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Dimension theory of flows: A survey
1. | Departamento de Matemática, Instituto Superior Técnico, UTL, 1049-001 Lisboa |
References:
[1] |
L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics,, Progress in Mathematics, (2008).
doi: 10.1007/978-3-7643-8882-9. |
[2] |
L. Barreira, Dimension Theory of Hyperbolic Flows,, Springer Monographs in Mathematics, (2013).
doi: 10.1007/978-3-319-00548-5. |
[3] |
L. Barreira and P. Doutor, Birkhoff averages for hyperbolic flows: Variational principles and applications,, J. Statist. Phys., 115 (2004), 1567.
doi: 10.1023/B:JOSS.0000028069.64945.65. |
[4] |
L. Barreira and P. Doutor, Dimension spectra of hyperbolic flows,, J. Stat. Phys., 136 (2009), 505.
doi: 10.1007/s10955-009-9790-5. |
[5] |
L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows,, Comm. Math. Phys., 214 (2000), 339.
doi: 10.1007/s002200000268. |
[6] |
L. Barreira and B. Saussol, Variational principles for hyperbolic flows,, in Differential Equations and Dynamical Systems (Lisbon, (2000), 43.
|
[7] |
L. Barreira and C. Wolf, Dimension and ergodic decompositions for hyperbolic flows,, Discrete Contin. Dyn. Syst., 17 (2007), 201.
doi: 10.3934/dcds.2007.17.201. |
[8] |
R. Bowen, Symbolic dynamics for hyperbolic flows,, Amer. J. Math., 95 (1973), 429.
doi: 10.2307/2373793. |
[9] |
R. Bowen, Hausdorff dimension of quasi-circles,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11.
doi: 10.1007/BF02684767. |
[10] |
R. Bowen and P. Walters, Expansive one-parameter flows,, J. Differential Equations, 12 (1972), 180.
doi: 10.1016/0022-0396(72)90013-7. |
[11] |
M. Brin and A. Katok, On local entropy,, in Geometric Dynamics (Rio de Janeiro, (1981), 30.
doi: 10.1007/BFb0061408. |
[12] |
K. Burns and K. Gelfert, Lyapunov spectrum for geodesic flows of rank 1 surfaces,, Discrete Contin. Dyn. Syst., 34 (2014), 1841.
doi: 10.3934/dcds.2014.34.1841. |
[13] |
P. Collet, J. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems,, J. Stat. Phys., 47 (1987), 609.
doi: 10.1007/BF01206149. |
[14] |
K. Falconer, Dimensions and measures of quasi self-similar sets,, Proc. Amer. Math. Soc., 106 (1989), 543.
doi: 10.1090/S0002-9939-1989-0969315-8. |
[15] |
T. Halsey, M. Jensen, L. Kadanoff, I. Procaccia and B. Shraiman, Fractal measures and their singularities: The characterization of strange sets,, Phys. Rev. A (3), 34 (1986).
doi: 10.1103/PhysRevA.33.1141. |
[16] |
B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations,, Ergodic Theory Dynam. Systems, 14 (1994), 645.
doi: 10.1017/S0143385700008105. |
[17] |
A. Lopes, The dimension spectrum of the maximal measure,, SIAM J. Math. Anal., 20 (1989), 1243.
doi: 10.1137/0520081. |
[18] |
H. McCluskey and A. Manning, Hausdorff dimension for horseshoes,, Ergodic Theory Dynam. Systems, 3 (1983), 251.
doi: 10.1017/S0143385700001966. |
[19] |
J. Palis and M. Viana, On the continuity of Hausdorff dimension and limit capacity for horseshoes,, in Dynamical Systems (Valparaiso, (1986), 150.
doi: 10.1007/BFb0083071. |
[20] |
Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications,, Chicago Lectures in Mathematics, (1997).
doi: 10.7208/chicago/9780226662237.001.0001. |
[21] |
Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows,, Comm. Math. Phys., 216 (2001), 277.
doi: 10.1007/s002200000329. |
[22] |
Ya. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Markov Moran geometric constructions,, J. Statist. Phys., 86 (1997), 233.
doi: 10.1007/BF02180206. |
[23] |
Ya. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples,, Chaos, 7 (1997), 89.
doi: 10.1063/1.166242. |
[24] |
M. Pollicott and H. Weiss, The dimensions of some self-affine limit sets in the plane and hyperbolic sets,, J. Statist. Phys., 77 (1994), 841.
doi: 10.1007/BF02179463. |
[25] |
F. Przytycki and M. Urbański, On the Hausdorff dimension of some fractal sets,, Studia Math., 93 (1989), 155.
|
[26] |
D. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters,, Ergodic Theory Dynam. Systems, 9 (1989), 527.
doi: 10.1017/S0143385700005162. |
[27] |
M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds,, Israel J. Math., 15 (1973), 92.
doi: 10.1007/BF02771776. |
[28] |
D. Ruelle, Repellers for real analytic maps,, Ergodic Theory Dynam. Systems, 2 (1982), 99.
doi: 10.1017/S0143385700009603. |
[29] |
F. Takens, Limit capacity and Hausdorff dimension of dynamically defined Cantor sets,, in Dynamical Systems (Valparaiso, (1986), 196.
doi: 10.1007/BFb0083074. |
[30] |
P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).
|
[31] |
L.-S. Young, Dimension, entropy and Lyapunov exponents,, Ergodic Theory Dynam. Systems, 2 (1982), 109.
doi: 10.1017/S0143385700009615. |
show all references
References:
[1] |
L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics,, Progress in Mathematics, (2008).
doi: 10.1007/978-3-7643-8882-9. |
[2] |
L. Barreira, Dimension Theory of Hyperbolic Flows,, Springer Monographs in Mathematics, (2013).
doi: 10.1007/978-3-319-00548-5. |
[3] |
L. Barreira and P. Doutor, Birkhoff averages for hyperbolic flows: Variational principles and applications,, J. Statist. Phys., 115 (2004), 1567.
doi: 10.1023/B:JOSS.0000028069.64945.65. |
[4] |
L. Barreira and P. Doutor, Dimension spectra of hyperbolic flows,, J. Stat. Phys., 136 (2009), 505.
doi: 10.1007/s10955-009-9790-5. |
[5] |
L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows,, Comm. Math. Phys., 214 (2000), 339.
doi: 10.1007/s002200000268. |
[6] |
L. Barreira and B. Saussol, Variational principles for hyperbolic flows,, in Differential Equations and Dynamical Systems (Lisbon, (2000), 43.
|
[7] |
L. Barreira and C. Wolf, Dimension and ergodic decompositions for hyperbolic flows,, Discrete Contin. Dyn. Syst., 17 (2007), 201.
doi: 10.3934/dcds.2007.17.201. |
[8] |
R. Bowen, Symbolic dynamics for hyperbolic flows,, Amer. J. Math., 95 (1973), 429.
doi: 10.2307/2373793. |
[9] |
R. Bowen, Hausdorff dimension of quasi-circles,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11.
doi: 10.1007/BF02684767. |
[10] |
R. Bowen and P. Walters, Expansive one-parameter flows,, J. Differential Equations, 12 (1972), 180.
doi: 10.1016/0022-0396(72)90013-7. |
[11] |
M. Brin and A. Katok, On local entropy,, in Geometric Dynamics (Rio de Janeiro, (1981), 30.
doi: 10.1007/BFb0061408. |
[12] |
K. Burns and K. Gelfert, Lyapunov spectrum for geodesic flows of rank 1 surfaces,, Discrete Contin. Dyn. Syst., 34 (2014), 1841.
doi: 10.3934/dcds.2014.34.1841. |
[13] |
P. Collet, J. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems,, J. Stat. Phys., 47 (1987), 609.
doi: 10.1007/BF01206149. |
[14] |
K. Falconer, Dimensions and measures of quasi self-similar sets,, Proc. Amer. Math. Soc., 106 (1989), 543.
doi: 10.1090/S0002-9939-1989-0969315-8. |
[15] |
T. Halsey, M. Jensen, L. Kadanoff, I. Procaccia and B. Shraiman, Fractal measures and their singularities: The characterization of strange sets,, Phys. Rev. A (3), 34 (1986).
doi: 10.1103/PhysRevA.33.1141. |
[16] |
B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations,, Ergodic Theory Dynam. Systems, 14 (1994), 645.
doi: 10.1017/S0143385700008105. |
[17] |
A. Lopes, The dimension spectrum of the maximal measure,, SIAM J. Math. Anal., 20 (1989), 1243.
doi: 10.1137/0520081. |
[18] |
H. McCluskey and A. Manning, Hausdorff dimension for horseshoes,, Ergodic Theory Dynam. Systems, 3 (1983), 251.
doi: 10.1017/S0143385700001966. |
[19] |
J. Palis and M. Viana, On the continuity of Hausdorff dimension and limit capacity for horseshoes,, in Dynamical Systems (Valparaiso, (1986), 150.
doi: 10.1007/BFb0083071. |
[20] |
Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications,, Chicago Lectures in Mathematics, (1997).
doi: 10.7208/chicago/9780226662237.001.0001. |
[21] |
Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows,, Comm. Math. Phys., 216 (2001), 277.
doi: 10.1007/s002200000329. |
[22] |
Ya. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Markov Moran geometric constructions,, J. Statist. Phys., 86 (1997), 233.
doi: 10.1007/BF02180206. |
[23] |
Ya. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples,, Chaos, 7 (1997), 89.
doi: 10.1063/1.166242. |
[24] |
M. Pollicott and H. Weiss, The dimensions of some self-affine limit sets in the plane and hyperbolic sets,, J. Statist. Phys., 77 (1994), 841.
doi: 10.1007/BF02179463. |
[25] |
F. Przytycki and M. Urbański, On the Hausdorff dimension of some fractal sets,, Studia Math., 93 (1989), 155.
|
[26] |
D. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters,, Ergodic Theory Dynam. Systems, 9 (1989), 527.
doi: 10.1017/S0143385700005162. |
[27] |
M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds,, Israel J. Math., 15 (1973), 92.
doi: 10.1007/BF02771776. |
[28] |
D. Ruelle, Repellers for real analytic maps,, Ergodic Theory Dynam. Systems, 2 (1982), 99.
doi: 10.1017/S0143385700009603. |
[29] |
F. Takens, Limit capacity and Hausdorff dimension of dynamically defined Cantor sets,, in Dynamical Systems (Valparaiso, (1986), 196.
doi: 10.1007/BFb0083074. |
[30] |
P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).
|
[31] |
L.-S. Young, Dimension, entropy and Lyapunov exponents,, Ergodic Theory Dynam. Systems, 2 (1982), 109.
doi: 10.1017/S0143385700009615. |
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