Multifractal Analysis for Gibbs Ground States

MESÓN, ALEJANDRO; VERICAT, FERNANDO

doi:10.1080/1726037X.2006.10698512

Open Access Original Articles

Multifractal Analysis for Gibbs Ground States

ALEJANDRO MESÓNmeson@iflysib.unlp.edu.arGrupo de Aplicaciones Matemáticas y Estadísticas de la Facultad de Ingeniería (GAMEFI)Instituto de Física de Líquidos y Sistemas Biológicos (IFLYSIB) CONICET-UNLPUNLP, La Plata, ArgentinaView full profile → , FERNANDO VERICATvericat@iflysib.unlp.edu.arGrupo de Aplicaciones Matemáticas y Estadísticas de la Facultad de Ingeniería (GAMEFI)Instituto de Física de Líquidos y Sistemas Biológicos (IFLYSIB) CONICET-UNLPUNLP, La Plata, ArgentinaView full profile →

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pp. 147–158Vol. 4Issue 2November 2006DOI: 10.1080/1726037X.2006.10698512XML

Received:: 22 Dec 2005
Published Online:: 03 Jun 2013
Article type:: Original Articles
Language:: EN
Article no.:: 1726037X.2006.10698512
Pages:: 147–158

Abstract

We consider a family of Gibbs states describing the multifractal spectrum of local entropies. An accumulation point μ∞ of this sequence may be called a Gibbs ground state or a zero temperature limit, because the interpretation of q as the inverse of the temperature. We prove that, in more general systems than symbolics, that μ∞ is a maximizing measure. We also do geometric and combinatorial approaches as well as some interpretations in a particular case.

Keywords

Gibbs ground states Zero limit temperature Boundary hyperbolic maps

References

Barreira, L., Pesin, Y. and Schmeling, J.1997 . On a General Concept of Multifractality: Multifractal Spectra for Dimension, Entropies and Lyapunov Exponents. Multifractal Rigidity . , 7 ( 1 ) : 27
Beardon, A.F.1983 . , Graduate texts in Mathematics New York: Springer .
Brémont, J.2003 . Gibbs measures at temperature zero . , 16 : 419 – 426 .
Brin and Katok, A.1983 . , Springer .
Bowen, R.E.1971 . Periodic points and measures for Axiom-A diffeomorphisms . , 154 : 377 – 397 .
Bowen, R.1973 . Topological entropy for non-compact sets . , 184 : 125 – 136 .
Bowen, R.E. and Series, C.1979 . Markov maps associated to Fuchsian groups . , 50 : 153 – 170 .
Cannon, J.1987 . Almost convex groups . , 22 : 197 – 210 .
Floyd, W. and Plotnick, S.1980 . Groups completations and limit sets of Kleinian groups . , 57 : 205 – 218 .
Katok, A. and Hasselblatt, B.1995 . , Cambridge Univ. Pres .
Jenkinson, O., Mauldin, R.D. and Urbanski, M.2005 . Zero temperature limits of Gibbs-equilibrium states for countable alphabet subshifts of finite type . , 119 : 765 – 776 .
Mesón, A.M. and Vericat, F.2002 . Geometric constructions and multifractal analysis for boundary hyperbolic maps . , 17 : 203 – 213 .
Mesón, A.M. and Vericat, F.2004 . Dimension Theory and Fuchsian Groups . , 80 : 95 – 121 .
Pollicott, M. and Weiss, H.2003 . Free energy as a dynamical and geometric invariant (or can you hear the shape of a potential) . , 240 : 457 – 482 .
Ruelle, D.1978 . “ Thermodybamic Formalism ” . In , Addison-Wesley .
Series, C.1991 . , Edited by: Bedford, T., Keane, M. and Series, C.Oxford: Oxford University Press .
Series, C.1986 . Geometrical Markov coding on surfaces of constant negative curvature . , 6
Takens, F. and Verbitski, E.1999 . Multifractal Analysis of Local Entropies for Expansive Homeo-morphisms with Specification . , 203 : 593
Walters, P.1982 . , Berlin: Springer-Verlag .

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[1] Barreira, L., Pesin, Y. and Schmeling, J.1997 . On a General Concept of Multifractality: Multifractal Spectra for Dimension, Entropies and Lyapunov Exponents. Multifractal Rigidity . , 7 ( 1 ) : 27

[2] Beardon, A.F.1983 . , Graduate texts in Mathematics New York: Springer .

[3] Brémont, J.2003 . Gibbs measures at temperature zero . , 16 : 419 – 426 .

[4] Brin and Katok, A.1983 . , Springer .

[5] Bowen, R.E.1971 . Periodic points and measures for Axiom-A diffeomorphisms . , 154 : 377 – 397 .

[6] Bowen, R.1973 . Topological entropy for non-compact sets . , 184 : 125 – 136 .

[7] Bowen, R.E. and Series, C.1979 . Markov maps associated to Fuchsian groups . , 50 : 153 – 170 .

[8] Cannon, J.1987 . Almost convex groups . , 22 : 197 – 210 .

[9] Floyd, W. and Plotnick, S.1980 . Groups completations and limit sets of Kleinian groups . , 57 : 205 – 218 .

[10] Katok, A. and Hasselblatt, B.1995 . , Cambridge Univ. Pres .

[11] Jenkinson, O., Mauldin, R.D. and Urbanski, M.2005 . Zero temperature limits of Gibbs-equilibrium states for countable alphabet subshifts of finite type . , 119 : 765 – 776 .

[12] Mesón, A.M. and Vericat, F.2002 . Geometric constructions and multifractal analysis for boundary hyperbolic maps . , 17 : 203 – 213 .

[13] Mesón, A.M. and Vericat, F.2004 . Dimension Theory and Fuchsian Groups . , 80 : 95 – 121 .

[14] Pollicott, M. and Weiss, H.2003 . Free energy as a dynamical and geometric invariant (or can you hear the shape of a potential) . , 240 : 457 – 482 .

[15] Ruelle, D.1978 . “ Thermodybamic Formalism ” . In , Addison-Wesley .

[16] Series, C.1991 . , Edited by: Bedford, T., Keane, M. and Series, C.Oxford: Oxford University Press .

[17] Series, C.1986 . Geometrical Markov coding on surfaces of constant negative curvature . , 6

[18] Takens, F. and Verbitski, E.1999 . Multifractal Analysis of Local Entropies for Expansive Homeo-morphisms with Specification . , 203 : 593

[19] Walters, P.1982 . , Berlin: Springer-Verlag .

Journal of Dynamical Systems and Geometric Theories

WoS JIF 2026 : 0.5 (Q3)

Multifractal Analysis for Gibbs Ground States

Abstract

Keywords

References