• Previous Article
    On the uniqueness of an ergodic measure of full dimension for non-conformal repellers
  • DCDS Home
  • This Issue
  • Next Article
    A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions
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. Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[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. Google Scholar

[23]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

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

[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. Google Scholar

[31]

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

[32]

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

[33]

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

[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/. Google Scholar

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. Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[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. Google Scholar

[23]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

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

[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. Google Scholar

[31]

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

[32]

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

[33]

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

[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/. Google Scholar

[1]

Yakov Pesin, Vaughn Climenhaga. Open problems in the theory of non-uniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 589-607. doi: 10.3934/dcds.2010.27.589

[2]

Boris Kalinin, Victoria Sadovskaya. Normal forms for non-uniform contractions. Journal of Modern Dynamics, 2017, 11: 341-368. doi: 10.3934/jmd.2017014

[3]

Markus Bachmayr, Van Kien Nguyen. Identifiability of diffusion coefficients for source terms of non-uniform sign. Inverse Problems & Imaging, 2019, 13 (5) : 1007-1021. doi: 10.3934/ipi.2019045

[4]

Zhong-Jie Han, Gen-Qi Xu. Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks. Networks & Heterogeneous Media, 2010, 5 (2) : 315-334. doi: 10.3934/nhm.2010.5.315

[5]

Donald L. DeAngelis, Bo Zhang. Effects of dispersal in a non-uniform environment on population dynamics and competition: A patch model approach. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3087-3104. doi: 10.3934/dcdsb.2014.19.3087

[6]

Zhong-Jie Han, Gen-Qi Xu. Exponential decay in non-uniform porous-thermo-elasticity model of Lord-Shulman type. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 57-77. doi: 10.3934/dcdsb.2012.17.57

[7]

Zhong-Jie Han, Gen-Qi Xu. Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Networks & Heterogeneous Media, 2011, 6 (2) : 297-327. doi: 10.3934/nhm.2011.6.297

[8]

Grigor Nika, Bogdan Vernescu. Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1553-1564. doi: 10.3934/dcdss.2016062

[9]

Hai Huyen Dam, Wing-Kuen Ling. Optimal design of finite precision and infinite precision non-uniform cosine modulated filter bank. Journal of Industrial & Management Optimization, 2019, 15 (1) : 97-112. doi: 10.3934/jimo.2018034

[10]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743

[11]

Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229

[12]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639

[13]

Tingting Zhang, Àngel Jorba, Jianguo Si. Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced maps in a degenerate case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6599-6622. doi: 10.3934/dcds.2016086

[14]

Alexander Zlotnik. The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis. Kinetic & Related Models, 2015, 8 (3) : 587-613. doi: 10.3934/krm.2015.8.587

[15]

Victor Churchill, Rick Archibald, Anne Gelb. Edge-adaptive $ \ell_2 $ regularization image reconstruction from non-uniform Fourier data. Inverse Problems & Imaging, 2019, 13 (5) : 931-958. doi: 10.3934/ipi.2019042

[16]

A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195

[17]

Dawei Yang, Shaobo Gan, Lan Wen. Minimal non-hyperbolicity and index-completeness. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1349-1366. doi: 10.3934/dcds.2009.25.1349

[18]

Boris Hasselblatt, Yakov Pesin, Jörg Schmeling. Pointwise hyperbolicity implies uniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2819-2827. doi: 10.3934/dcds.2014.34.2819

[19]

Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048

[20]

Pedro J. Torres. Non-collision periodic solutions of forced dynamical systems with weak singularities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 693-698. doi: 10.3934/dcds.2004.11.693

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (16)
  • HTML views (24)
  • Cited by (0)

Other articles
by authors

[Back to Top]