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December  2015, 20(10): 3345-3362. doi: 10.3934/dcdsb.2015.20.3345

Dimension theory of flows: A survey

Luis Barreira 1,

1. 

Departamento de Matemática, Instituto Superior Técnico, UTL, 1049-001 Lisboa

Received  November 2014 Revised  March 2015 Published  September 2015

This is a survey on recent developments of the dimension theory of flows, with emphasis on hyperbolic flows. In particular, we describe various results of the dimension theory and multifractal analysis of flows, including the dimension of hyperbolic sets, the dimension of invariant measures, the multifractal analysis of equilibrium measures, conditional variational principles, multidimensional spectra, and dimension spectra taking both into account past and future. The dimension theory and the multifractal analysis of dynamical systems grew out exponentially during the last two decades, but for various reasons flows have initially been given less attention than maps. We emphasize that this is not the case anymore and the survey is also an invitation to the theory.
Keywords: hyperbolic flows, multifractal analysis, Dimension theory, thermodynamic formalism, variational principles..
Mathematics Subject Classification: Primary: 37C45, 37D20, 37D3.
Citation: Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345
References:
[1]

L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics, 272, Birkhäuser, 2008. doi: 10.1007/978-3-7643-8882-9.  Google Scholar

[2]

L. Barreira, Dimension Theory of Hyperbolic Flows, Springer Monographs in Mathematics, Springer, 2013. doi: 10.1007/978-3-319-00548-5.  Google Scholar

[3]

L. Barreira and P. Doutor, Birkhoff averages for hyperbolic flows: Variational principles and applications, J. Statist. Phys., 115 (2004), 1567-1603. doi: 10.1023/B:JOSS.0000028069.64945.65.  Google Scholar

[4]

L. Barreira and P. Doutor, Dimension spectra of hyperbolic flows, J. Stat. Phys., 136 (2009), 505-525. doi: 10.1007/s10955-009-9790-5.  Google Scholar

[5]

L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows, Comm. Math. Phys., 214 (2000), 339-371. doi: 10.1007/s002200000268.  Google Scholar

[6]

L. Barreira and B. Saussol, Variational principles for hyperbolic flows, in Differential Equations and Dynamical Systems (Lisbon, 2000), Fields Inst. Commun., 31, Amer. Math. Soc., Providence, RI, 2002, 43-63.  Google Scholar

[7]

L. Barreira and C. Wolf, Dimension and ergodic decompositions for hyperbolic flows, Discrete Contin. Dyn. Syst., 17 (2007), 201-212. doi: 10.3934/dcds.2007.17.201.  Google Scholar

[8]

R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. doi: 10.2307/2373793.  Google Scholar

[9]

R. Bowen, Hausdorff dimension of quasi-circles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25. doi: 10.1007/BF02684767.  Google Scholar

[10]

R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193. doi: 10.1016/0022-0396(72)90013-7.  Google Scholar

[11]

M. Brin and A. Katok, On local entropy, in Geometric Dynamics (Rio de Janeiro, 1981) (ed. J. Palis), Lect. Notes in Math., 1007, Springer, 1983, 30-38. doi: 10.1007/BFb0061408.  Google Scholar

[12]

K. Burns and K. Gelfert, Lyapunov spectrum for geodesic flows of rank 1 surfaces, Discrete Contin. Dyn. Syst., 34 (2014), 1841-1872. doi: 10.3934/dcds.2014.34.1841.  Google Scholar

[13]

P. Collet, J. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems, J. Stat. Phys., 47 (1987), 609-644. doi: 10.1007/BF01206149.  Google Scholar

[14]

K. Falconer, Dimensions and measures of quasi self-similar sets, Proc. Amer. Math. Soc., 106 (1989), 543-554. doi: 10.1090/S0002-9939-1989-0969315-8.  Google Scholar

[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), p1601. doi: 10.1103/PhysRevA.33.1141.  Google Scholar

[16]

B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory Dynam. Systems, 14 (1994), 645-666. doi: 10.1017/S0143385700008105.  Google Scholar

[17]

A. Lopes, The dimension spectrum of the maximal measure, SIAM J. Math. Anal., 20 (1989), 1243-1254. doi: 10.1137/0520081.  Google Scholar

[18]

H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergodic Theory Dynam. Systems, 3 (1983), 251-260. doi: 10.1017/S0143385700001966.  Google Scholar

[19]

J. Palis and M. Viana, On the continuity of Hausdorff dimension and limit capacity for horseshoes, in Dynamical Systems (Valparaiso, 1986) (eds. R. Bamón, R. Labarca and J. Palis), Lect. Notes in Math., 1331, Springer, 1988, 150-160. doi: 10.1007/BFb0083071.  Google Scholar

[20]

Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics, Chicago University Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar

[21]

Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows, Comm. Math. Phys., 216 (2001), 277-312. doi: 10.1007/s002200000329.  Google Scholar

[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), 233-275. doi: 10.1007/BF02180206.  Google Scholar

[23]

Ya. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples, Chaos, 7 (1997), 89-106. doi: 10.1063/1.166242.  Google Scholar

[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), 841-866. doi: 10.1007/BF02179463.  Google Scholar

[25]

F. Przytycki and M. Urbański, On the Hausdorff dimension of some fractal sets, Studia Math., 93 (1989), 155-186.  Google Scholar

[26]

D. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters, Ergodic Theory Dynam. Systems, 9 (1989), 527-541. doi: 10.1017/S0143385700005162.  Google Scholar

[27]

M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math., 15 (1973), 92-114. doi: 10.1007/BF02771776.  Google Scholar

[28]

D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems, 2 (1982), 99-107. doi: 10.1017/S0143385700009603.  Google Scholar

[29]

F. Takens, Limit capacity and Hausdorff dimension of dynamically defined Cantor sets, in Dynamical Systems (Valparaiso, 1986) (eds. R. Bamón, R. Labarca and J. Palis), Lect. Notes in Math., 1331, Springer, 1988, 196-212. doi: 10.1007/BFb0083074.  Google Scholar

[30]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer, 1982.  Google Scholar

[31]

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

show all references

References:
[1]

L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics, 272, Birkhäuser, 2008. doi: 10.1007/978-3-7643-8882-9.  Google Scholar

[2]

L. Barreira, Dimension Theory of Hyperbolic Flows, Springer Monographs in Mathematics, Springer, 2013. doi: 10.1007/978-3-319-00548-5.  Google Scholar

[3]

L. Barreira and P. Doutor, Birkhoff averages for hyperbolic flows: Variational principles and applications, J. Statist. Phys., 115 (2004), 1567-1603. doi: 10.1023/B:JOSS.0000028069.64945.65.  Google Scholar

[4]

L. Barreira and P. Doutor, Dimension spectra of hyperbolic flows, J. Stat. Phys., 136 (2009), 505-525. doi: 10.1007/s10955-009-9790-5.  Google Scholar

[5]

L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows, Comm. Math. Phys., 214 (2000), 339-371. doi: 10.1007/s002200000268.  Google Scholar

[6]

L. Barreira and B. Saussol, Variational principles for hyperbolic flows, in Differential Equations and Dynamical Systems (Lisbon, 2000), Fields Inst. Commun., 31, Amer. Math. Soc., Providence, RI, 2002, 43-63.  Google Scholar

[7]

L. Barreira and C. Wolf, Dimension and ergodic decompositions for hyperbolic flows, Discrete Contin. Dyn. Syst., 17 (2007), 201-212. doi: 10.3934/dcds.2007.17.201.  Google Scholar

[8]

R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. doi: 10.2307/2373793.  Google Scholar

[9]

R. Bowen, Hausdorff dimension of quasi-circles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25. doi: 10.1007/BF02684767.  Google Scholar

[10]

R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193. doi: 10.1016/0022-0396(72)90013-7.  Google Scholar

[11]

M. Brin and A. Katok, On local entropy, in Geometric Dynamics (Rio de Janeiro, 1981) (ed. J. Palis), Lect. Notes in Math., 1007, Springer, 1983, 30-38. doi: 10.1007/BFb0061408.  Google Scholar

[12]

K. Burns and K. Gelfert, Lyapunov spectrum for geodesic flows of rank 1 surfaces, Discrete Contin. Dyn. Syst., 34 (2014), 1841-1872. doi: 10.3934/dcds.2014.34.1841.  Google Scholar

[13]

P. Collet, J. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems, J. Stat. Phys., 47 (1987), 609-644. doi: 10.1007/BF01206149.  Google Scholar

[14]

K. Falconer, Dimensions and measures of quasi self-similar sets, Proc. Amer. Math. Soc., 106 (1989), 543-554. doi: 10.1090/S0002-9939-1989-0969315-8.  Google Scholar

[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), p1601. doi: 10.1103/PhysRevA.33.1141.  Google Scholar

[16]

B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory Dynam. Systems, 14 (1994), 645-666. doi: 10.1017/S0143385700008105.  Google Scholar

[17]

A. Lopes, The dimension spectrum of the maximal measure, SIAM J. Math. Anal., 20 (1989), 1243-1254. doi: 10.1137/0520081.  Google Scholar

[18]

H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergodic Theory Dynam. Systems, 3 (1983), 251-260. doi: 10.1017/S0143385700001966.  Google Scholar

[19]

J. Palis and M. Viana, On the continuity of Hausdorff dimension and limit capacity for horseshoes, in Dynamical Systems (Valparaiso, 1986) (eds. R. Bamón, R. Labarca and J. Palis), Lect. Notes in Math., 1331, Springer, 1988, 150-160. doi: 10.1007/BFb0083071.  Google Scholar

[20]

Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics, Chicago University Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar

[21]

Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows, Comm. Math. Phys., 216 (2001), 277-312. doi: 10.1007/s002200000329.  Google Scholar

[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), 233-275. doi: 10.1007/BF02180206.  Google Scholar

[23]

Ya. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples, Chaos, 7 (1997), 89-106. doi: 10.1063/1.166242.  Google Scholar

[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), 841-866. doi: 10.1007/BF02179463.  Google Scholar

[25]

F. Przytycki and M. Urbański, On the Hausdorff dimension of some fractal sets, Studia Math., 93 (1989), 155-186.  Google Scholar

[26]

D. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters, Ergodic Theory Dynam. Systems, 9 (1989), 527-541. doi: 10.1017/S0143385700005162.  Google Scholar

[27]

M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math., 15 (1973), 92-114. doi: 10.1007/BF02771776.  Google Scholar

[28]

D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems, 2 (1982), 99-107. doi: 10.1017/S0143385700009603.  Google Scholar

[29]

F. Takens, Limit capacity and Hausdorff dimension of dynamically defined Cantor sets, in Dynamical Systems (Valparaiso, 1986) (eds. R. Bamón, R. Labarca and J. Palis), Lect. Notes in Math., 1331, Springer, 1988, 196-212. doi: 10.1007/BFb0083074.  Google Scholar

[30]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer, 1982.  Google Scholar

[31]

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

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