Ian Putnam: Research
"If we knew what it was we were doing,
it would not be called research, would it?"
 Albert Einstein.
Contents:
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, Ktheory 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 CuntzKrieger 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 Ktheory provides an interesting invariant,
rather like homology of spaces. In my work on minimal dynamical systems on Cantor sets, I (and my coworkers) used
Ktheory 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.
(Indicated papers may be obtained as PDF
and/or postscript 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 Ktheory of C*algebras, with
applications to groupoid C*algebras, Munster Mathematics Journal 14 (2021), 349402
PDF
 With R.J. Deeley and K.R. Strung, Nonhomogeneous extensions of Cantor minimal systems,
Proc. A.M.S. 149 (2021), 20812089
PDF
 With R.J. Deeley and K.R. Strung, Minimal homeomorphism and topological Ktheory, to appear,
Groups, Geometry and Dynamics.
PDF
 With R.J. Deeley and K.R. Strung, Classifiable C*algebras from minimal Zactions and their orbitbreaking subalgebras,
to appear, Mathematische Annalen,
PDF
 With T. Giordano and C.F. Skau, Z^dodometers and cohomology,
PDFGroups, Geometry, and Dynamics, 13 (2019), 909–938.
 Some classifiable groupoid C*algebras with prescribed Ktheory,
Math. Ann., 370 (2018), 13611387.
PDF
 With R.J. Deeley and K.R. Strung, Constructing minimal homeomorphisms of pointlike spaces and
a dynamical presentation of the JiangSu algebra, Journal fur
die Reine und Angewandte Mathematik 742 (2018), 241261.
PDF
 With M. Amini and S. Saeidi Gholikandi, Order on the homology groups of Smale
spaces, Pacific Journal of Mathematics, 288(2017), 219242,
PDF
 With N.D. Burke, Markov partitions and homology for n/msolenoids,
Ergodic Theory and Dynamical Systems, 37(2017), 716738,
PDF
 With M. Amini and S. Saeidi Gholikandi, Homology for onedimensional solenoids,
Mathematica Scandinavica, 121 (2017), 219242,
PDF
 With A. Julien, Spectral triples for subshifts, Journal of
Functional Analysis, 270 (2016), 10311063,
PDF
 With B. Killough, Trace asymptotics for C*algebras from Smale spaces,
Proceedings of the American Mathematical Society, 143
(2015), 317325,
PDF
 With B. Killough, Bowen measure from heteroclinic points, Can. J. Math., 64 (2012), 13411358.
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), 13701399.
PDF
 With J. Kaminker and M.F. Whittaker, Ktheoretic duality for hyperbolic dynamical systems, Journal fur
die Reine und Angewandte Mathematik, 730(2017), 257288.
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), 16371681.
PDF
 With T. Giordano, H. Matui and C.F. Skau, Orbit
equivalence for
Cantor minimal Z^dsystems, Invent. Math., 179(2010),
119158. PDF
 Orbit equivalence of Cantor minimal systems: A survey
and a new
proof, Expos. Math., 28(2010), 101131.
PDF
 Noncommutative methods for the Ktheory of C*algebras
of aperiodic patterns from cutandproject systems, Comm. Math.
Phys., 294(2010), 703729.
PDF
 A homology theory for Smale spaces: a summary
(Research
announcement), preprint,
PDF
 With T. Giordano and C.F. Skau, Cocycles
for Cantor minimal
Z^dsystems,
Internat. J. Math., 20(2009), 11071135. PDF
 With T. Giordano, H. Matui and C.F. Skau, The
absorption
theorem for
affable equivalence relations, Ergod. Th. and Dynam. Sys.,
28(2008), 15091531.
, PDF
 With T. Giordano, H. Matui and C.F. Skau, Orbit
equivalence
for Cantor minimal Z^2actions,Journal A.M.S.,
21(2008), 863892. PDF
 With J. Kellendonk, The RuelleSullivan
map for actions of
R^d, Math. Ann. 334(2006), 693711.
PDF
 With T. Giordano and C.F. Skau, The orbit
structure of
Cantor minimal Z^2actions,
preprint, PDF
or
PS
 Lifting factor maps to resolving maps,
Israel J. Math. 146(2005), 253280.
PDF
 With N. Brownlowe, N. S. Larsen and I.
Raeburn,
Subquotients of Hecke C*algebras,
Ergod. Th. and Dynam. Sys. 25(2005), 15031520. 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), 441476.
PDF
 With J. Kaminker,
A proof of the gap labeling conjecture,
Mich. Math. J., 51(2003), 537546.
PDF
or
PS
 With N. S. Larsen and I. Raeburn,
The twoprime analogue of the
Hecke C*algebra of Bost and Connes,
Indiana U. Math. J., 51 (2002), 171186.
PDF
or
PS
 With T. Giordano and C.F. Skau,
Ktheory and
asymptotic index for certain almost onetoone
factors
, Math. Scand., 89(2001), 297319. PDF or
PS
 With J. Kellendonk,
Tilings, C*algebras and Ktheory
in Directions in Mathematical Quasicrystals,
M. Baake and R.V. Moody, editors, CRM Monograph
Series, American Mathematical Society,
Providence, R.I., 2000.
 The ordered Ktheory
of C*algebras associated with substitution
tilings, Comm. Math. Phys., 214 (2000),
593605.
PDF
or
PS
 The functoriality of
the C*algebras associated with hyperbolic dynamical
systems, J. London Math. Soc., (2) 62 (2000),
873884.
PDF
or
PS
 With J. Spielberg, The
structure of C*algebras
associated with hyperbolic dynamical systems,
J. Functional Analysis, 163(1999), 279299.
PDF or PS
 With T. Giordano and C.F. Skau,
Full groups of Cantor
minimal systems, Israel J. Math., 111(1999),
285320.
PDF or
PS
 On the Ktheory of C*algebras of
principal
groupoids, Rocky Mountain J. Math. 28(1998), 14831518.
 With J. Anderson, Topological invariants
for
substitution tilings and their C*algebras
, Ergodic Th. and Dynam. Sys. 18(1998), 509537.
 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, Ktheoretic
duality for shifts of finite type,
Comm. Math. Phys. 187(1997), 509522.
PDF or
PS
 An excision theorem for the
Ktheory of C*algebras
, J. Operator Th. 38(1997), 151171.
PDF or
PS
 A survey of recent Ktheoretic invariants
for
dynamical systems, The Dynamics of Zdactions, M.
Pollicott and K. Schmidt, ed., Cambridge University Press, 1996.
 C*algebras from Smale
spaces, Canad. J. Math
48(1996), 175195.PDF or
PS
 With T. Giordano and C.F. Skau, Topological
orbit
equivalence and C*crossed products, J. Reine Angew.
Math. 469(1995), 51111.
 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), 231250.
 With R.H. Herman and C.F. Skau, Ordered
Bratteli
diagrams, dimension groups and topological dynamics
, International J. Math. 6(1992), 827864.
 With P.S. Muhly and J. Xia, On the
Ktheory of some
C*algebras of Toeplitz and singular integral operators
, J. of Functional Analysis110(1992), 161225.
 The invertible elements are dense in the
irrational
rotation C*algebras, J. Reine Angew. Math. 410(1990),
160166.
 On the topological stable rank of certain
transformation group C*algebras, Ergodic Th. and
Dynam. Sys. 10(1990), 197207.
 With P.S. Muhly and J. Xia, Contributions
to the
Ktheory of C*algebras of Toeplitz and singular integral
operators, Bull. Amer. Math. Soc. 21(1) (1989),4750.
 The C*algebras associated with minimal
homeomorphisms
of the Cantor set, Pacific J. Math.136(2) (1989), 329353.
 With M. Rordam, The maximum unitary rank
of some
C*algebras, Math. Scand. 63(1988), 297304.
 Strong Morita equivalence for the Denjoy
C*algebras
, Can. Math. Bull. 31(4) (1988), 439447.
 With K. Schmidt and C. Skau, C*algebras
associated with Denjoy homeomorphisms of the circle
, J. Operator Th. 16(1986), 99126.
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
CuntzKrieger
algebras, Lecture Notes from the Summer School in
Operator Algeras, Odense, 1996.
 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 WilhelmsWestfalische 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 twolecture miniseries 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, postdoctoral 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.