**Biographical Data**

**Education**:

B.A. University of Texas 1948

Ph.D. University of Virginia 1954

**Professional Experience:**

- 1987- Professor, The University of Texas, Austin
- 1967-87 Professor, Northwestern University
- 1963-67 Associate Professor, Northwestern University
- 1961-63 Assistant Professor, University of Chicago
- 1959-61 Associate Member, Institute for Advanced Study
- 1956-59 Assistant Professor, Purdue University
- 1954-55 Assistant Professor, Florida State University
*Visiting Positions* - 1992 Visitor, Mathematics Institute, University of Warwick
- 1990-91, 92-93 Visitor, Aspen Institute for Theoretical Physics
- 1989-90 Visitor, IBM J.T. Watson Research Center, White Plains, NY
- 1986-87 Invited Visitor, Institute des Hautes Etudes Scientifique, Paris
- 1985-86 (Winter) Visiting, University of California, La Jolla
- 1985 (Winter) Visiting, Instituto Matematica Pura y Applicada, Rio de Janeiro
- 1983 (Spring) Visiting Professor, Boston University
- 1982 (Fall) Visiting Professor, University of California, Berkeley
- 1974 (Fall) Stanislav Ulam Visiting Chair, University of Colorado, Boulder
- 1972-73 Associate Member, Institute des Hautes Etudes Scientifique, Paris
- 1971 Visiting, Instituto Matematica Pura y Applicada, Rio de Janeiro
- 1955-56 Visiting Lecturer, University of Wisconsin, Madison

**Professional Society
Memberships:**

- American Mathematical Society
- Mathematical Association of America
- National Association of Mathematicians
- American Association for the Advancement of Science

**Awards and Honors:**

- 1994-96 Nominating Committee of Math Section, AAAS
- 1990-92 Nominating Committee of the American Mathematical Society
- 1986-91 Editorial Board of Ergodic Theory and Dynamical Systems
- 1986-89 Member at Large, Council of the American Mathematical Society
- 1977 Stanislav Ulam Visiting Chair, University of Colorado
- 1971 Invited address, A.M.S. Meeting, Las Vegas
- 1970 Invited address, International Congress of Mathematicians, Nice
- 1959-61 National Science Foundation Fellowship

**Publications**:

- 2D continued fractions and positive matrices.
- Penrose, Ammann, and DA tilings are Cantor set fiber bundles.
- The universal templates of Ghrist, BAMS 35, No. 2, April, 1998, 145-156
- Classification of Denjoy continua, to appear in Topology and its applications,
- Pisot-Vijayarghavan numbers and positive matrices, Proceedings of the International Conference on Dynamical Systems and Chaos, vol. 1 World Scientific, 1995.
- Spaces that won't say no, International conference on dynamical systems, Montevideo 1995, Pitman Research notes in mathematics 362, 1996 236-246.
- (with Z. Coelho and W. Parry)A note on Liv\^sic's periodic point theorem, Contemporarty Mathematics, AMS, vol. 215, 223-230.
- Local contractions of compact metric spaces, Proc. Amer. Math. Soc., 5(1954), 652-654.
- A note on unstable homeomorphism, Proc. Amer. Math. Soc., 6(1955), 308-309.
- Local properties of open mappings, Duke Math. J., 22(1955), 339-346.
- Reduction of open mappings, Proc. Amer. Math. Soc., 7(1956), 312-318.
- The effect of maps upon the dimension of subsets of the domain, Proc. Amer. Math. Soc., 8(1957), 580-583.
- Local contractions and the size of metric spaces, Duke Math. J., 26(1959), 33-46.
- Lebesgue area of maps from Hausdorff spaces, Acta Math., 102(1959), 33-46.
- Open mappings and solenoids, Notices Amer. Math. Soc., 6(1959), 867.
- Lebesgue area zero, dimension and fine cyclic elements, Riv. Mat. Univ. Parma, 10(1959), 313-143.
- (with Frank Raymond) Examples of p-adic transformation groups, Bull. Amer. Math. Soc., 66(1960), 392-394.
- (with G. Bredon and F. Raymond) p-adic groups of transformations, Trans. Amer. Math. Soc., 99(19691), 488-498.
- (with F. Raymond) Examples of p-adic transformation groups, Ann. of Math., 78(1963), 92-106.
- A useful functor and three famous examples in topology, Trans. Amer. Math. Soc., 105(1963), 319-329.
- (with M. Mahowald) The stable homotopy of K(Z,n), Bol. Soc. Mat. Mexicana (1967), 22-28.
- Concerning the construction of certain zero-dimensional transformation groups, Trans. Amer. Math. Soc., 129(1967), 140-156.
- One dimensional non-wandering sets, Topology, 6(1967), 473-487.
- The zeta function of an attractor, Conference on the Topology of Manifolds, (John C. Hocking, Prindle, Weber and Schmidt, eds.) Boston (1968), 155-161.
- The action of compact non-Lie groups, Proceedings of the Conference on Transformation Groups, Springer, Berlin (1968), 366-369.
- The action of non-compact Lie groups, Proceedings of the Conference on Transformation Groups, Springer, Berlin (1968), 441-445.
- (with Michael Shub) Future stability is not generic, Proc. Amer. Math. Soc., 22(1969), 483-484.
- Expanding attractors, Colloque de Topologie Differentielle (Mont-Aiguall 1969), Universit\'e de Montpellier (1969), 79-89.
- Classification of 1-dimensional attractors, Global Analysis, Proceedings of the Symposium on Pure Math., 14(1970), 341-361.
- The DA maps of Smale and structural stability, Global Analysis, Proceedings of the Symposium on Pure Math., 14(1970), 329-334.
- The zeta function in global analysis, Global Analysis, Proceedings of the Symposium on Pure Math., 14(1970), 335-339.
- The structure of attractors, (address at the International Congress, Nice 1970), Actes, Congres Internationale Math., 2(1970), 947-951.
- Classification of shifts of finite type, Bull. Amer. Math. Soc., 77(1971), 439-443.
- Composition of contractions, Bol. Soc. Brasileira Mat., (2) 2(1971), 55-60.
- Classification of sub-shifts of finite type, Ann. of Math., 68(1973), 120-153. (Errata: 99[1974], 380-381).
- (with Clark Robinson) Finite stability is not generic, Dynamical Systems, Proceedings of a Symposium, University of Bahia, Salvador 1971, Academic Press (1973), 451-462.
- Expanding attractors, Publ. Inst. Hautes Etudes Sci., 43(1974), 169-203.
- (with Michael Shub) Entropy and Stability, Topology, 14(1975), 329-338.
- (with Denis Sullivan) The homology of attractors, Topology, 15(1976), 259-262.
- (with Clark Robinson) Classification of expanding attractors: an example, Topology, 15(1976), 321-323.
- (with S. Smale) The qualitative analysis of a difference equation of population growth, J. Math. Biology, 3(1976), 1-4.
- (with W. Parry) Block coding and a zeta function for finite Markov chains, London Math. Society, 25(1977), 483-495.
- The Lorenz Attractor, Turbulence Seminar, Springer Lecture Notes #615 (1977), 94-112.
- The structure of Lorenz attractors, Publ. I.H.E.S., 50(1979), 321-342.
- The bifurcation space of the Lorenz attractor, Annals of the New York Academy of Science, 316(1979), 393-399.
- (with John Guckenheimer) Structural stability of Lorenz Attractors, Publ. I.H.E.S., 50(1979), 307-320.
- (with John Franks) Anomalous Anosov flows, in Global Theory of Dynamical Systems, Springer Lecture Notes #819 (1980), 158-174.
- (with Joan Birman) Knotted periodic orbits I: Lorenz knots, Topology, 22(1983), 47-82.
- (with Joan Birman) Knotted periodic orbits II: Fibered knots, Low Dimensional Topology (ed. S. Lomonaco), Contemporary Mathematics 20, A.M.S. (1983), 1-60.
- Lorenz Knots are prime, Ergodic Theory and Dynamical Systems, 4(1983), 147-163.
- Attractors, strange and perverse, Proceedings of the 1981 Symposium on Differential Geometry and Differential Equations, Shanghai-Hefei, (1984), Science Press, Beijing, 473-497.
- (with John Franks) Entropy and knots, Trans. Amer. Math. Soc., 291(1985), 241-253.
- (with Phil Holmes) Knotted periodic orbits in suspension of Smale's horseshoe: Torus knots and bifurcations sequences, Arch. Rational Mech. Anal., 90(1985), 115-194.
- (with John Franks) Braids and the Jones-Conway Polynomial, Trans. Amer. Math. Soc., 303(1987), 97-108.
- (with Mike Shub and Dave Tischler) The Newtonian Graph of a complex polynomial, SIAM J. Math. Anal., (1) 19(1988), 246-256.
- The braid index of an algebraic link, Contemporary Math., Vol. 78, AMS (1988), 697-703.
- The geometric theory of dynamical systems, Workshop on dynamical systems, Pitman Research Notes in Math #221, (1990) 67-75.
- How big is the intersection of two thick Cantor sets? Continuum Theory and Dynamical Systems Contemporary Mathematics, AMS, 117(1991), 163-175.
- The braid index of generalized cables, Pacific J. of Mathematics, (2), 155(1992).
- Strong shift equivalence of matrices in GL(2,Z), Contemporary Mathematics, 135(1992), 445-451.
- (with Charles Tresser) Splitting sequences and positive entropy for Lorenz knots, Physica D 62(1993) 15-21.
- A new zeta function, natural for links, From Topology to Computation, (Smalefest volume)(1993) 270-278.
- (with Steve Kennedy and Mat Stafford) A new proof of the Cayley Hamilton theorem, Global Analysis in Modern Mathematics, Publish or Perish, (1993), 247-252.

**Invited Lectures and Service to Other
Universities;**

- Organizer, Dynamical Systems Special Session, A.M.S. Meeting in San Antonio (January), 1980
- Organizer, Midwestern Dynamical Systems Meetings Evanston (1968-1986).
- Organizer, Dynamical Systems Ergodic Theory Special Session, A.M.S. Summer Meeting Laramie (August).
- Organizer, Relationship between Dynamical Systems and Continua Theory, Arcata, 1989.
- Organizing Committee, Dynamic Days, Austin, January 3-6, 1990.
- Invited address, Symposium in Honor of Ren\'e Thom's 65th Birthday, Trieste, 1988.
- Fellowship Committee of National Academy of Science for NSF Graduate Fellowships, 1988, 1989, 1990
- Address, Dynamic Days, Houston, 1989.
- Colloquium, Rice, 1989.
- Invited speaker, ``Short Course on Chaotic Dynamical Systems", American Association for the Advancement of Science, Annual
- Meeting, New Orleans, February 15-20, 1990
- Organizer, Special session of AMS, invited lecture, Denton, Texas, 1990.
- Organized panel for Smalefest, July 1990.
- Colloquium, U. North Carolina, February 1991.
- Address, International Meeting for Dynamical Systems, Northwestern University, March, 1991.
- Seminar, at Session on Chaos, Aspen Center for Physics, June 1991.
- Invited speaker, AMS Special Session in Orono (Maine), August 1991.
- Invited speaker, AMS Special Session in Santa Barbara, November 1991.
- Colloquium, University of Warwick, February 1992.
- Topology Seminar, University of Warwick, March 1992.
- Dynamics Seminar, University of Warwick, March 1992.
- Invited address, University of South Wales, March 1992.
- Lecture to High School Teachers, Regional Geometry Institute, Park City, Utah, July 1992.
- Five Lectures, the Boston University Regional Geometry Institute, July 6-10, 1992.
- Lecture to High School Students, Boston, July 8, 1992.
- Invited Speaker, International workshop on Symbolic Dynamics, Mathematical Sciences Research Institute, Berkeley, November 5,1992.
- Invited Speaker, International Conference on Dynamical Systems and Chaos,Tokyo 1994.
- EPSCOR Speaker, ``Positive Matrices and Pisot-Vijayargharan Numbers, Montana State University, Bozeman.

**University and Department
Committees:**

Department Committee Service

- 1987-90 Subcommittee on Chairs
- 1988-90 Review Committee
- 1988-90 ASGSC
- 1993-94 Graduate Studies Committee
- 1993-94 Undergraduate Studies Committee
- 1993-94 Subcommittee on Chairs

**Significant Activities in Student
Affairs:**

- Lecture, Freshman Honors Seminar, March 19, 1991
- Designed a M375 course - undergraduate Dynamical Systems
- Supervised research projects for 2 undergraduates: 1991 Spring, W. Wesley Wilson, Michael Betz
- Lecture Pi Mu Epsilon, October 1991
- Supervised research for undergraduate 1993 Fall, Brian Marbensen
- Supervised research for undergraduate 1992, Abe Anderson,
- Josic Kresimir
- Lecture, Math Club, UT-Austin, November 1993
- Lecture, Dean's Scholars, April 1994
- Supervised research for undergraduate, Brian Martensen, 1993-94

**Courses Taught:**

- M427K Advanced Calculus for Applications I
- M427L Advanced Calculus for Applications II
- M392C Conference Course in Geometry and Topology
- M393C Beginning Dynamical Systems
- M393C Conference Course in Applied Mathematics
- M393C Dynamical Systems
- M427KH Honors Advanced Calculus for Applications
- M365C Introduction to Analysis
- M367K Introduction to Topology
- M367L Introduction to Topology, second semester

**Graduate Student Supervision and Committee
Participation:**

PhD Degrees Supervised

- Joel Gibbon, Northwestern University
- Carl Simon, Northwestern University
- Lewis Block, Northwestern University
- John Franke, Northwestern University
- Heinrick Köllmer, Northwestern University
- Lan Wen, 1986, Northwestern University
- Steve Kennedy, 1988, Northwestern University
- James Ward, 1989, Northwestern University
- Mike Sullivan, 1992, University of Texas
*Master's Degrees Supervised* - Emily B. Ragsdale, Florida State University, 1955
- Bill Kennedy, 1989
*Membership on PhD Committees* - Paul Sundheim
- Mike Sullivan
- Stanley Young, Electrical Engineering
- Lillian Xu, Electrical Engineering
- Phil Shafer
- Stathis Tompaidis

**Research in Progress:**

- Bifurcation of Lorenz-like systems
- Gaps in symbolic sequences
- Two dimensional continued fractions and a partial converse to the Perron-Frobenius theorem.

**Research Support:**

- NSF grants, 1963-1988
- NSF grants, 1990-1991, 1991-1992
- Texas Applied Research Grant, 1987-89
- Grant from Mathematical Sciences Research Institute to attend a workshop in Berkeley for a week.
- Grant from the University Institute for a sabbatical, Spring 1992.