Dimension theory of flows: A survey

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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

Abstract
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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 and 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.
[2]	L. Barreira, Dimension Theory of Hyperbolic Flows, Springer Monographs in Mathematics, Springer, 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), 1567-1603. doi: 10.1023/B:JOSS.0000028069.64945.65.
[4]	L. Barreira and P. Doutor, Dimension spectra of hyperbolic flows, J. Stat. Phys., 136 (2009), 505-525. doi: 10.1007/s10955-009-9790-5.
[5]	L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows, Comm. Math. Phys., 214 (2000), 339-371. doi: 10.1007/s002200000268.
[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.
[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.
[8]	R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. doi: 10.2307/2373793.
[9]	R. Bowen, Hausdorff dimension of quasi-circles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25. doi: 10.1007/BF02684767.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory Dynam. Systems, 14 (1994), 645-666. doi: 10.1017/S0143385700008105.
[17]	A. Lopes, The dimension spectrum of the maximal measure, SIAM J. Math. Anal., 20 (1989), 1243-1254. doi: 10.1137/0520081.
[18]	H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergodic Theory Dynam. Systems, 3 (1983), 251-260. 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) (eds. R. Bamón, R. Labarca and J. Palis), Lect. Notes in Math., 1331, Springer, 1988, 150-160. doi: 10.1007/BFb0083071.
[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.
[21]	Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows, Comm. Math. Phys., 216 (2001), 277-312. 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), 233-275. doi: 10.1007/BF02180206.
[23]	Ya. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples, Chaos, 7 (1997), 89-106. 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), 841-866. doi: 10.1007/BF02179463.
[25]	F. Przytycki and M. Urbański, On the Hausdorff dimension of some fractal sets, Studia Math., 93 (1989), 155-186.
[26]	D. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters, Ergodic Theory Dynam. Systems, 9 (1989), 527-541. doi: 10.1017/S0143385700005162.
[27]	M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math., 15 (1973), 92-114. doi: 10.1007/BF02771776.
[28]	D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems, 2 (1982), 99-107. doi: 10.1017/S0143385700009603.
[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.
[30]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer, 1982.
[31]	L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems, 2 (1982), 109-124. doi: 10.1017/S0143385700009615.

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.
[2]	L. Barreira, Dimension Theory of Hyperbolic Flows, Springer Monographs in Mathematics, Springer, 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), 1567-1603. doi: 10.1023/B:JOSS.0000028069.64945.65.
[4]	L. Barreira and P. Doutor, Dimension spectra of hyperbolic flows, J. Stat. Phys., 136 (2009), 505-525. doi: 10.1007/s10955-009-9790-5.
[5]	L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows, Comm. Math. Phys., 214 (2000), 339-371. doi: 10.1007/s002200000268.
[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.
[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.
[8]	R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. doi: 10.2307/2373793.
[9]	R. Bowen, Hausdorff dimension of quasi-circles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25. doi: 10.1007/BF02684767.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory Dynam. Systems, 14 (1994), 645-666. doi: 10.1017/S0143385700008105.
[17]	A. Lopes, The dimension spectrum of the maximal measure, SIAM J. Math. Anal., 20 (1989), 1243-1254. doi: 10.1137/0520081.
[18]	H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergodic Theory Dynam. Systems, 3 (1983), 251-260. 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) (eds. R. Bamón, R. Labarca and J. Palis), Lect. Notes in Math., 1331, Springer, 1988, 150-160. doi: 10.1007/BFb0083071.
[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.
[21]	Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows, Comm. Math. Phys., 216 (2001), 277-312. 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), 233-275. doi: 10.1007/BF02180206.
[23]	Ya. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples, Chaos, 7 (1997), 89-106. 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), 841-866. doi: 10.1007/BF02179463.
[25]	F. Przytycki and M. Urbański, On the Hausdorff dimension of some fractal sets, Studia Math., 93 (1989), 155-186.
[26]	D. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters, Ergodic Theory Dynam. Systems, 9 (1989), 527-541. doi: 10.1017/S0143385700005162.
[27]	M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math., 15 (1973), 92-114. doi: 10.1007/BF02771776.
[28]	D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems, 2 (1982), 99-107. doi: 10.1017/S0143385700009603.
[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.
[30]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer, 1982.
[31]	L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems, 2 (1982), 109-124. doi: 10.1017/S0143385700009615.

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