Anosov diffeomorphisms

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

Anosov diffeomorphisms

João P. Almeida ^1, , Albert M. Fisher ^2, , Alberto Adrego Pinto ^3, and David A. Rand ^4,

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

Abstract
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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.

Keywords: self-renormalizable structures, train-tracks, circle diffeomorphisms., Anosov diffeomorphisms, Markov maps.

Mathematics Subject Classification: Primary: 37D20; Secondary: 37E1.

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. Google Scholar
[2]	J. P. Almeida, A. M. Fisher, A. A. Pinto and D. A. Rand, Anosov and circle diffeomorphisms,, in, (2011), 11. Google Scholar
[3]	J. P. Almeida, A. A. Pinto and D. A. Rand, Renormalization of circle diffeomorphism sequences and Markov sequences,, to appear in, (2012). Google Scholar
[4]	V. I. Arnol'd, Small denominators I: On the mapping of a circle into itself,, Investijia Akad. Nauk. Math., 25 (1961), 21. Google Scholar
[5]	E. Cawley, The Teichmüller space of an Anosov diffeomorphism of $T^2$,, Inventiones Mathematicae, 112 (1993), 351. Google Scholar
[6]	P. Coullet and C. Tresser, Itération d'endomorphismes et groupe de renormalisation,, Journal de Physique Colloques, 39 (1978), 5. Google Scholar
[7]	J. Franks, Anosov diffeomorphisms,, in, 14 (1970), 61. Google Scholar
[8]	E. Ghys, Rigidité différentiable des groupes Fuchsiens,, Publ. IHES, 78 (1993), 163. Google Scholar
[9]	M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,, Publ. IHES, 49 (1979), 5. Google Scholar
[10]	Y. Jiang, Teichmüller structures and dual geometric Gibbs type measure theory for continuous potentials,, preprint, (2008), 1. Google Scholar
[11]	Y. Jiang, Metric invariants in dynamical systems,, Journal of Dynamics and Differentiable Equations, 17 (2005), 51. Google Scholar
[12]	O. Lanford, Renormalization group methods for critical circle mappings with general rotation number,, in, (1987), 532. Google Scholar
[13]	R. de la Llave, Invariants for smooth conjugacy of hyperbolic dynamical systems II,, Commun. Math. Phys., 109 (1987), 369. Google Scholar
[14]	A. Manning, There are no new Anosov diffeomorphisms on tori,, Amer. J. Math., 96 (1974), 422. Google Scholar
[15]	R. Manẽ, "Ergodic Theory and Differentiable Dynamics,", Springer-Verlag, (1987). Google Scholar
[16]	J. M. Marco, and R. Moriyon, Invariants for Smooth conjugacy of hyperbolic dynamical systems I,, Commun. Math. Phys., 109 (1987), 681. Google Scholar
[17]	J. M. Marco, and R. Moriyon, Invariants for Smooth conjugacy of hyperbolic dynamical systems III,, Commun. Math. Phys., 112 (1989), 317. Google Scholar
[18]	H. Masur, Interval exchange transformations and measured foliations,, The Annals of Mathematics. 2nd Ser., 115 (1982), 169. Google Scholar
[19]	W. de Melo and S. van Strien, "One-dimensional Dynamics,", A series of Modern Surveys in Mathematics, (1993). Google Scholar
[20]	R. C. Penner and J. L. Harer, "Combinatorics of Train-Tracks,", Princeton University Press, (1992). Google Scholar
[21]	A. A. Pinto, J. P. Almeida and A. Portela, Golden tilings,, Transactions of the American Mathematical Society, 364 (2012), 2261. Google Scholar
[22]	A. A. Pinto, J. P. Almeida and D. A. Rand, Anosov and renormalized circle diffeomorphisms,, submitted, (2012), 1. Google Scholar
[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. Google Scholar
[24]	A. A. Pinto and D. A. Rand, Solenoid functions for hyperbolic sets on surfaces,, in, (2007), 145. Google Scholar
[25]	A. A. Pinto and D. A. Rand, Rigidity of hyperbolic sets on surfaces,, J. London Math. Soc., 71 (2004), 481. Google Scholar
[26]	A. A. Pinto and D. A. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics,, Bull. London Math. Soc., 34 (2002), 341. Google Scholar
[27]	A. A. Pinto and D. A. Rand, Teichmüller spaces and HR structures for hyperbolic surface dynamics,, Ergodic Theory & Dynamical Systems, 22 (2002), 1905. Google Scholar
[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. Google Scholar
[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). Google Scholar
[30]	A. A. Pinto, D. A. Rand and F. Ferreira, Arc exchange systems and renormalization,, Journal of Difference Equations and Applications, 16 (2010), 347. Google Scholar
[31]	A. A. Pinto, D. A. Rand and F. Ferreira, Cantor exchange systems and renormalization,, Journal of Differential Equations, 243 (2007), 593. Google Scholar
[32]	A. A. Pinto, D. A. Rand and F. Ferreira, "Fine structures of hyperbolic diffeomorphisms,", Springer Monographs in Mathematics, (2009). Google Scholar
[33]	A. A. Pinto and D. Sullivan, The circle and the solenoid,, Dedicated to Anatole Katok On the Occasion of his 60th Birthday, 16 (2006), 463. Google Scholar
[34]	M. Shub, "Global Stability of Dynamical Systems,", Springer-Verlag, (1987). Google Scholar
[35]	Ya. Sinai, Markov Partitions and C-diffeomorphisms,, Anal. and Appl., 2 (1968), 70. Google Scholar
[36]	W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces,, Bull. Amer. Math. Soc., 19 (1988), 417. Google Scholar
[37]	W. Veech, Gauss measures for transformations on the space of interval exchange maps,, The Annals of Mathematics, 115 (1982), 201. Google Scholar
[38]	R. F. Williams, Expanding attractors,, Publ. I.H.E.S., 43 (1974), 169. Google Scholar
[39]	R. F. Williams, The "DA" maps of Smale and structural stability,, in, (1970), 329. Google Scholar
[40]	J. C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne,, Ann. Scient. Éc. Norm. Sup., 17 (1984), 333. Google Scholar

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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. Google Scholar
[2]	J. P. Almeida, A. M. Fisher, A. A. Pinto and D. A. Rand, Anosov and circle diffeomorphisms,, in, (2011), 11. Google Scholar
[3]	J. P. Almeida, A. A. Pinto and D. A. Rand, Renormalization of circle diffeomorphism sequences and Markov sequences,, to appear in, (2012). Google Scholar
[4]	V. I. Arnol'd, Small denominators I: On the mapping of a circle into itself,, Investijia Akad. Nauk. Math., 25 (1961), 21. Google Scholar
[5]	E. Cawley, The Teichmüller space of an Anosov diffeomorphism of $T^2$,, Inventiones Mathematicae, 112 (1993), 351. Google Scholar
[6]	P. Coullet and C. Tresser, Itération d'endomorphismes et groupe de renormalisation,, Journal de Physique Colloques, 39 (1978), 5. Google Scholar
[7]	J. Franks, Anosov diffeomorphisms,, in, 14 (1970), 61. Google Scholar
[8]	E. Ghys, Rigidité différentiable des groupes Fuchsiens,, Publ. IHES, 78 (1993), 163. Google Scholar
[9]	M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,, Publ. IHES, 49 (1979), 5. Google Scholar
[10]	Y. Jiang, Teichmüller structures and dual geometric Gibbs type measure theory for continuous potentials,, preprint, (2008), 1. Google Scholar
[11]	Y. Jiang, Metric invariants in dynamical systems,, Journal of Dynamics and Differentiable Equations, 17 (2005), 51. Google Scholar
[12]	O. Lanford, Renormalization group methods for critical circle mappings with general rotation number,, in, (1987), 532. Google Scholar
[13]	R. de la Llave, Invariants for smooth conjugacy of hyperbolic dynamical systems II,, Commun. Math. Phys., 109 (1987), 369. Google Scholar
[14]	A. Manning, There are no new Anosov diffeomorphisms on tori,, Amer. J. Math., 96 (1974), 422. Google Scholar
[15]	R. Manẽ, "Ergodic Theory and Differentiable Dynamics,", Springer-Verlag, (1987). Google Scholar
[16]	J. M. Marco, and R. Moriyon, Invariants for Smooth conjugacy of hyperbolic dynamical systems I,, Commun. Math. Phys., 109 (1987), 681. Google Scholar
[17]	J. M. Marco, and R. Moriyon, Invariants for Smooth conjugacy of hyperbolic dynamical systems III,, Commun. Math. Phys., 112 (1989), 317. Google Scholar
[18]	H. Masur, Interval exchange transformations and measured foliations,, The Annals of Mathematics. 2nd Ser., 115 (1982), 169. Google Scholar
[19]	W. de Melo and S. van Strien, "One-dimensional Dynamics,", A series of Modern Surveys in Mathematics, (1993). Google Scholar
[20]	R. C. Penner and J. L. Harer, "Combinatorics of Train-Tracks,", Princeton University Press, (1992). Google Scholar
[21]	A. A. Pinto, J. P. Almeida and A. Portela, Golden tilings,, Transactions of the American Mathematical Society, 364 (2012), 2261. Google Scholar
[22]	A. A. Pinto, J. P. Almeida and D. A. Rand, Anosov and renormalized circle diffeomorphisms,, submitted, (2012), 1. Google Scholar
[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. Google Scholar
[24]	A. A. Pinto and D. A. Rand, Solenoid functions for hyperbolic sets on surfaces,, in, (2007), 145. Google Scholar
[25]	A. A. Pinto and D. A. Rand, Rigidity of hyperbolic sets on surfaces,, J. London Math. Soc., 71 (2004), 481. Google Scholar
[26]	A. A. Pinto and D. A. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics,, Bull. London Math. Soc., 34 (2002), 341. Google Scholar
[27]	A. A. Pinto and D. A. Rand, Teichmüller spaces and HR structures for hyperbolic surface dynamics,, Ergodic Theory & Dynamical Systems, 22 (2002), 1905. Google Scholar
[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. Google Scholar
[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). Google Scholar
[30]	A. A. Pinto, D. A. Rand and F. Ferreira, Arc exchange systems and renormalization,, Journal of Difference Equations and Applications, 16 (2010), 347. Google Scholar
[31]	A. A. Pinto, D. A. Rand and F. Ferreira, Cantor exchange systems and renormalization,, Journal of Differential Equations, 243 (2007), 593. Google Scholar
[32]	A. A. Pinto, D. A. Rand and F. Ferreira, "Fine structures of hyperbolic diffeomorphisms,", Springer Monographs in Mathematics, (2009). Google Scholar
[33]	A. A. Pinto and D. Sullivan, The circle and the solenoid,, Dedicated to Anatole Katok On the Occasion of his 60th Birthday, 16 (2006), 463. Google Scholar
[34]	M. Shub, "Global Stability of Dynamical Systems,", Springer-Verlag, (1987). Google Scholar
[35]	Ya. Sinai, Markov Partitions and C-diffeomorphisms,, Anal. and Appl., 2 (1968), 70. Google Scholar
[36]	W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces,, Bull. Amer. Math. Soc., 19 (1988), 417. Google Scholar
[37]	W. Veech, Gauss measures for transformations on the space of interval exchange maps,, The Annals of Mathematics, 115 (1982), 201. Google Scholar
[38]	R. F. Williams, Expanding attractors,, Publ. I.H.E.S., 43 (1974), 169. Google Scholar
[39]	R. F. Williams, The "DA" maps of Smale and structural stability,, in, (1970), 329. Google Scholar
[40]	J. C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne,, Ann. Scient. Éc. Norm. Sup., 17 (1984), 333. Google Scholar

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