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A Markov partition that reflects the geometry of a hyperbolic toral automorphism
Author(s):
Anthony
Manning
Journal:
Trans. Amer. Math. Soc.
354
(2002),
2849-2863.
MSC (2000):
Primary 37D20, 37B10;
Secondary 28A80, 37B40
Posted:
February 26, 2002
MathSciNet review:
1895206
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Abstract:
We show how to construct a Markov partition that reflects the geometrical action of a hyperbolic automorphism of the  -torus. The transition matrix is the transpose of the matrix induced by the automorphism in  -dimensional homology, provided this is non-negative. (Here  denotes the expanding dimension.) That condition is satisfied, at least for some power of the original automorphism, under a certain non-degeneracy condition on the Galois group of the characteristic polynomial. The  rectangles are constructed by an iterated function system, and they resemble the product of the projection of a  -dimensional face of the unit cube onto the unstable subspace and the projection of minus the orthogonal  -dimensional face onto the stable subspace.
References:
-
- 1.
- R Adler and B Weiss, Similarity of automorphisms of the torus, Memoirs Amer. Math. Soc., 98 (1970). MR 41:1966
- 2.
- T Bedford, Generating special Markov partitions for hyperbolic toral automorphisms using fractals, Ergodic Theory Dynam. Systems 6 (1986), 325-333. MR 88c:58037
- 3.
- R Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics 470, Springer, New York 1975. MR 56:1364
- 4.
- R Bowen, On axiom A diffeomorphisms, CBMS Regional Conference Series in Mathematics 35, Amer. Math. Soc. 1978. MR 58:2888
- 5.
- R Bowen, Markov partitions are not smooth, Proc. Amer. Math. Soc. 71 (1978), 130-132. MR 57:14055
- 6.
- R Devaney, An introduction to chaotic dynamical systems, Benjamin, Reading, MA, 1986. MR 87e:58142
- 7.
- K Falconer, Fractal geometry: mathematical foundations and applications, Wiley, Chichester 1990. MR 92j:28008
- 8.
- J Franks, Homology and dynamical systems, CBMS Regional Conference Series in Mathematics 49, Amer. Math. Soc. 1982. MR 84f:58067
- 9.
- D. Garling. A course in Galois theory, Cambridge University Press, Cambridge 1986. MR 88d:12007
- 10.
- P Hilton and S Wylie, Homology Theory: An introduction to algebraic topology, Cambridge University Press, Cambridge 1960. MR 22:5963
- 11.
- M W Hirsch, Expanding maps and transformation groups, Proc. Symp. Pure Math. 14 (1970), 125-131. MR 45:7750
- 12.
- J E Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 82h:49026
- 13.
- S Ito and M Ohtsuki, Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms, Tokyo J. Math. 16 (1993), 441-472. MR 95g:58167
- 14.
- A Katok and B Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, Cambridge 1995. MR 96c:58055
- 15.
- R Kenyon, Self-similar tilings, Thesis, Princeton, 1990.
- 16.
- R Kenyon, Self-replicating tilings, Contemp. Math. 135 (1992), 239-263. MR 94a:52043
- 17.
- R Kenyon and A Vershik, Arithmetic construction of sofic partitions of hyperbolic toral automorphisms, Ergodic Theory Dynam. Systems 18 (1998), 357-372. MR 99g:58092
- 18.
- J C Lagarias and Y Wang, Self-affine tiles in
 , Adv. Math. 121 (1996), 21-49. MR 97d:52034 - 19.
- S Le Borgne, Un codage sofique des automorphismes hyperboliques du tore, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 1123-1128. MR 97k:58095
- 20.
- R Mañé, Ergodic theory and differentiable dynamics, Springer, Berlin 1987. MR 88c:58040
- 21.
- A Manning, Axiom A diffeomorphisms have rational zeta functions, Bull. London Math. Soc. 3 (1971), 215-220. MR 44:5982
- 22.
- A Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. MR 50:11324
- 23.
- A Manning, Irrationality of linear combinations of eigenvectors, Proc. Indian Acad. Sci. Math. Sci., 105 (1995), 269-271. MR 96h:15009
- 24.
- W Massey, A Basic Course in Algebraic Topology, Springer, New York 1991. MR 92c:55001
- 25.
- R D Mauldin and S C Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829. MR 89i:28003
- 26.
- B Praggastis, Numeration systems and Markov partitions from self-similar tilings, Trans. Amer. Math. Soc. 351 (1999), 3315-3349. MR 99m:11009
- 27.
- D Ruelle, One-dimensional Gibbs states and axiom A diffeomorphisms, J. Differential Geom. 25 (1987), 117-137. MR 88j:58099
- 28.
- E Seneta, Nonnegative matrices and Markov chains, 2nd ed., Springer, New York 1981. MR 85i:60058
- 29.
- Ya G Sinai, Markov partitions and
 -diffeomorphisms, Func. Anal. Appl. 2 (1968), 61-82. MR 38:1361 - 30.
- S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 37:3598
- 31.
- P. Walters, An introduction to ergodic theory, Springer, New York-Berlin 1982. MR 84e:28017
- 32.
- R F Williams, Classification of subshifts of finite type, Ann. of Math. 98 (1973), 120-153. Errata, ibid. 99 (1974), 380-381. MR 48:9769
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Additional Information:
Anthony
Manning
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
Email:
akm@maths.warwick.ac.uk
DOI:
10.1090/S0002-9947-02-03003-9
PII:
S 0002-9947(02)03003-9
Keywords:
Markov partition,
hyperbolic toral automorphism,
iterated function system
Received by editor(s):
September 4, 2001
Posted:
February 26, 2002
Copyright of article:
Copyright
2002,
American Mathematical Society
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