Ian Putnam: Research

"If we knew what it was we were doing,
it would not be called research, would it?"
- Albert Einstein.

Description of research interests

My research concerns dynamical systems and operator algebras and the close connections between these fields. In particular, I am interested in topological dynamical systems, particularly minimal systems on the Cantor set, systems arising from aperiodic tilings, symbolic dynamical systems and the hyperbolic systems associated with Smale's program for Axiom A dynamics. In operator algebras, I am mainly interested in C*-algebras, groupoid C*-algebras, K-theory for C*-algebras and other aspects of Alain Connes' program in noncommutative geometry.

From a topological dynamical system, there are several constructions of C*-algebras. This provides a fascinating link between the two subjects. On the one hand, it produces interesting examples of C*-algebras. Some, such as the Cuntz-Krieger algebras constructed from subshifts of finite type and the C*-algebras associated with irrational rotations of the circle, have had a fundamental impact on the subject. It is insightful to see how dynamical properties manifest themselves in the C*-algebras. On the other hand, techniques which are used in the study of C*-algebras can often give new information about the dynamics. As a specific example, starting from a dynamical system, constructing the C*-algebra and then taking its K-theory provides an interesting invariant, rather like homology of spaces. In my work on minimal dynamical systems on Cantor sets, I (and my co-workers) used K-theory to obtain a complete invariant for orbit equivalence for many systems. In addition, our work brought many C*-algebra ideas (such as Bratteli diagrams) to effective use in dynamics.

List of Publications

(Indicated papers may be obtained as PDF and/or post-script files)

Binary factors of shifts of finite type, , PDF

With R. Trevino, Bratteli diagrams, translation flows and their C*-algebras, , PDF

An excision theorem for the K-theory of C*-algebras, with applications to groupoid C*-algebras, Munster Mathematics Journal 14 (2021), 349-402 PDF

With R.J. Deeley and K.R. Strung, Non-homogeneous extensions of Cantor minimal systems, Proc. A.M.S. 149 (2021), 2081-2089 PDF

With R.J. Deeley and K.R. Strung, Minimal homeomorphism and topological K-theory, to appear, Groups, Geometry and Dynamics. PDF

With R.J. Deeley and K.R. Strung, Classifiable C*-algebras from minimal Z-actions and their orbit-breaking subalgebras, to appear, Mathematische Annalen, PDF

With T. Giordano and C.F. Skau, Z^d-odometers and cohomology, PDFGroups, Geometry, and Dynamics, 13 (2019), 909–938.

Some classifiable groupoid C*-algebras with prescribed K-theory, Math. Ann., 370 (2018), 1361-1387. PDF

With R.J. Deeley and K.R. Strung, Constructing minimal homeomorphisms of point-like spaces and a dynamical presentation of the Jiang-Su algebra, Journal fur die Reine und Angewandte Mathematik 742 (2018), 241-261. PDF

With M. Amini and S. Saeidi Gholikandi, Order on the homology groups of Smale spaces, Pacific Journal of Mathematics, 288(2017), 219-242, PDF

With N.D. Burke, Markov partitions and homology for n/m-solenoids, Ergodic Theory and Dynamical Systems, 37(2017), 716-738, PDF

With M. Amini and S. Saeidi Gholikandi, Homology for one-dimensional solenoids, Mathematica Scandinavica, 121 (2017), 219-242, PDF

With A. Julien, Spectral triples for subshifts, Journal of Functional Analysis, 270 (2016), 1031-1063, PDF

With B. Killough, Trace asymptotics for C*-algebras from Smale spaces, Proceedings of the American Mathematical Society, 143 (2015), 317-325, PDF

With B. Killough, Bowen measure from heteroclinic points, Can. J. Math., 64 (2012), 1341-1358. PDF

With B. Killough, Ring and module structures on dimension groups associated with a shift of finite type, Ergod. Th. and Dynam. Sys., 32 (2012), 1370-1399. PDF

With J. Kaminker and M.F. Whittaker, K-theoretic duality for hyperbolic dynamical systems, Journal fur die Reine und Angewandte Mathematik, 730(2017), 257-288. PDF

A homology theory for Smale spaces, Memoirs A.M.S., 232 (2014), Online version, Older PDF version

With A.L. Carey, J. Phillips and A. Rennie, Type III KMS states on a class of C*-algebras containing O_n and Q_N and their modular index, in Perspectives on Noncommutative Geometry, M. Khalkali and G. Yu, editors, Fields Institute Comminucations, American Mathematical Society, Providence, R.I., 2011. PDF

With A.L. Carey, J. Phillips and A. Rennie, Families of Type III KMS states on a class of C*-algebras containing O_n and Q_N, J. Func. Anal., 260 (2011), 1637-1681. PDF

With T. Giordano, H. Matui and C.F. Skau, Orbit equivalence for Cantor minimal Z^d-systems, Invent. Math., 179(2010), 119-158. PDF

Orbit equivalence of Cantor minimal systems: A survey and a new proof, Expos. Math., 28(2010), 101-131. PDF

Non-commutative methods for the K-theory of C*-algebras of aperiodic patterns from cut-and-project systems, Comm. Math. Phys., 294(2010), 703-729. PDF

A homology theory for Smale spaces: a summary (Research announcement), pre-print, PDF

With T. Giordano and C.F. Skau, Cocycles for Cantor minimal Z^d-systems, Internat. J. Math., 20(2009), 1107-1135. PDF

With T. Giordano, H. Matui and C.F. Skau, The absorption theorem for affable equivalence relations, Ergod. Th. and Dynam. Sys., 28(2008), 1509-1531. , PDF

With T. Giordano, H. Matui and C.F. Skau, Orbit equivalence for Cantor minimal Z^2-actions,Journal A.M.S., 21(2008), 863-892. PDF

With J. Kellendonk, The Ruelle-Sullivan map for actions of R^d, Math. Ann. 334(2006), 693-711. PDF

With T. Giordano and C.F. Skau, The orbit structure of Cantor minimal Z^2-actions, preprint, PDF or PS

Lifting factor maps to resolving maps, Israel J. Math. 146(2005), 253-280. PDF

With N. Brownlowe, N. S. Larsen and I. Raeburn, Subquotients of Hecke C*-algebras, Ergod. Th. and Dynam. Sys. 25(2005), 1503-1520. PDF

With T. Giordano and C. F. Skau,, Affable equivalence relations and orbit structure of Cantor dynamical systems, Ergod. Th. and Dynam. Sys., 24 (2004), 441-476. PDF

With J. Kaminker, A proof of the gap labeling conjecture, Mich. Math. J., 51(2003), 537-546. PDF or PS

With N. S. Larsen and I. Raeburn, The two-prime analogue of the Hecke C*-algebra of Bost and Connes, Indiana U. Math. J., 51 (2002), 171-186. PDF or PS

With T. Giordano and C.F. Skau, K-theory and asymptotic index for certain almost one-to-one factors , Math. Scand., 89(2001), 297-319. PDF or PS

With J. Kellendonk, Tilings, C*-algebras and K-theory in Directions in Mathematical Quasicrystals, M. Baake and R.V. Moody, editors, CRM Monograph Series, American Mathematical Society, Providence, R.I., 2000.

The ordered K-theory of C*-algebras associated with substitution tilings, Comm. Math. Phys., 214 (2000), 593-605. PDF or PS

The functoriality of the C*-algebras associated with hyperbolic dynamical systems, J. London Math. Soc., (2) 62 (2000), 873-884. PDF or PS

With J. Spielberg, The structure of C*-algebras associated with hyperbolic dynamical systems, J. Functional Analysis, 163(1999), 279-299. PDF or PS

With T. Giordano and C.F. Skau, Full groups of Cantor minimal systems, Israel J. Math., 111(1999), 285-320. PDF or PS

On the K-theory of C*-algebras of principal groupoids, Rocky Mountain J. Math. 28(1998), 1483-1518.

With J. Anderson, Topological invariants for substitution tilings and their C*-algebras , Ergodic Th. and Dynam. Sys. 18(1998), 509-537.

With J. Kaminker and J. Spielberg, Operator algebras and hyperbolic dynamics, Operator Algebras and Quantum Field Theory, S. Doplicher, R. Longo, J.E.Roberts and L. Zsido, Eds., International Press, 1997.PDF or PS

With J. Kaminker, K-theoretic duality for shifts of finite type, Comm. Math. Phys. 187(1997), 509-522. PDF or PS

An excision theorem for the K-theory of C*-algebras , J. Operator Th. 38(1997), 151-171. PDF or PS

A survey of recent K-theoretic invariants for dynamical systems, The Dynamics of Zd-actions, M. Pollicott and K. Schmidt, ed., Cambridge University Press, 1996.

C*-algebras from Smale spaces, Canad. J. Math 48(1996), 175-195.PDF or PS

With T. Giordano and C.F. Skau, Topological orbit equivalence and C*-crossed products, J. Reine Angew. Math. 469(1995), 51-111.

Topological orbit equivalence, in Operator Algebras, Mathematical Physics and Low Dimensional Topology, AK Peters Publishing, 1993.

C*-algebras arising from interval exchange transformations, J. Operator Th. 27(1992), 231-250.

With R.H. Herman and C.F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics , International J. Math. 6(1992), 827-864.

With P.S. Muhly and J. Xia, On the K-theory of some C*-algebras of Toeplitz and singular integral operators , J. of Functional Analysis110(1992), 161-225.

The invertible elements are dense in the irrational rotation C*-algebras, J. Reine Angew. Math. 410(1990), 160-166.

On the topological stable rank of certain transformation group C*-algebras, Ergodic Th. and Dynam. Sys. 10(1990), 197-207.

With P.S. Muhly and J. Xia, Contributions to the K-theory of C*-algebras of Toeplitz and singular integral operators, Bull. Amer. Math. Soc. 21(1) (1989),47-50.

The C*-algebras associated with minimal homeomorphisms of the Cantor set, Pacific J. Math.136(2) (1989), 329-353.

With M. Rordam, The maximum unitary rank of some C*-algebras, Math. Scand. 63(1988), 297-304.

Strong Morita equivalence for the Denjoy C*-algebras , Can. Math. Bull. 31(4) (1988), 439-447.

With K. Schmidt and C. Skau, C*-algebras associated with Denjoy homeomorphisms of the circle , J. Operator Th. 16(1986), 99-126.

Unpublished items

Lecture Notes from the Winter School in Operator Algebras, Kyoto, 2011, PDF This version was produced December 29, 2011. This material is subsumed by the Lecture Notes on Cantor minimal dynamics above.

Hyperbolic dynamics and generalized Cuntz-Krieger algebras, Lecture Notes from the Summer School in Operator Algeras, Odense, 1996.

Presentations at conferences and workshops

Spectral triples for subshifts This was from a talk at the special session in honour of the 90th birthday of Richard V. Kadison at the 2015 Joint Mathematics Meetings in San Antonio, Texas. The audience was researchers in operator algebras

A homology theory for Smale spaces This was from a talk at the Workshop on Groups, Dynamics and C*-algebras held at Wilhelms-Westfalische Universitat, Muenster in August 2013. The audience was researchers in operator algebras

Aperiodic order with a little topology and dynamical systemsThis was from a colloquium talk at the University of Nevada, Reno, March 31, 2011.

Some interactions between operator algebras and dynamical systems This was given at the Pingree Park workshop in August 2010. The audience was researchers in dynamical systems.

C*-algebras for projection method tilings This was given at GPOTS at Denver University in June 2010 and also at the Workshop on Selected Topics in Noncommutative Geometry at the University of Victoria in June 2010. The audiences were researchers in C*-algebras.

A homology theory for basic sets This was given at the Northwest Dynamics Symposium held at the University of Victoria, August, 2008. The audience was researchers in dynamical systems.

A brief survey of tiling cohomology This was a short talk at the workshop on "Combinatorial and computational aspects of tilings" held at Imperial College London, July, 2008. The audience was researchers in various aspects of tilings.

Orbit equivalence for Cantor minimal systems A plenary talk at the 2008 Spring Topology and Dynamical Systems Conference, Milwaukee, March 2008.

C*-algebras and topological dynamics: From dynamical systems to operator algebras and Topological orbit equivalence Two colloquium presentations at Wayne State University, February, 2008.

C*-algebras and tilings. This is a two-lecture mini-series given at the workshop "Aperodic order: new ideas and old problems revisited" held at C.I.R.M., Luminy, France in September, 2007. The intended audience was mainly graduate students, post-doctoral fellows and other researchers having little knowledge of C*-algebras.

C*-algebras for hyperbolic dynamical systems. This was given at the Candian Operator Symposium at the University of Guelph and also at the Barcelona Workshop on C*-algebras in June 2007. The audience was researchers working in C*-algebras.