• 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 and Continuous Dynamical Systems, 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/.

[1]

Yakov Pesin, Vaughn Climenhaga. Open problems in the theory of non-uniform hyperbolicity. Discrete and Continuous Dynamical Systems, 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]

Pablo G. Barrientos, Abbas Fakhari. Ergodicity of non-autonomous discrete systems with non-uniform expansion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1361-1382. doi: 10.3934/dcdsb.2019231

[4]

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

[5]

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 and Heterogeneous Media, 2010, 5 (2) : 315-334. doi: 10.3934/nhm.2010.5.315

[6]

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

[7]

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

[8]

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

[9]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400

[10]

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

[11]

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

[12]

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

[13]

Ruilin Li, Xin Wang, Hongyuan Zha, Molei Tao. Improving sampling accuracy of stochastic gradient MCMC methods via non-uniform subsampling of gradients. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021157

[14]

Zhenglin Wang. Fast non-uniform Fourier transform based regularization for sparse-view large-size CT reconstruction. STEM Education, 2022, 2 (2) : 121-139. doi: 10.3934/steme.2022009

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

Xueli Song, Jianhua Wu. Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor. Evolution Equations and Control Theory, 2022, 11 (1) : 41-65. doi: 10.3934/eect.2020102

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (168)
  • HTML views (70)
  • Cited by (0)

Other articles
by authors

[Back to Top]