On the ergodicity of partially hyperbolic systems

Abstract

Pugh and Shub have conjectured that essential accessibility implies ergodicity for a $C^2$, partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satisfied in particular by all partially hyperbolic systems with $1$-dimensional center bundle. We also obtain ergodicity results for $C^{1+\delta}$ partially hyperbolic systems.

  • [viana] Go to document F. Abdenur and M. Viana, "Flavors of partial hyperbolicity," , Preprint , 2008.
    @techreport{viana,
      author = {Abdenur, F. and Viana, M.},
      TITLE = {Flavors of partial hyperbolicity},
      TYPE = {Preprint},
      YEAR={2008},
      URL={http://www.impa.br/opencms/pt/publicacoes/index.html},
      }
  • [An] D. V. Anosov, "Geodesic flows on closed Riemannian manifolds of negative curvature," Trudy Mat. Inst. Steklov., vol. 90, p. 209, 1967.
    @article{An, MRKEY = {0224110},
      AUTHOR = {Anosov, D. V.},
      TITLE = {Geodesic flows on closed {R}iemannian manifolds of negative curvature},
      JOURNAL = {Trudy Mat. Inst. Steklov.},
      FJOURNAL = {Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova},
      NOTE = {In Russian},
      VOLUME = {90},
      YEAR = {1967},
      PAGES = {209},
      ISSN = {0371-9685},
      MRCLASS = {57.36 (28.00)},
      MRNUMBER = {36 \#7157},
      ZBLNUMBER = {0176.19101},
      MRREVIEWER = {M. Eisenberg},
      }
  • [AS] D. V. Anosov and J. G. Sinaui, "Certain smooth ergodic systems," Uspehi Mat. Nauk, vol. 22, iss. 5 (137), pp. 107-172, 1967.
    @article{AS, MRKEY = {0224771},
      AUTHOR = {Anosov, D. V. and Sina{\u\i},
      Ja. G.},
      TITLE = {Certain smooth ergodic systems},
      JOURNAL = {Uspehi Mat. Nauk},
      FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
      VOLUME = {22},
      NOTE = {In Russian},
      YEAR = {1967},
      NUMBER = {5 (137)},
      PAGES = {107--172},
      ISSN = {0042-1316},
      MRCLASS = {28.70},
      MRNUMBER = {37 \#370},
      MRREVIEWER = {M. Eisenberg},
      ZBLNUMBER={0177.42002},
      }
  • [Mishapp] M. Brin, "Ergodicity of the geodesic flow." Basel: Birkhäuser, 1995, p. viii.
    @incollection{Mishapp, MRKEY = {1377265},
      TITLE = {Ergodicity of the geodesic flow},
      AUTHOR = {Misha Brin},
      NOTE = {Appendix to {\em Lectures on spaces of nonpositive curvature},
      by W. Ballmann},
      SERIES = {DMV Seminar},
      NUMBER = {25},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {1995},
      PAGES = {viii+112},
      ISBN = {3-7643-5242-6},
      MRCLASS = {53C21 (58F17)},
      MRNUMBER = {97a:53053},
      ZBLNUMBER = {0834.53003},
      MRREVIEWER = {Boris Hasselblatt},
      }
  • [BP1] M. I. Brin and J. B. Pesin, "Partially hyperbolic dynamical systems," Izv. Akad. Nauk SSSR Ser. Mat., vol. 38, pp. 170-212, 1974.
    @article{BP1, MRKEY = {0343316},
      AUTHOR = {Brin, M. I. and Pesin, Ja. B.},
      TITLE = {Partially hyperbolic dynamical systems},
      JOURNAL = {Izv. Akad. Nauk {\rm SSSR} Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      NOTE = {In Russian, translated in {\it Math. {\rm USSR} Izv.} {\bf 8} (1974), 177--218},
      VOLUME = {38},
      YEAR = {1974},
      PAGES = {170--212},
      ISSN = {0373-2436},
      MRCLASS = {58F15 (28A65)},
      MRNUMBER = {49 \#8058},
      ZBLNUMBER = {0304.58017},
      MRREVIEWER = {A. Morimoto},
      }
  • [BW] Go to document K. Burns and A. Wilkinson, "Better center bunching," , Preprint , 2005.
    @techreport{BW,
      author = {Burns, K. and Wilkinson, A.},
      TITLE = {Better center bunching},
      TYPE = {Preprint},
      YEAR={2005},
      URL={http://www.math.northwestern.edu/~wilkinso/papers/papers.html},
      }
  • [BWhold] Go to document K. Burns and A. Wilkinson, "A note on unstable holonomy between centers," , Preprint , 2006.
    @techreport{BWhold,
      author = {Burns, K. and Wilkinson, A.},
      TITLE = {A note on unstable holonomy between centers},
      TYPE = {Preprint},
      YEAR={2006},
      URL={http://www.math.northwestern.edu/~wilkinso/papers/papers.html},
      }
  • [BWnondc] Go to document K. Burns and A. Wilkinson, "Dynamical coherence and center bunching," , Preprint , 2006.
    @techreport{BWnondc,
      author = {Burns, K. and Wilkinson, A.},
      TITLE = {Dynamical coherence and center bunching},
      TYPE = {Preprint},
      YEAR={2006},
      URL={http://www.math.northwestern.edu/~wilkinso/papers/papers.html},
     }
  • [BPSW] K. Burns, C. Pugh, M. Shub, and A. Wilkinson, "Recent results about stable ergodicity," in Smooth Ergodic Theory and its Applications, Katok, A., de la Llave, R., Pesin, Y., and Weiss, H., Eds., Amer. Math. Soc., 2001, pp. 327-366.
    @incollection{BPSW, MRKEY = {1858538},
      AUTHOR = {Burns, Keith and Pugh, Charles and Shub, Michael and Wilkinson, Amie},
      EDITOR = {Katok, Anatole and de la Llave, Rafael and Pesin, Yakov and Weiss, Howard},
      TITLE = {Recent results about stable ergodicity},
      BOOKTITLE = {Smooth Ergodic Theory and its Applications},
      VENUE = {Seattle, 1999},
      SERIES = {Proc. Sympos. Pure Math.},
      NUMBER = {69},
      PAGES = {327--366},
      PUBLISHER = {Amer. Math. Soc.},
      INVADDRESS = {Providence, RI},
      YEAR = {2001},
      MRCLASS = {37D30 (37C20 37C40)},
      MRNUMBER = {2002m:37042},
      ZBLNUMBER = {1012.37019},
      MRREVIEWER = {Pei Dong Liu},
      }
  • [D] D. Dolgopyat, "On differentiability of SRB states for partially hyperbolic systems," Invent. Math., vol. 155, iss. 2, pp. 389-449, 2004.
    @article{D, MRKEY = {2031432},
      AUTHOR = {Dolgopyat, Dmitry},
      TITLE = {On differentiability of {SRB} states for partially hyperbolic systems},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {155},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {389--449},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {37D30 (37C40)},
      MRNUMBER = {2005h:37070},
      ZBLNUMBER = {1059.37021},
      MRREVIEWER = {Lorenzo J. D{\'ı}az},
      }
  • [DW] D. Dolgopyat and A. Wilkinson, "Stable accessibility is $C^1$ dense," Astérisque, iss. 287, p. xvii, 33-60, 2003.
    @article{DW, MRKEY = {2039999},
      AUTHOR = {Dolgopyat, Dmitry and Wilkinson, Amie},
      TITLE = {Stable accessibility is {$C\sp 1$} dense},
      INVNOTE = {Geometric methods in dynamics. II},
      JOURNAL = {Astérisque},
      FJOURNAL = {Astérisque},
      NUMBER = {287},
      YEAR = {2003},
      PAGES = {xvii, 33--60},
      ISSN = {0303-1179},
      MRCLASS = {37D30 (37C20 37J10)},
      MRNUMBER = {2005d:37058},
      MRREVIEWER = {Lorenzo J. D{\'ı}az},
      ZBLNUMBER = {02066298},
      }
  • [GPS] M. Grayson, C. Pugh, and M. Shub, "Stably ergodic diffeomorphisms," Ann. of Math., vol. 140, iss. 2, pp. 295-329, 1994.
    @article{GPS, MRKEY = {1298715},
      AUTHOR = {Grayson, Matthew and Pugh, Charles and Shub, Michael},
      TITLE = {Stably ergodic diffeomorphisms},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {140},
      YEAR = {1994},
      NUMBER = {2},
      PAGES = {295--329},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {58F11 (58F17 58F30)},
      MRNUMBER = {95g:58128},
      ZBLNUMBER = {0824.58032},
      MRREVIEWER = {Remo Badii},
      }
  • [hass] B. Hasselblatt, "Periodic bunching and invariant foliations," Math. Res. Lett., vol. 1, iss. 5, pp. 597-600, 1994.
    @article{hass, MRKEY = {1295553},
      AUTHOR = {Hasselblatt, Boris},
      TITLE = {Periodic bunching and invariant foliations},
      JOURNAL = {Math. Res. Lett.},
      FJOURNAL = {Mathematical Research Letters},
      VOLUME = {1},
      YEAR = {1994},
      NUMBER = {5},
      PAGES = {597--600},
      ISSN = {1073-2780},
      MRCLASS = {58F15 (58F17 58F18)},
      MRNUMBER = {95h:58097},
      ZBLNUMBER = {0845.58043},
      MRREVIEWER = {Gabriel P. Paternain},
      }
  • [HaW] B. Hasselblatt and A. Wilkinson, "Prevalence of non-Lipschitz Anosov foliations," Ergodic Theory Dynam. Systems, vol. 19, iss. 3, pp. 643-656, 1999.
    @article{HaW, MRKEY = {1695913},
      AUTHOR = {Hasselblatt, Boris and Wilkinson, Amie},
      TITLE = {Prevalence of non-{L}ipschitz {A}nosov foliations},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {19},
      YEAR = {1999},
      NUMBER = {3},
      PAGES = {643--656},
      ISSN = {0143-3857},
      MRCLASS = {37D20 (37C20 37D05 37D30 37J05)},
      MRNUMBER = {2000f:37035},
      ZBLNUMBER = {1069.37031},
      MRREVIEWER = {Viorel Ni{\c{t}}ic{\u{a}}},
      }
  • [HP] M. Hirayama and Y. Pesin, "Non-absolutely continuous foliations," Israel J. Math., vol. 160, pp. 173-187, 2007.
    @article{HP, MRKEY = {2342495},
      AUTHOR = {Hirayama, Michihiro and Pesin, Yakov},
      TITLE = {Non-absolutely continuous foliations},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {160},
      YEAR = {2007},
      PAGES = {173--187},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {37Dxx},
      MRNUMBER = {2342495},
      ZBLNUMBER = {1137.37014},
      }
  • [hopf] E. Hopf, "Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung," Ber. Verh. Sächs. Akad. Wiss. Leipzig, vol. 91, pp. 261-304, 1939.
    @article{hopf, MRKEY = {0001464},
      AUTHOR = {Hopf, Eberhard},
      TITLE = {Statistik der geodätischen {L}inien in {M}annigfaltigkeiten negativer {K}rümmung},
      JOURNAL = {Ber. Verh. Sächs. Akad. Wiss. Leipzig},
      VOLUME = {91},
      YEAR = {1939},
      PAGES = {261--304},
      MRCLASS = {46.3X},
      MRNUMBER = {1,243a},
      ZBLNUMBER = {0024.08003},
      JFMNUMBER = {65.1413.02},
      MRREVIEWER = {G. A. Hedlund},
      }
  • [HHU] F. Rodríguez Hertz, M. A. Rodríguez Hertz, and R. Ures, "Accessibility and stable ergodicity for parially hyperbolic diffeomorphisms with 1D-center bundle," Invent. Math., vol. 172, pp. 353-381, 2008.
    @article{HHU,
      author = {Rodr\'iguez Hertz, F. and Rodr\'iguez Hertz, M. A. and Ures, R.},
      TITLE = {Accessibility and stable ergodicity for parially hyperbolic diffeomorphisms with 1D-center bundle},
      JOURNAL={Invent. Math.},
      VOLUME={172},
      YEAR={2008},
      PAGES={353--381},
      MRNUMBER={2390288},
      ZBLNUMBER={1136.37020},
      }
  • [HPS] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, New York: Springer-Verlag, 1977.
    @book{HPS, MRKEY = {0501173},
      AUTHOR = {Hirsch, M. W. and Pugh, C. C. and Shub, M.},
      TITLE = {Invariant manifolds},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {583},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {ii+149},
      MRCLASS = {58F15 (58F10)},
      MRNUMBER = {58 \#18595},
      ZBLNUMBER = {0355.58009},
      MRREVIEWER = {M. C. Irwin},
      }
  • [KH] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge: Cambridge Univ. Press, 1995.
    @book{KH, MRKEY = {1326374},
      AUTHOR = {Katok, Anatole and Hasselblatt, Boris},
      TITLE = {Introduction to the Modern Theory of Dynamical Systems},
      SERIES = {Encycl. Math. Appl.},
      NUMBER = {54},
      INVNOTE = {With a supplementary chapter by Katok and Leonardo Mendoza},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1995},
      PAGES = {xviii+802},
      ISBN = {0-521-34187-6},
      MRCLASS = {58Fxx (34Cxx 34Dxx 58-01 58F11 58F15)},
      MRNUMBER = {96c:58055},
      ZBLNUMBER = {0878.58020},
      MRREVIEWER = {Edoh Amiran},
      }
  • [Mi] J. Milnor, "Fubini foiled: Katok’s paradoxical example in measure theory," Math. Intelligencer, vol. 19, iss. 2, pp. 30-32, 1997.
    @article{Mi, MRKEY = {1457445},
      AUTHOR = {Milnor, John},
      TITLE = {Fubini foiled: {K}atok's paradoxical example in measure theory},
      JOURNAL = {Math. Intelligencer},
      FJOURNAL = {The Mathematical Intelligencer},
      VOLUME = {19},
      YEAR = {1997},
      NUMBER = {2},
      PAGES = {30--32},
      ISSN = {0343-6993},
      CODEN = {MAINDC},
      MRCLASS = {28A35},
      MRNUMBER = {98g:28005},
      ZBLNUMBER = {0883.28004},
      MRREVIEWER = {Nathaniel F. G. Martin},
      }
  • [P] Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zürich: European Mathematical Society, 2004.
    @book{P, MRKEY = {2068774},
      AUTHOR = {Pesin, Yakov B.},
      TITLE = {Lectures on Partial Hyperbolicity and Stable Ergodicity},
      SERIES = {Zur. Lect. Adv. Math.},
      PUBLISHER = {European Mathematical Society},
      ADDRESS = {Zürich},
      YEAR = {2004},
      PAGES = {vi+122},
      ISBN = {3-03719-003-5},
      MRCLASS = {37D30 (37C40 37D10)},
      MRNUMBER = {2005j:37039},
      ZBLNUMBER = {1098.37024},
      MRREVIEWER = {Amie Wilkinson},
      }
  • [PSanosovactions] C. Pugh and M. Shub, "Ergodicity of Anosov actions," Invent. Math., vol. 15, pp. 1-23, 1972.
    @article{PSanosovactions, MRKEY = {0295390},
      AUTHOR = {Pugh, Charles and Shub, Michael},
      TITLE = {Ergodicity of {A}nosov actions},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {15},
      YEAR = {1972},
      PAGES = {1--23},
      ISSN = {0020-9910},
      MRCLASS = {58F10},
      MRNUMBER = {45 \#4456},
      ZBLNUMBER = {0236.58007},
      MRREVIEWER = {Karl Sigmund},
      }
  • [PSa] C. Pugh and M. Shub, "Stably ergodic dynamical systems and partial hyperbolicity," J. Complexity, vol. 13, iss. 1, pp. 125-179, 1997.
    @article{PSa, MRKEY = {1449765},
      AUTHOR = {Pugh, Charles and Shub, Michael},
      TITLE = {Stably ergodic dynamical systems and partial hyperbolicity},
      JOURNAL = {J. Complexity},
      FJOURNAL = {Journal of Complexity},
      VOLUME = {13},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {125--179},
      ISSN = {0885-064X},
      MRCLASS = {58F11 (58F15)},
      MRNUMBER = {98e:58110},
      ZBLNUMBER = {0883.58025},
      MRREVIEWER = {Christian Bonatti},
      }
  • [PS] C. Pugh and M. Shub, "Stable ergodicity and julienne quasi-conformality," J. Eur. Math. Soc. (JEMS), vol. 2, iss. 1, pp. 1-52, 2000.
    @article{PS, MRKEY = {1750453},
      AUTHOR = {Pugh, Charles and Shub, Michael},
      TITLE = {Stable ergodicity and julienne quasi-conformality},
      JOURNAL = {J. Eur. Math. Soc. {\rm (JEMS)}},
      FJOURNAL = {Journal of the European Mathematical Society $($JEMS$)$},
      VOLUME = {2},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {1--52},
      ISSN = {1435-9855},
      MRCLASS = {37D30 (37A25 37C20 37C40)},
      MRNUMBER = {2003j:37046},
      ZBLNUMBER = {0964.37017},
      MRREVIEWER = {Boris Hasselblatt},
      }
  • [PSW] C. Pugh, M. Shub, and A. Wilkinson, "Hölder foliations," Duke Math. J., vol. 86, iss. 3, pp. 517-546, 1997.
    @article{PSW, MRKEY = {MR1432307},
      AUTHOR = {Pugh, Charles and Shub, Michael and Wilkinson, Amie},
      TITLE = {Hölder foliations},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {86},
      YEAR = {1997},
      NUMBER = {3},
      PAGES = {517--546},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {58F18 (58F15)},
      MRNUMBER = {97m:58155},
      MRREVIEWER = {Boris Hasselblatt},
      ZBLNUMBER = {0877.58045},
      }
  • [PSWc] C. Pugh, M. Shub, and A. Wilkinson, "Correction to \citePSW," Duke Math. J., vol. 105, iss. 1, pp. 105-106, 2000.
    @article{PSWc, MRKEY = {1788044},
      AUTHOR = {Pugh, Charles and Shub, Michael and Wilkinson, Amie},
      TITLE = {Correction to \cite{PSW}},
      FTITLE = {Correction to ``{H}ölder foliations'' [{D}uke {M}ath. {J}. {\bf 86} (1997), no. 3, 517--546; {MR}1432307 (97m:58155)]},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {105},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {105--106},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {37D30},
      MRNUMBER = {2001h:37057},
      }
  • [ss] J. Schmeling and R. Siegmund-Schultze, "Hölder continuity of the holonomy maps for hyperbolic basic sets, I," in Ergodic Theory and Related Topics, III, Krengel, U., Richter, K., and Warstat, V., Eds., New York: Springer-Verlag, 1992, pp. 174-191.
    @incollection{ss, MRKEY = {1179182},
      AUTHOR = {Schmeling, J. and Siegmund-Schultze, Ra.},
      EDITOR = {Krengel, U. and Richter, K. and Warstat, V.},
      TITLE = {Hölder continuity of the holonomy maps for hyperbolic basic sets, {I}},
      BOOKTITLE = {Ergodic Theory and Related Topics, {\rm III}},
      VENUE = {Güstrow, 1990},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {1514},
      PAGES = {174--191},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1992},
      MRCLASS = {58F15},
      MRNUMBER = {93j:58104},
      MRREVIEWER = {Hu Yi Hu},
      ZBLNUMBER = {0766.58043},
      }
  • [SW1] M. Shub and A. Wilkinson, "Stably ergodic approximation: Two examples," Ergodic Theory Dynam. Systems, vol. 20, iss. 3, pp. 875-893, 2000.
    @article{SW1, MRKEY = {1764933},
      AUTHOR = {Shub, Michael and Wilkinson, Amie},
      TITLE = {Stably ergodic approximation: {T}wo examples},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {20},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {875--893},
      ISSN = {0143-3857},
      MRCLASS = {37D20 (37A25 37C20)},
      MRNUMBER = {2001d:37032},
      ZBLNUMBER = {0970.37022},
      MRREVIEWER = {Nikolai Chernov},
      }
  • [SW2] M. Shub and A. Wilkinson, "Pathological foliations and removable zero exponents," Invent. Math., vol. 139, iss. 3, pp. 495-508, 2000.
    @article{SW2, MRKEY = {1738057},
      AUTHOR = {Shub, Michael and Wilkinson, Amie},
      TITLE = {Pathological foliations and removable zero exponents},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {139},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {495--508},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {37D25 (28A50)},
      MRNUMBER = {2001c:37030},
      ZBLNUMBER = {0976.37013},
      MRREVIEWER = {Boris Hasselblatt},
      }
  • [amiethesis] A. Wilkinson, "Stable ergodicity of the time-one map of a geodesic flow," Ergodic Theory Dynamical Systems, vol. 18, iss. 6, pp. 1545-1587, 1998.
    @article{amiethesis, MRKEY = {MR1658611},
      AUTHOR = {Wilkinson, Amie},
      TITLE = {Stable ergodicity of the time-one map of a geodesic flow},
      JOURNAL = {Ergodic Theory Dynamical Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {18},
      YEAR = {1998},
      NUMBER = {6},
      PAGES = {1545--1587},
      ISSN = {0143-3857},
      MRCLASS = {58F11 (58F15 58F17)},
      MRNUMBER = {99m:58129},
      MRREVIEWER = {Edoh Amiran},
      ZBLNUMBER = {0915.58079},
      }

Authors

Keith Burns

Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston, IL 60208-2730
United States

Amie Wilkinson

Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston, IL 60208-2730
United States