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November  2017, 37(11): 5747-5761. doi: 10.3934/dcds.2017249

Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent

1. 

Institute of Mathematics, Friedrich-Schiller-Universität Jena, Jena 07743, Germany

2. 

Department of Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China

* Corresponding author: Jing Wang

Received  June 2016 Revised  June 2017 Published  July 2017

We study order-preserving $\mathcal{C}^1$-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures of nearly all family members are one-rectifiable, that is, absolutely continuous with respect to the restriction of the one-dimensional Hausdorff measure to a countable union of Lipschitz graphs.

Citation: Gabriel Fuhrmann, Jing Wang. Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5747-5761. doi: 10.3934/dcds.2017249
References:
[1]

L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann., 318 (2000), 527-555. doi: 10.1007/s002080000122.

[2]

V. I. Arnold, Cardiac arrhythmias and circle mappings, Chaos, 1 (1991), 20-24. doi: 10.1063/1.165812.

[3]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math., 133 (1991), 73-169. doi: 10.2307/2944326.

[4]

K. Bjerklöv, Positive lyapunov exponent and minimality for a class of one-dimensional quasi-periodic schrödinger equations, Ergodic Theory Dynam. Systems, 25 (2005), 1015-1045. doi: 10.1017/S0143385704000999.

[5]

K. Bjerklöv, Dynamics of the quasi-periodic Schrödinger cocycle at the lowest energy in the spectrum, Comm. Math. Phys., 272 (2007), 397-442. doi: 10.1007/s00220-007-0238-y.

[6]

S. Coombes and P. C. Bressloff, Mode locking and arnold tongues in integrate-and-fire neural oscillators, Phys. Rev. E, 60 (1999), 2086-2096. doi: 10.1103/PhysRevE.60.2086.

[7]

E. J. Ding, Analytic treatment of a driven oscillator with a limit cycle, Phys. Rev. A, 35 (1987), 2669-2683. doi: 10.1103/PhysRevA.35.2669.

[8]

P. FredericksonJ. KaplanE. Yorke and J. Yorke, The Liapunov dimension of strange attractors, J. Differential Equations, 49 (1983), 185-207. doi: 10.1016/0022-0396(83)90011-6.

[9]

G. Fuhrmann, Non-smooth saddle-node bifurcations Ⅰ: Existence of an SNA, Ergodic Theory Dynam. Systems, 36 (2016), 1130-1155, URL http://journals.cambridge.org/article_S0143385714000923. doi: 10.1017/etds.2014.92.

[10]

G. Fuhrmann, M. Gröger and T. Jäger, Non-smooth saddle-node bifurcations Ⅱ: Dimensions of strange attractors, Ergodic Theory Dynam. Systems, 1-23.

[11]

H. Fürstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601. doi: 10.2307/2372899.

[12]

C. GrebogiE. OttS. Pelikan and J. A. Yorke, Strange attractors that are not chaotic, Phys. D, 13 (1984), 261-268. doi: 10.1016/0167-2789(84)90282-3.

[13]

M. Gröger and T. Jäger, Dimensions of attractors in pinched skew products, Comm. Math. Phys., 320 (2013), 101-119. doi: 10.1007/s00220-013-1713-2.

[14]

M. R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractére local d'un théoréme d'Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helv., 58 (1983), 453-502. doi: 10.1007/BF02564647.

[15]

T. Jäger, Quasiperiodically forced interval maps with negative Schwarzian derivative, Nonlinearity, 16 (2003), 1239-1255. doi: 10.1088/0951-7715/16/4/303.

[16]

T. Jäger, On the structure of strange non-chaotic attractors in pinched skew products, Ergodic Theory Dynam. Systems, 27 (2007), 493-510. doi: 10.1017/S0143385706000745.

[17]

T. Jäger, The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations, Mem. Amer. Math. Soc., 201 (2009), ⅵ+106 pp. doi: 10.1090/memo/0945.

[18]

T. Jäger, Strange non-chaotic attractors in quasiperiodically forced circle maps, Comm. Math. Phys., 289 (2009), 253-289. doi: 10.1007/s00220-009-0753-0.

[19]

T. Jäger, Strange non-chaotic attractors in quasi-periodically forced circle maps: Diophantine forcing, Ergodic Theory Dynam. Systems, 33 (2013), 1477-1501. doi: 10.1017/S0143385712000375.

[20]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[21]

G. Keller, A note on strange nonchaotic attractors, Fund. Math., 151 (1996), 139-148, URL http://eudml.org/doc/212186.

[22]

J. -W. Kim, S. -Y. Kim, B. Hunt and E. Ott, Fractal properties of robust strange nonchaotic attractors in maps of two or more dimensions, Phys. Rev. E, 67 (2003), 036211, 8pp. doi: 10.1103/PhysRevE.67.036211.

[23]

F. Ledrappier, Some relations between dimension and Lyapounov exponents, Comm. Math. Phys., 81 (1981), 229-238. doi: 10.1007/BF01208896.

[24]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms: Part Ⅱ: Relations between entropy, exponents and dimension, Ann. of Math., 122 (1985), 540-574. doi: 10.2307/1971329.

[25]

V. M. Millionščikov, Proof of the existence of irregular systems of linear differential equations with almost periodic coefficients, Differ. Equ., 4 (1968), 391-396.

[26]

D. H. Perkel, J. H. Schulman, T. H. Bullock, G. P. Moore and J. P. Segundo, How do Brains Work? Papers of a Comparative Neurophysiologist, chapter Pacemaker Neurons: Effects of Regularly Spaced Synaptic Input, 112-115, Birkhäuser Boston, Boston, MA, 1993. doi: 10.1007/978-1-4684-9427-3_12.

[27]

Y. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions, Journal of Statistical Physics, 71 (1993), 529-547. doi: 10.1007/BF01058436.

[28]

R. Sturman and J. Stark, Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity, 13 (2000), 113-143. doi: 10.1088/0951-7715/13/1/306.

[29]

R. E. Vinograd, A problem suggested by N.R. Erugin, Differ. Equ., 11 (1975), 632-638.

[30]

J. Wang and T. Jäger, Abundance of mode-locking for quasiperiodically forced circle maps, Comm. Math. Phys., 353 (2017), 1-36. doi: 10.1007/s00220-017-2870-5.

[31]

J. Wang and T. Jäger, Genericity of mode-locking for quasiperiodically forced circle maps, in prep.

[32]

L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems, 2 (1982), 109-124. doi: 10.1017/S0143385700009615.

[33]

L.-S. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergodic Theory Dynam. Systems, 17 (1997), 483-504. doi: 10.1017/S0143385797079170.

[34]

O. Zindulka, Hentschel-Procaccia spectra in separable metric spaces, Real Analysis Exchange, Summer Symposium in Real Analysis, 26 (2002), 115-119, See also http://mat.fsv.cvut.cz/zindulka/.

show all references

References:
[1]

L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann., 318 (2000), 527-555. doi: 10.1007/s002080000122.

[2]

V. I. Arnold, Cardiac arrhythmias and circle mappings, Chaos, 1 (1991), 20-24. doi: 10.1063/1.165812.

[3]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math., 133 (1991), 73-169. doi: 10.2307/2944326.

[4]

K. Bjerklöv, Positive lyapunov exponent and minimality for a class of one-dimensional quasi-periodic schrödinger equations, Ergodic Theory Dynam. Systems, 25 (2005), 1015-1045. doi: 10.1017/S0143385704000999.

[5]

K. Bjerklöv, Dynamics of the quasi-periodic Schrödinger cocycle at the lowest energy in the spectrum, Comm. Math. Phys., 272 (2007), 397-442. doi: 10.1007/s00220-007-0238-y.

[6]

S. Coombes and P. C. Bressloff, Mode locking and arnold tongues in integrate-and-fire neural oscillators, Phys. Rev. E, 60 (1999), 2086-2096. doi: 10.1103/PhysRevE.60.2086.

[7]

E. J. Ding, Analytic treatment of a driven oscillator with a limit cycle, Phys. Rev. A, 35 (1987), 2669-2683. doi: 10.1103/PhysRevA.35.2669.

[8]

P. FredericksonJ. KaplanE. Yorke and J. Yorke, The Liapunov dimension of strange attractors, J. Differential Equations, 49 (1983), 185-207. doi: 10.1016/0022-0396(83)90011-6.

[9]

G. Fuhrmann, Non-smooth saddle-node bifurcations Ⅰ: Existence of an SNA, Ergodic Theory Dynam. Systems, 36 (2016), 1130-1155, URL http://journals.cambridge.org/article_S0143385714000923. doi: 10.1017/etds.2014.92.

[10]

G. Fuhrmann, M. Gröger and T. Jäger, Non-smooth saddle-node bifurcations Ⅱ: Dimensions of strange attractors, Ergodic Theory Dynam. Systems, 1-23.

[11]

H. Fürstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601. doi: 10.2307/2372899.

[12]

C. GrebogiE. OttS. Pelikan and J. A. Yorke, Strange attractors that are not chaotic, Phys. D, 13 (1984), 261-268. doi: 10.1016/0167-2789(84)90282-3.

[13]

M. Gröger and T. Jäger, Dimensions of attractors in pinched skew products, Comm. Math. Phys., 320 (2013), 101-119. doi: 10.1007/s00220-013-1713-2.

[14]

M. R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractére local d'un théoréme d'Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helv., 58 (1983), 453-502. doi: 10.1007/BF02564647.

[15]

T. Jäger, Quasiperiodically forced interval maps with negative Schwarzian derivative, Nonlinearity, 16 (2003), 1239-1255. doi: 10.1088/0951-7715/16/4/303.

[16]

T. Jäger, On the structure of strange non-chaotic attractors in pinched skew products, Ergodic Theory Dynam. Systems, 27 (2007), 493-510. doi: 10.1017/S0143385706000745.

[17]

T. Jäger, The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations, Mem. Amer. Math. Soc., 201 (2009), ⅵ+106 pp. doi: 10.1090/memo/0945.

[18]

T. Jäger, Strange non-chaotic attractors in quasiperiodically forced circle maps, Comm. Math. Phys., 289 (2009), 253-289. doi: 10.1007/s00220-009-0753-0.

[19]

T. Jäger, Strange non-chaotic attractors in quasi-periodically forced circle maps: Diophantine forcing, Ergodic Theory Dynam. Systems, 33 (2013), 1477-1501. doi: 10.1017/S0143385712000375.

[20]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[21]

G. Keller, A note on strange nonchaotic attractors, Fund. Math., 151 (1996), 139-148, URL http://eudml.org/doc/212186.

[22]

J. -W. Kim, S. -Y. Kim, B. Hunt and E. Ott, Fractal properties of robust strange nonchaotic attractors in maps of two or more dimensions, Phys. Rev. E, 67 (2003), 036211, 8pp. doi: 10.1103/PhysRevE.67.036211.

[23]

F. Ledrappier, Some relations between dimension and Lyapounov exponents, Comm. Math. Phys., 81 (1981), 229-238. doi: 10.1007/BF01208896.

[24]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms: Part Ⅱ: Relations between entropy, exponents and dimension, Ann. of Math., 122 (1985), 540-574. doi: 10.2307/1971329.

[25]

V. M. Millionščikov, Proof of the existence of irregular systems of linear differential equations with almost periodic coefficients, Differ. Equ., 4 (1968), 391-396.

[26]

D. H. Perkel, J. H. Schulman, T. H. Bullock, G. P. Moore and J. P. Segundo, How do Brains Work? Papers of a Comparative Neurophysiologist, chapter Pacemaker Neurons: Effects of Regularly Spaced Synaptic Input, 112-115, Birkhäuser Boston, Boston, MA, 1993. doi: 10.1007/978-1-4684-9427-3_12.

[27]

Y. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions, Journal of Statistical Physics, 71 (1993), 529-547. doi: 10.1007/BF01058436.

[28]

R. Sturman and J. Stark, Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity, 13 (2000), 113-143. doi: 10.1088/0951-7715/13/1/306.

[29]

R. E. Vinograd, A problem suggested by N.R. Erugin, Differ. Equ., 11 (1975), 632-638.

[30]

J. Wang and T. Jäger, Abundance of mode-locking for quasiperiodically forced circle maps, Comm. Math. Phys., 353 (2017), 1-36. doi: 10.1007/s00220-017-2870-5.

[31]

J. Wang and T. Jäger, Genericity of mode-locking for quasiperiodically forced circle maps, in prep.

[32]

L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems, 2 (1982), 109-124. doi: 10.1017/S0143385700009615.

[33]

L.-S. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergodic Theory Dynam. Systems, 17 (1997), 483-504. doi: 10.1017/S0143385797079170.

[34]

O. Zindulka, Hentschel-Procaccia spectra in separable metric spaces, Real Analysis Exchange, Summer Symposium in Real Analysis, 26 (2002), 115-119, See also http://mat.fsv.cvut.cz/zindulka/.

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