Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Expansive Subdynamics


Authors: Mike Boyle and Douglas Lind
Journal: Trans. Amer. Math. Soc. 349 (1997), 55-102
MSC (1991): Primary 54H20, 58F03; Secondary 28D20, 28D15, 28F15, 58F11, 58F08
DOI: https://doi.org/10.1090/S0002-9947-97-01634-6
MathSciNet review: 1355295
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let $\alpha $ be a continuous action of ${\mathbb Z}^d $ on an infinite compact metric space. For each subspace $V$ of ${\mathbb R}^d $ we introduce a notion of expansiveness for $\alpha $ along $V$, and show that there are nonexpansive subspaces in every dimension $\le d-1$. For each $k\le d$ the set ${\mathbb E} _k(\alpha )$ of expansive $k$-dimensional subspaces is open in the Grassmann manifold of all $k$-dimensional subspaces of ${\mathbb R}^d $. Various dynamical properties of $\alpha $ are constant, or vary nicely, within a connected component of ${\mathbb E} _k(\alpha )$, but change abruptly when passing from one expansive component to another. We give several examples of this sort of ``phase transition,'' including the topological and measure-theoretic directional entropies studied by Milnor, zeta functions, and dimension groups. For $d=2$ we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an ${\mathbb E} _1(\alpha )$. The unresolved case is that of the complement of a single irrational direction. Algebraic examples using commuting automorphisms of compact abelian groups are an important source of phenomena, and we study several instances in detail. We conclude with a set of problems and research directions suggested by our analysis.


References [Enhancements On Off] (What's this?)

  • [AKM] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965) 309-319. MR 30:5291
  • [A] N. Aoki, Topological dynamics, in Topics in General Topology, North-Holland, Amsterdam (1989) 625-740. MR 91m:58120
  • [AM] N. Aoki and K. Moriyasu, Expansive homeomorphisms of solenoidal groups Hokkaido Math. J. 18 (1989), 301-319. MR 90i:58148
  • [BW] R. E. Bowen and P. Walters, Expansive one-parameter flows, J. Diff. Equations 12 (1972), 180-193. MR 49:6202
  • [BK] M. Boyle and W. Krieger, Periodic points and automorphisms of the shift, Trans. Amer. Math. Soc. 302 (1987), 125-149. MR 88g:54065
  • [BLR] M. Boyle, D. Lind, and D. Rudolph, The automorphism group of a shift of finite type, Trans. Amer. Math. Soc. 306 (1988), 71-114. MR 89m:54051
  • [BMT] M. Boyle, B. Marcus and P. Trow, Resolving maps and the dimension group for shifts of finite type, Memoirs of the Amer. Math. Soc. 377 (1987). MR 89c:28019
  • [DGS] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces in Springer Lecture Notes in Math 527, Springer-Verlag, (1976). MR 56:15879
  • [Fa] A. Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Commun. Math. Phys. 126 (1989), 249-262. MR 90m:58159
  • [Fr1] D. Fried, Metriques naturelles sur les espaces de Smale, C. R. Acad. Sc. Paris 297 (1983), 77-79. MR 85c:58085
  • [Fr2] D. Fried, Entropy for smooth abelian actions, Proc. Amer. Math. Soc. 87 (1983), 111-117. MR 83m:54078
  • [Fr3] D. Fried, Rationality for isolated expansive sets, Advances in Math. 65 (1987), 35-38. MR 88i:58144
  • [Fr4] D. Fried, Finitely presented dynamical systems, Ergod. Th. & Dyn. Syst. 7 (1987), 489-507. MR 89h:58157
  • [FR] D. B. Fuks and V. A. Rokhlin, Beginner's Course in Topology, Springer-Verlag, New York, (1984). MR 86a:57001
  • [Go] L. W. Goodwyn, Some counterexamples in topological entropy, Topology 11 (1972), 377-385. MR 47:2575
  • [GH] W. Gottschalk and G. Hedlund, Topological Dynamics, AMS Colloq. Publ., 36 Providence (1955). MR 17:650e
  • [G] Branko Grünbaum, Convex Polytopes, Interscience, London (1967). MR 37:2085
  • [H1] K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math. 27 (1990), 117-162. MR 91b:58184
  • [H2] K. Hiraide, Dynamical systems of expansive maps on compact manifolds, Sugaku Expositions 5 (No.2) (1992), 133-154. MR 91d:58197
  • [IT] Sh. Ito and Y.Takahashi, Markov subshifts and realization of beta-expansions, J. Math. Soc. Japan 26 (1974), 33-55. MR 49:10860
  • [K] Irving Kaplansky, Commutative Rings, Univ. of Chicago Press, Chicago (1974). MR 49:10674
  • [Ka] H. Kato, Expansive homeomorphisms in continuum theory, Topology Appl. 45 (1992), 223-243. MR 93j:54023
  • [KaSp] A. Katok and R. J. Spatzier, Invariant measures for higher rank hyperbolic abelian actions, Ergod. Th. & Dyn. Syst., to appear.
  • [KR] K. H. Kim and F. W. Roush, Williams' conjecture is false for reducible subshifts, Jour. Amer. Math. Soc. 5 (1992), 213-215. MR 92j:54055
  • [KRW] K. H. Kim, F. W. Roush and J. B. Wagoner, Automorphisms of the dimension group and gyration numbers of automorphisms of the shift, Jour. Amer. Math. Soc. 5 (1992), 191-212. MR 93h:54026
  • [Ki] J. L. King, A map with topological minimal self joinings in the sense of del Junco, Erg. Th. & Dyn. Syst. 10 (1990), 745-761. MR 92a:54036
  • [KS1] B. Kitchens and K. Schmidt, Automorphisms of compact groups, Ergod. Th. & Dyn. Syst. 9 (1989), 691-735. MR 91g:22008
  • [KS2] B. Kitchens and K. Schmidt, Markov subgroups of $(\mathbb Z/2\mathbb Z)^{\mathbb Z^2}$, in Symbolic Dynamics and its Applications, American Math. Soc., Providence (1992). MR 93k:58136
  • [Kr1] W. Krieger, On dimension functions and topological Markov chains, Inventiones Math. 56 (1980), 239-250. MR 81m:28018
  • [Kr2] W. Krieger, On a dimension for a class of homeomorphism groups, Math. Ann. 252 (1980), 87-95. MR 82b:46083
  • [La] W. Lawton, The structure of compact connected groups which admit an expansive automorphism, Springer Lecture Notes Math. 318 (1973), 182-196. MR 52:11873
  • [Led] F. Ledrappier, Un champ markovian peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris, Ser. A 287 (1978), 561-563. MR 80b:28030
  • [Lew] J. Lewowicz, Expansive homeomorphisms of surfaces Bull. Soc. Brasil Math. 20 (1989), 113-133. MR 92i:58139
  • [L] D. A. Lind, Entropies of automorphisms of a topological Markov shift, Proc. Amer. Math. Soc. 99 (1987), 589-595. MR 88c:54034
  • [LSW] D. Lind, K. Schmidt, and T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (1990), 593-629. MR 92j:22013
  • [LW] D. Lind and T. Ward, Automorphisms of solenoids and $p$-adic entropy, Ergod. Th. & Dynam. Sys. 8 (1988), 411-419. MR 90a:28031
  • [Man1 ] Ricardo Mañé, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252 (1979), 313-319. MR 80i:58032
  • [Man2] Ricardo Mañé, Ergodic theory and differentiable dynamics, Springer-Verlag, (1987). MR 88c:58040
  • [Mt ] H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, Cambridge (1989). MR 90i:13001
  • [Mi] John Milnor, On the entropy geometry of cellular automata, Complex Systems 2 (1988), 357-386. MR 90c:54026
  • [Mis] M. Misiurewicz, A short proof of the variational principle for a $\mathbb Z_+^N$ action on a compact space, Asterisque 40 (1976), 147-187. MR 56:3250
  • [N1] M. Nasu, Lecture on textile systems, CBMS Conference on Symbolic Dynamics, University of Washington (1989).
  • [N2] M. Nasu, Textile systems for endomorphisms and automorphisms of the shift, Memoirs Amer. Math. Soc. 546 (1995). MR 95i:54051
  • [P] K. Park, Continuity of directional entropy, Osaka J. Math. 31 (1994), 613-628. MR 95m:28021
  • [Pe] J. B. Pesin, Lyapunov characteristic exponents and smooth ergodic theory, Russian Math. Surveys 32 (1977), 55-114.
  • [R] W. Reddy, Expansive canonical coordinates are hyperbolic, Topology Appl. 15 (1983), 205-210. MR 84a:54076
  • [S1] K. Schmidt, Automorphisms of compact abelian groups and affine varieties, Proc. London Math. Soc. 61 (1990), 480-496. MR 91j:28015
  • [S2] K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Math., 128, Birkhäuser, 1995. CMP 95:16
  • [SW] K. Schmidt and T. Ward, Mixing automorphisms of compact groups and a theorem of Schlickewei, Invent. Math. 111 (1993), 69-76. MR 95c:22011
  • [Sh] M. Shereshevsky, Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms, Indag. Math. (2) 4 (1993), 203-210. MR 94c:54074
  • [Si1] Ya. Sinai, An answer to a question of J. Milnor, Comment. Math. Helv. 60 (1985), 173-178. MR 86m:28012
  • [Si2] Ya. Sinai, Topics in Ergodic Theory (1994), Princeton Univ. Press, Princeton, N.J. MR 95j:28017
  • [Sma] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 37:3598
  • [Smi] J. Smillie, Properties of the directional entropy function for cellular automata, Springer Lecture Notes in Math 1342 (1988), 689-705. MR 90b:58150
  • [W] Peter Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York (1982). MR 84e:28017

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 54H20, 58F03, 28D20, 28D15, 28F15, 58F11, 58F08

Retrieve articles in all journals with MSC (1991): 54H20, 58F03, 28D20, 28D15, 28F15, 58F11, 58F08


Additional Information

Mike Boyle
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: mmb@math.umd.edu

Douglas Lind
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195–4350
Email: lind@math.washington.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01634-6
Keywords: Expansive, subdynamics, symbolic dynamics, entropy, directional entropy, shift of finite type, group automorphism.
Received by editor(s): May 6, 1994
Additional Notes: The first author was supported in part by NSF Grants DMS-8802593, DMS-9104134, and DMS-9401538.
The second author was supported in part by NSF Grants DMS-9004253 and DMS-9303240.
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society