# American Institute of Mathematical Sciences

2013, 2013(special): 837-845. doi: 10.3934/proc.2013.2013.837

## Anosov diffeomorphisms

 1 LIAAD-INESC TEC and Department of Mathematics, School of Technology and Management, Polytechnic Institute of Bragança, Campus de Santa Apolónia, Ap. 1134, 5301-857 Bragança, Portugal 2 Departamento de Matemática, IME-USP, Caixa Postal 66281, CEP 05315-970 São Paulo, Brazil 3 LIAAD-INESC TEC and Department of Mathematics, Faculty of Sciences, University of Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal 4 Warwick Systems Biology & Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Received  September 2012 Revised  October 2013 Published  November 2013

We use Adler, Tresser and Worfolk decomposition of Anosov automorphisms to give an explicit construction of the stable and unstable $C^{1+}$ self-renormalizable sequences.
Citation: João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837
##### References:
 [1] R. Adler, C. Tresser and P. A. Worfolk, Topological conjugacy of linear endomorphisms of the 2-torus, Trans. Amer. Math. Soc., 349 (1997), 1633-1652. [2] J. P. Almeida, A. M. Fisher, A. A. Pinto and D. A. Rand, Anosov and circle diffeomorphisms, in "Dynamics Games and Science I" (eds. M. Peixoto, A. Pinto and D. Rand), Springer Proceedings in Mathematics, Springer Verlag, 2011, 11-23. [3] J. P. Almeida, A. A. Pinto and D. A. Rand, Renormalization of circle diffeomorphism sequences and Markov sequences, to appear in "Proceedings of the Conference NOMA11," Évora, Portugal, Springer Proceedings in Mathematics, Springer Verlag, 2012. [4] V. I. Arnol'd, Small denominators I: On the mapping of a circle into itself, Investijia Akad. Nauk. Math., 25 (1961), 21-96; Transl. A.M.S., 2nd series, 46, 213-284. [5] E. Cawley, The Teichmüller space of an Anosov diffeomorphism of $T^2$, Inventiones Mathematicae, 112 (1993), 351-376. [6] P. Coullet and C. Tresser, Itération d'endomorphismes et groupe de renormalisation, Journal de Physique Colloques, 39 (1978), C5-25-C5-28. [7] J. Franks, Anosov diffeomorphisms, in "Global Analysis" (ed. S. Smale), Proc. Sympos. Pure Math., 14, Amer. Math. Soc., Providence, R.I., 1970, 61-93. [8] E. Ghys, Rigidité différentiable des groupes Fuchsiens, Publ. IHES, 78 (1993), 163-185. [9] M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. IHES, 49 (1979), 5-233. [10] Y. Jiang, Teichmüller structures and dual geometric Gibbs type measure theory for continuous potentials, preprint, (2008), 1-67. [11] Y. Jiang, Metric invariants in dynamical systems, Journal of Dynamics and Differentiable Equations, 17 (2005), 51-71. [12] O. Lanford, Renormalization group methods for critical circle mappings with general rotation number, in "VIIIth International Congress on Mathematical Physics," World Sci. Publishing, Singapore, 1987, 532-536. [13] R. de la Llave, Invariants for smooth conjugacy of hyperbolic dynamical systems II, Commun. Math. Phys., 109 (1987), 369-378. [14] A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. [15] R. Manẽ, "Ergodic Theory and Differentiable Dynamics," Springer-Verlag, Berlin, 1987. [16] J. M. Marco, and R. Moriyon, Invariants for Smooth conjugacy of hyperbolic dynamical systems I, Commun. Math. Phys., 109 (1987), 681-689. [17] J. M. Marco, and R. Moriyon, Invariants for Smooth conjugacy of hyperbolic dynamical systems III, Commun. Math. Phys., 112 (1989), 317-333. [18] H. Masur, Interval exchange transformations and measured foliations, The Annals of Mathematics. 2nd Ser., 115 (1982), 169-200. [19] W. de Melo and S. van Strien, "One-dimensional Dynamics," A series of Modern Surveys in Mathematics, Springer-Verlag, New York, 1993. [20] R. C. Penner and J. L. Harer, "Combinatorics of Train-Tracks," Princeton University Press, Princeton, New Jersey, 1992. [21] A. A. Pinto, J. P. Almeida and A. Portela, Golden tilings, Transactions of the American Mathematical Society, 364 (2012), 2261-2280. [22] A. A. Pinto, J. P. Almeida and D. A. Rand, Anosov and renormalized circle diffeomorphisms, submitted, (2012), 1-33. [23] A. A. Pinto and D. A. Rand, Train-tracks with $C^{1+}$ self-renormalisable structures, Journal of Difference Equations and Applications, 16 (2010), 945-962. [24] A. A. Pinto and D. A. Rand, Solenoid functions for hyperbolic sets on surfaces, in "Dynamics, Ergodic Theory and Geometry" (ed. Boris Hasselblatt), 54, MSRI Publications, 2007, 145-178. [25] A. A. Pinto and D. A. Rand, Rigidity of hyperbolic sets on surfaces, J. London Math. Soc., 71 (2004), 481-502. [26] A. A. Pinto and D. A. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics, Bull. London Math. Soc., 34 (2002), 341-352. [27] A. A. Pinto and D. A. Rand, Teichmüller spaces and HR structures for hyperbolic surface dynamics, Ergodic Theory & Dynamical Systems, 22 (2002), 1905-1931. [28] A. A. Pinto and D. A. Rand, Existence, uniqueness and ratio decomposition for Gibbs states via duality, Ergodic Theory & Dynamical Systems, 21 (2001), 533-543. [29] A. A. Pinto and D. A. Rand, Characterising rigidity and flexibility of pseudo-Anosov and other transversally laminated dynamical systems on surfaces, Warwick preprint, 1995. [30] A. A. Pinto, D. A. Rand and F. Ferreira, Arc exchange systems and renormalization, Journal of Difference Equations and Applications, 16 (2010), 347-371. [31] A. A. Pinto, D. A. Rand and F. Ferreira, Cantor exchange systems and renormalization, Journal of Differential Equations, 243 (2007), 593-616. [32] A. A. Pinto, D. A. Rand and F. Ferreira, "Fine structures of hyperbolic diffeomorphisms," Springer Monographs in Mathematics, Springer, 2009. [33] A. A. Pinto and D. Sullivan, The circle and the solenoid, Dedicated to Anatole Katok On the Occasion of his 60th Birthday, DCDS A, 16 (2006), 463-504. [34] M. Shub, "Global Stability of Dynamical Systems," Springer-Verlag, 1987. [35] Ya. Sinai, Markov Partitions and C-diffeomorphisms, Anal. and Appl., 2 (1968), 70-80. [36] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc., 19 (1988), 417-431. [37] W. Veech, Gauss measures for transformations on the space of interval exchange maps, The Annals of Mathematics, 2nd Ser., 115 (1982), 201-242. [38] R. F. Williams, Expanding attractors, Publ. I.H.E.S., 43 (1974), 169-203. [39] R. F. Williams, The "DA" maps of Smale and structural stability, in "Global Analysis" (ed. S. Smale), Proc. Symp. in Pure Math., 14, Amer. Math. Soc., Providence, RI, 1970, 329-334. [40] J. C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Scient. Éc. Norm. Sup., 4 série, t., 17 (1984), 333-359.

show all references

##### References:
 [1] R. Adler, C. Tresser and P. A. Worfolk, Topological conjugacy of linear endomorphisms of the 2-torus, Trans. Amer. Math. Soc., 349 (1997), 1633-1652. [2] J. P. Almeida, A. M. Fisher, A. A. Pinto and D. A. Rand, Anosov and circle diffeomorphisms, in "Dynamics Games and Science I" (eds. M. Peixoto, A. Pinto and D. Rand), Springer Proceedings in Mathematics, Springer Verlag, 2011, 11-23. [3] J. P. Almeida, A. A. Pinto and D. A. Rand, Renormalization of circle diffeomorphism sequences and Markov sequences, to appear in "Proceedings of the Conference NOMA11," Évora, Portugal, Springer Proceedings in Mathematics, Springer Verlag, 2012. [4] V. I. Arnol'd, Small denominators I: On the mapping of a circle into itself, Investijia Akad. Nauk. Math., 25 (1961), 21-96; Transl. A.M.S., 2nd series, 46, 213-284. [5] E. Cawley, The Teichmüller space of an Anosov diffeomorphism of $T^2$, Inventiones Mathematicae, 112 (1993), 351-376. [6] P. Coullet and C. Tresser, Itération d'endomorphismes et groupe de renormalisation, Journal de Physique Colloques, 39 (1978), C5-25-C5-28. [7] J. Franks, Anosov diffeomorphisms, in "Global Analysis" (ed. S. Smale), Proc. Sympos. Pure Math., 14, Amer. Math. Soc., Providence, R.I., 1970, 61-93. [8] E. Ghys, Rigidité différentiable des groupes Fuchsiens, Publ. IHES, 78 (1993), 163-185. [9] M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. IHES, 49 (1979), 5-233. [10] Y. Jiang, Teichmüller structures and dual geometric Gibbs type measure theory for continuous potentials, preprint, (2008), 1-67. [11] Y. Jiang, Metric invariants in dynamical systems, Journal of Dynamics and Differentiable Equations, 17 (2005), 51-71. [12] O. Lanford, Renormalization group methods for critical circle mappings with general rotation number, in "VIIIth International Congress on Mathematical Physics," World Sci. Publishing, Singapore, 1987, 532-536. [13] R. de la Llave, Invariants for smooth conjugacy of hyperbolic dynamical systems II, Commun. Math. Phys., 109 (1987), 369-378. [14] A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. [15] R. Manẽ, "Ergodic Theory and Differentiable Dynamics," Springer-Verlag, Berlin, 1987. [16] J. M. Marco, and R. Moriyon, Invariants for Smooth conjugacy of hyperbolic dynamical systems I, Commun. Math. Phys., 109 (1987), 681-689. [17] J. M. Marco, and R. Moriyon, Invariants for Smooth conjugacy of hyperbolic dynamical systems III, Commun. Math. Phys., 112 (1989), 317-333. [18] H. Masur, Interval exchange transformations and measured foliations, The Annals of Mathematics. 2nd Ser., 115 (1982), 169-200. [19] W. de Melo and S. van Strien, "One-dimensional Dynamics," A series of Modern Surveys in Mathematics, Springer-Verlag, New York, 1993. [20] R. C. Penner and J. L. Harer, "Combinatorics of Train-Tracks," Princeton University Press, Princeton, New Jersey, 1992. [21] A. A. Pinto, J. P. Almeida and A. Portela, Golden tilings, Transactions of the American Mathematical Society, 364 (2012), 2261-2280. [22] A. A. Pinto, J. P. Almeida and D. A. Rand, Anosov and renormalized circle diffeomorphisms, submitted, (2012), 1-33. [23] A. A. Pinto and D. A. Rand, Train-tracks with $C^{1+}$ self-renormalisable structures, Journal of Difference Equations and Applications, 16 (2010), 945-962. [24] A. A. Pinto and D. A. Rand, Solenoid functions for hyperbolic sets on surfaces, in "Dynamics, Ergodic Theory and Geometry" (ed. Boris Hasselblatt), 54, MSRI Publications, 2007, 145-178. [25] A. A. Pinto and D. A. Rand, Rigidity of hyperbolic sets on surfaces, J. London Math. Soc., 71 (2004), 481-502. [26] A. A. Pinto and D. A. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics, Bull. London Math. Soc., 34 (2002), 341-352. [27] A. A. Pinto and D. A. Rand, Teichmüller spaces and HR structures for hyperbolic surface dynamics, Ergodic Theory & Dynamical Systems, 22 (2002), 1905-1931. [28] A. A. Pinto and D. A. Rand, Existence, uniqueness and ratio decomposition for Gibbs states via duality, Ergodic Theory & Dynamical Systems, 21 (2001), 533-543. [29] A. A. Pinto and D. A. Rand, Characterising rigidity and flexibility of pseudo-Anosov and other transversally laminated dynamical systems on surfaces, Warwick preprint, 1995. [30] A. A. Pinto, D. A. Rand and F. Ferreira, Arc exchange systems and renormalization, Journal of Difference Equations and Applications, 16 (2010), 347-371. [31] A. A. Pinto, D. A. Rand and F. Ferreira, Cantor exchange systems and renormalization, Journal of Differential Equations, 243 (2007), 593-616. [32] A. A. Pinto, D. A. Rand and F. Ferreira, "Fine structures of hyperbolic diffeomorphisms," Springer Monographs in Mathematics, Springer, 2009. [33] A. A. Pinto and D. Sullivan, The circle and the solenoid, Dedicated to Anatole Katok On the Occasion of his 60th Birthday, DCDS A, 16 (2006), 463-504. [34] M. Shub, "Global Stability of Dynamical Systems," Springer-Verlag, 1987. [35] Ya. Sinai, Markov Partitions and C-diffeomorphisms, Anal. and Appl., 2 (1968), 70-80. [36] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc., 19 (1988), 417-431. [37] W. Veech, Gauss measures for transformations on the space of interval exchange maps, The Annals of Mathematics, 2nd Ser., 115 (1982), 201-242. [38] R. F. Williams, Expanding attractors, Publ. I.H.E.S., 43 (1974), 169-203. [39] R. F. Williams, The "DA" maps of Smale and structural stability, in "Global Analysis" (ed. S. Smale), Proc. Symp. in Pure Math., 14, Amer. Math. Soc., Providence, RI, 1970, 329-334. [40] J. C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Scient. Éc. Norm. Sup., 4 série, t., 17 (1984), 333-359.
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