314551501.  Google Scholar

[25]

A. Katok and A. Zemljakov, Topological transitivity of billiards in polygons, (Russian), Mat.Zametki, 18 (1975), 291-300.   Google Scholar

[26]

S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2), 124 (1986), 293-311.  doi: 10.2307/1971280.  Google Scholar

[27]

D. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2), 148 (1998), 339-360.  doi: 10.2307/120997.  Google Scholar

[28]

D. Kleinbock, R. Shi and B. Weiss, Pointwise equidistribution with an error rate and with respect to unbounded functions, Math. Ann., 367 (2017), 857-879.  doi: 10.1007/s00208-016-1404-3.  Google Scholar

[29]

R. Lyons, Strong laws of large numbers for weakly correlated random variables, Michigan Math. J., 35 (1988), 353-359.  doi: 10.1307/mmj/1029003816.  Google Scholar

[30]

H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J., 66 (1992), 387-442.   Google Scholar

[31]

Y. Minsky and B. Weiss, Nondivergence of horocyclic flows on moduli space, J. Reine Angew. Math., 552 (2002), 131-177.   Google Scholar

[32]

Y. Minsky and B. Weiss, Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 245-284.  doi: 10.24033/asens.2214.  Google Scholar

[33]

S. Mozes, Epimorphic subgroups and invariant measures, Ergodic Theory Dynam. Systems, 15 (1995), 1207-1210.  doi: 10.1017/S0143385700009871.  Google Scholar

[34]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1991), 545-607.  doi: 10.2307/2944357.  Google Scholar

[35]

D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 275-306.   Google Scholar

[36]

N. Shah, Equidistribution of expanding translates of curves and Dirichlet's theorem on diophantine approximation, Invent. Math, 177 (2009), 509-532.  doi: 10.1007/s00222-009-0186-6.  Google Scholar

[37]

N. Shah, Limiting distributions of curves under geodesic flow on hyperbolic manifolds, Duke Math. J., 148 (2009), 251-279.  doi: 10.1215/00127094-2009-026.  Google Scholar

[38]

R. Shi, Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space, arXiv: 1405.2067. Google Scholar

[39]

R. Shi, Expanding cone and applications to homogeneous dynamics, arXiv: 1510.05256. Google Scholar

[40]

A. N. Starkov, Dynamical Systems on Homogeneous Spaces, Translations of Mathematical Monographs, 190, American Mathematical Society, Providence, RI, 2000.  Google Scholar

[41]

S. Tabachnikov, Geometry and Billiards, Student Mathematical Library, 30, American Mathematical Society, Providence, RI, 2005.  Google Scholar

[42]

A. Zorich, Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499.  doi: 10.1017/S0143385797086215.  Google Scholar

[43]

A. Zorich, How do the leaves of a closed 1-form wind around a surface?, Amer. Math. Soc. Transl. Ser. 2, 197 (1999), 135-178.   Google Scholar

show all references

References:
[1]

A. Avila, A. Eskin and M. Möller, Symplectic and isometric SL(2, R)-invariant subbundles of the Hodge bundle, J. Reine Angew. Math., 732 (2017), 1-20.   Google Scholar

[2]

A. Avila, S. Gouëzel and J.-Ch. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211.   Google Scholar

[3]

A. Avila and P. Hubert, Recurrence for the wind-tree model, Ann. Inst. H. Poincaré Anal. Non Linéaire, (2018). inéaire, (2018). doi: 10.1016/j.anihpc.2017.11.006.  Google Scholar

[4]

M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol., 11 (2007), 1887-2073.  doi: 10.2140/gt.2007.11.1887.  Google Scholar

[5]

T. Browning and I. Vinogradov, Effective Ratner theorem for $SL(2, \mathbb R) <\ltimes \mathbb R^2$ and gaps in $\sqrt n$ modulo $1$, J. Lond. Math. oc. (2), 94 (2016), 61-84.  doi: 10.1112/jlms/jdw025.  Google Scholar

[6]

Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (Ⅱ), J. Amer. Math. Soc., 26 (2013), 659-734.  doi: 10.1090/S0894-0347-2013-00760-2.  Google Scholar

[7]

J. Chaika and A. Eskin, Every flat surface is Birkhoff and Oseledets generic in almost every direction, J. Mod. Dyn., 9 (2015), 1-23.  doi: 10.3934/jmd.2015.9.1.  Google Scholar

[8]

V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 1085-1110.  doi: 10.24033/asens.2234.  Google Scholar

[9]

V. Dragović and M. Radnović, Pseudo-integrable billiards and arithmetic dynamics, J. Mod. Dyn., 8 (2014), 109-132.  doi: 10.3934/jmd.2014.8.109.  Google Scholar

[10]

M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010,155–241.  Google Scholar

[11]

N. D. Elkies and C. T. McMullen, Gaps in $\sqrt{n}$ mod $1$ and ergodic theory, Duke Math. J., 123 (2004), 95-139.  doi: 10.1215/S0012-7094-04-12314-0.  Google Scholar

[12]

A. Eskin, S. Filip and A. Wright, The algebraic hull of the Kontsevich-Zorich cocycle, arXiv: 1702.02074. Google Scholar

[13]

A. Eskin, G. A. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141.  doi: 10.2307/120984.  Google Scholar

[14]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL_2(\mathbb{R})$ action on moduli space, arXiv: 1302.3320. Google Scholar

[15]

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution and orbit closures for the $SL(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673-721.   Google Scholar

[16]

S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, Invent. Math., 205 (2016), 617-670.  doi: 10.1007/s00222-015-0643-3.  Google Scholar

[17]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.  doi: 10.2307/3062150.  Google Scholar

[18]

R. Fox and R. Kershner, Concerning the transitive properties of geodesics on a rational polyhedron, Duke Math. J., 2 (1936), 147-150.  doi: 10.1215/S0012-7094-36-00213-2.  Google Scholar

[19]

K. Frączek and P. Hubert, Recurrence and non-ergodicity in generalized wind-tree models, Math. Nachr., to appear, arXiv: 1506.05884. Google Scholar

[20]

K. Frączek and M. Schmoll, Directional localization of light rays in a periodic array of retro-reflector lenses, Nonlinearity, 27 (2014), 1689-1707.  doi: 10.1088/0951-7715/27/7/1689.  Google Scholar

[21]

K. Frączek and C. Ulcigrai, Non-ergodic $\mathbb{Z}$-periodic billiards and infinite translation surfaces, Invent. Math., 197 (2014), 241-298.  doi: 10.1007/s00222-013-0482-z.  Google Scholar

[22]

I. Ya. Goldsheid and G. A. Margulis, Lyapunov exponents of a product of random matrices, Russian Math. Surveys, 44 (1989), 11-71.   Google Scholar

[23]

Y. Guivarc'h and A. N. Starkov, Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms, Ergodic Theory Dynam. Systems, 24 (2004), 767-802.  doi: 10.1017/S0143385703000440.  Google Scholar

[24]

J. H. Hannay and T. M. Haeusserab, Retroreflection by refraction, J. Mod. Opt., 40 (1993), 1437-1442.  doi: 10.1080/09500349314551501.  Google Scholar

[25]

A. Katok and A. Zemljakov, Topological transitivity of billiards in polygons, (Russian), Mat.Zametki, 18 (1975), 291-300.   Google Scholar

[26]

S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2), 124 (1986), 293-311.  doi: 10.2307/1971280.  Google Scholar

[27]

D. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2), 148 (1998), 339-360.  doi: 10.2307/120997.  Google Scholar

[28]

D. Kleinbock, R. Shi and B. Weiss, Pointwise equidistribution with an error rate and with respect to unbounded functions, Math. Ann., 367 (2017), 857-879.  doi: 10.1007/s00208-016-1404-3.  Google Scholar

[29]

R. Lyons, Strong laws of large numbers for weakly correlated random variables, Michigan Math. J., 35 (1988), 353-359.  doi: 10.1307/mmj/1029003816.  Google Scholar

[30]

H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J., 66 (1992), 387-442.   Google Scholar

[31]

Y. Minsky and B. Weiss, Nondivergence of horocyclic flows on moduli space, J. Reine Angew. Math., 552 (2002), 131-177.   Google Scholar

[32]

Y. Minsky and B. Weiss, Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 245-284.  doi: 10.24033/asens.2214.  Google Scholar

[33]

S. Mozes, Epimorphic subgroups and invariant measures, Ergodic Theory Dynam. Systems, 15 (1995), 1207-1210.  doi: 10.1017/S0143385700009871.  Google Scholar

[34]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1991), 545-607.  doi: 10.2307/2944357.  Google Scholar

[35]

D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 275-306.   Google Scholar

[36]

N. Shah, Equidistribution of expanding translates of curves and Dirichlet's theorem on diophantine approximation, Invent. Math, 177 (2009), 509-532.  doi: 10.1007/s00222-009-0186-6.  Google Scholar

[37]

N. Shah, Limiting distributions of curves under geodesic flow on hyperbolic manifolds, Duke Math. J., 148 (2009), 251-279.  doi: 10.1215/00127094-2009-026.  Google Scholar

[38]

R. Shi, Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space, arXiv: 1405.2067. Google Scholar

[39]

R. Shi, Expanding cone and applications to homogeneous dynamics, arXiv: 1510.05256. Google Scholar

[40]

A. N. Starkov, Dynamical Systems on Homogeneous Spaces, Translations of Mathematical Monographs, 190, American Mathematical Society, Providence, RI, 2000.  Google Scholar

[41]

S. Tabachnikov, Geometry and Billiards, Student Mathematical Library, 30, American Mathematical Society, Providence, RI, 2005.  Google Scholar

[42]

A. Zorich, Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499.  doi: 10.1017/S0143385797086215.  Google Scholar

[43]

A. Zorich, How do the leaves of a closed 1-form wind around a surface?, Amer. Math. Soc. Transl. Ser. 2, 197 (1999), 135-178.   Google Scholar

Figure 1.  The table $\mathscr{D}_{\lambda_0}$ and two types of invariant sets for the billiard flow
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Eaton lens and a parallel family of light rays
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The system of lenses $L(\Lambda,R)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  A translation surface $(M, \omega)$ in the space $\mathscr{M}^{dc}$ of double covers of tori.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The billiard flow on $\mathscr{S}_{\lambda}$ and the corresponding table $\mathscr{P}_\lambda$ for $\lambda_0 <\lambda< b$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The billiard flow on $\mathscr{S}_{\lambda}$ and the corresponding table $\mathscr{P}_\lambda$ for $b<\lambda< a$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  The surface $({M}_\lambda, \omega_\lambda)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  The system of lenses $F(\Lambda,R,\theta)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  The surface $M(\Lambda,R)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  The triangle $\triangle ABC$ and the line $x = 2sy+s^2$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Artur Avila, Carlos Matheus, Jean-Christophe Yoccoz. The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces. Journal of Modern Dynamics, 2019, 14: 21-54. doi: 10.3934/jmd.2019002
[2]

Vladimir Dragović, Milena Radnović. Pseudo-integrable billiards and arithmetic dynamics. Journal of Modern Dynamics, 2014, 8 (1) : 109-132. doi: 10.3934/jmd.2014.8.109
[3]

Rodolfo Gutiérrez-Romo. A family of quaternionic monodromy groups of the Kontsevich–Zorich cocycle. Journal of Modern Dynamics, 2019, 14: 227-242. doi: 10.3934/jmd.2019008
[4]

Benjamin Dozier. Equidistribution of saddle connections on translation surfaces. Journal of Modern Dynamics, 2019, 14: 87-120. doi: 10.3934/jmd.2019004
[5]

Giovanni Forni. A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle. Journal of Modern Dynamics, 2011, 5 (2) : 355-395. doi: 10.3934/jmd.2011.5.355
[6]

Eugene Gutkin. Insecure configurations in lattice translation surfaces, with applications to polygonal billiards. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 367-382. doi: 10.3934/dcds.2006.16.367
[7]

David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477
[8]

Alexander I. Bufetov. Hölder cocycles and ergodic integrals for translation flows on flat surfaces. Electronic Research Announcements, 2010, 17: 34-42. doi: 10.3934/era.2010.17.34
[9]

Gary Froyland, Simon Lloyd, Anthony Quas. A semi-invertible Oseledets Theorem with applications to transfer operator cocycles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3835-3860. doi: 10.3934/dcds.2013.33.3835
[10]

Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237
[11]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43
[12]

Yannan Liu, Linfen Cao. Lifespan theorem and gap lemma for the globally constrained Willmore flow. Communications on Pure & Applied Analysis, 2014, 13 (2) : 715-728. doi: 10.3934/cpaa.2014.13.715
[13]

Jon Fickenscher. A combinatorial proof of the Kontsevich-Zorich-Boissy classification of Rauzy classes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1983-2025. doi: 10.3934/dcds.2016.36.1983
[14]

Yuri Kifer. Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2687-2716. doi: 10.3934/dcds.2018113
[15]

Alex Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2377-2403. doi: 10.3934/dcds.2016.36.2377
[16]

Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247
[17]

Kariane Calta, John Smillie. Algebraically periodic translation surfaces. Journal of Modern Dynamics, 2008, 2 (2) : 209-248. doi: 10.3934/jmd.2008.2.209
[18]

Elismar R. Oliveira. Generic properties of Lagrangians on surfaces: The Kupka-Smale theorem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 551-569. doi: 10.3934/dcds.2008.21.551
[19]

Matilde Martínez, Shigenori Matsumoto, Alberto Verjovsky. Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem. Journal of Modern Dynamics, 2016, 10: 113-134. doi: 10.3934/jmd.2016.10.113
[20]

Zbigniew Gomolka, Boguslaw Twarog, Jacek Bartman. Improvement of image processing by using homogeneous neural networks with fractional derivatives theorem. Conference Publications, 2011, 2011 (Special) : 505-514. doi: 10.3934/proc.2011.2011.505

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (145)
  • HTML views (1265)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences