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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

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), 1633-1652.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

V. I. Arnol'd, Small denominators I: On the mapping of a circle into itself, Investijia Akad. Nauk. Math., 25 (1961), 21-96; Transl. A.M.S., 2nd series, 46, 213-284.  Google Scholar

[5]

E. Cawley, The Teichmüller space of an Anosov diffeomorphism of $T^2$, Inventiones Mathematicae, 112 (1993), 351-376.  Google Scholar

[6]

P. Coullet and C. Tresser, Itération d'endomorphismes et groupe de renormalisation, Journal de Physique Colloques, 39 (1978), C5-25-C5-28. Google Scholar

[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.  Google Scholar

[8]

E. Ghys, Rigidité différentiable des groupes Fuchsiens, Publ. IHES, 78 (1993), 163-185.  Google Scholar

[9]

M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. IHES, 49 (1979), 5-233.  Google Scholar

[10]

Y. Jiang, Teichmüller structures and dual geometric Gibbs type measure theory for continuous potentials, preprint, (2008), 1-67. Google Scholar

[11]

Y. Jiang, Metric invariants in dynamical systems, Journal of Dynamics and Differentiable Equations, 17 (2005), 51-71.  Google Scholar

[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.  Google Scholar

[13]

R. de la Llave, Invariants for smooth conjugacy of hyperbolic dynamical systems II, Commun. Math. Phys., 109 (1987), 369-378.  Google Scholar

[14]

A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429.  Google Scholar

[15]

R. Manẽ, "Ergodic Theory and Differentiable Dynamics," Springer-Verlag, Berlin, 1987.  Google Scholar

[16]

J. M. Marco, and R. Moriyon, Invariants for Smooth conjugacy of hyperbolic dynamical systems I, Commun. Math. Phys., 109 (1987), 681-689.  Google Scholar

[17]

J. M. Marco, and R. Moriyon, Invariants for Smooth conjugacy of hyperbolic dynamical systems III, Commun. Math. Phys., 112 (1989), 317-333.  Google Scholar

[18]

H. Masur, Interval exchange transformations and measured foliations, The Annals of Mathematics. 2nd Ser., 115 (1982), 169-200.  Google Scholar

[19]

W. de Melo and S. van Strien, "One-dimensional Dynamics," A series of Modern Surveys in Mathematics, Springer-Verlag, New York, 1993.  Google Scholar

[20]

R. C. Penner and J. L. Harer, "Combinatorics of Train-Tracks," Princeton University Press, Princeton, New Jersey, 1992.  Google Scholar

[21]

A. A. Pinto, J. P. Almeida and A. Portela, Golden tilings, Transactions of the American Mathematical Society, 364 (2012), 2261-2280.  Google Scholar

[22]

A. A. Pinto, J. P. Almeida and D. A. Rand, Anosov and renormalized circle diffeomorphisms, submitted, (2012), 1-33. 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), 945-962.  Google Scholar

[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.  Google Scholar

[25]

A. A. Pinto and D. A. Rand, Rigidity of hyperbolic sets on surfaces, J. London Math. Soc., 71 (2004), 481-502.  Google Scholar

[26]

A. A. Pinto and D. A. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics, Bull. London Math. Soc., 34 (2002), 341-352.  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), 1905-1931.  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), 533-543.  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), 347-371.  Google Scholar

[31]

A. A. Pinto, D. A. Rand and F. Ferreira, Cantor exchange systems and renormalization, Journal of Differential Equations, 243 (2007), 593-616.  Google Scholar

[32]

A. A. Pinto, D. A. Rand and F. Ferreira, "Fine structures of hyperbolic diffeomorphisms," Springer Monographs in Mathematics, Springer, 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, DCDS A, 16 (2006), 463-504.  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), 70-80. Google Scholar

[36]

W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc., 19 (1988), 417-431.  Google Scholar

[37]

W. Veech, Gauss measures for transformations on the space of interval exchange maps, The Annals of Mathematics, 2nd Ser., 115 (1982), 201-242.  Google Scholar

[38]

R. F. Williams, Expanding attractors, Publ. I.H.E.S., 43 (1974), 169-203.  Google Scholar

[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.  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., 4 série, t., 17 (1984), 333-359.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

V. I. Arnol'd, Small denominators I: On the mapping of a circle into itself, Investijia Akad. Nauk. Math., 25 (1961), 21-96; Transl. A.M.S., 2nd series, 46, 213-284.  Google Scholar

[5]

E. Cawley, The Teichmüller space of an Anosov diffeomorphism of $T^2$, Inventiones Mathematicae, 112 (1993), 351-376.  Google Scholar

[6]

P. Coullet and C. Tresser, Itération d'endomorphismes et groupe de renormalisation, Journal de Physique Colloques, 39 (1978), C5-25-C5-28. Google Scholar

[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.  Google Scholar

[8]

E. Ghys, Rigidité différentiable des groupes Fuchsiens, Publ. IHES, 78 (1993), 163-185.  Google Scholar

[9]

M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. IHES, 49 (1979), 5-233.  Google Scholar

[10]

Y. Jiang, Teichmüller structures and dual geometric Gibbs type measure theory for continuous potentials, preprint, (2008), 1-67. Google Scholar

[11]

Y. Jiang, Metric invariants in dynamical systems, Journal of Dynamics and Differentiable Equations, 17 (2005), 51-71.  Google Scholar

[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.  Google Scholar

[13]

R. de la Llave, Invariants for smooth conjugacy of hyperbolic dynamical systems II, Commun. Math. Phys., 109 (1987), 369-378.  Google Scholar

[14]

A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429.  Google Scholar

[15]

R. Manẽ, "Ergodic Theory and Differentiable Dynamics," Springer-Verlag, Berlin, 1987.  Google Scholar

[16]

J. M. Marco, and R. Moriyon, Invariants for Smooth conjugacy of hyperbolic dynamical systems I, Commun. Math. Phys., 109 (1987), 681-689.  Google Scholar

[17]

J. M. Marco, and R. Moriyon, Invariants for Smooth conjugacy of hyperbolic dynamical systems III, Commun. Math. Phys., 112 (1989), 317-333.  Google Scholar

[18]

H. Masur, Interval exchange transformations and measured foliations, The Annals of Mathematics. 2nd Ser., 115 (1982), 169-200.  Google Scholar

[19]

W. de Melo and S. van Strien, "One-dimensional Dynamics," A series of Modern Surveys in Mathematics, Springer-Verlag, New York, 1993.  Google Scholar

[20]

R. C. Penner and J. L. Harer, "Combinatorics of Train-Tracks," Princeton University Press, Princeton, New Jersey, 1992.  Google Scholar

[21]

A. A. Pinto, J. P. Almeida and A. Portela, Golden tilings, Transactions of the American Mathematical Society, 364 (2012), 2261-2280.  Google Scholar

[22]

A. A. Pinto, J. P. Almeida and D. A. Rand, Anosov and renormalized circle diffeomorphisms, submitted, (2012), 1-33. 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), 945-962.  Google Scholar

[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.  Google Scholar

[25]

A. A. Pinto and D. A. Rand, Rigidity of hyperbolic sets on surfaces, J. London Math. Soc., 71 (2004), 481-502.  Google Scholar

[26]

A. A. Pinto and D. A. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics, Bull. London Math. Soc., 34 (2002), 341-352.  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), 1905-1931.  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), 533-543.  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), 347-371.  Google Scholar

[31]

A. A. Pinto, D. A. Rand and F. Ferreira, Cantor exchange systems and renormalization, Journal of Differential Equations, 243 (2007), 593-616.  Google Scholar

[32]

A. A. Pinto, D. A. Rand and F. Ferreira, "Fine structures of hyperbolic diffeomorphisms," Springer Monographs in Mathematics, Springer, 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, DCDS A, 16 (2006), 463-504.  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), 70-80. Google Scholar

[36]

W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc., 19 (1988), 417-431.  Google Scholar

[37]

W. Veech, Gauss measures for transformations on the space of interval exchange maps, The Annals of Mathematics, 2nd Ser., 115 (1982), 201-242.  Google Scholar

[38]

R. F. Williams, Expanding attractors, Publ. I.H.E.S., 43 (1974), 169-203.  Google Scholar

[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.  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., 4 série, t., 17 (1984), 333-359.  Google Scholar

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