On a Multifractal Pressure for Countable Markov Shifts

MESÓN, ALEJANDRO; VERICAT, FERNANDO

doi:10.1080/1726037X.2019.1668150

Open Access Article

On a Multifractal Pressure for Countable Markov Shifts

^*ALEJANDRO MESÓNCorresponding authormeson@iflysib.unlp.edu.arConicet-UNLP and Grupo De Aplicaciones Matemáticas Y Estadísticas De La Facultad De Ingeniería (Gamefi) UnlpInstituto De Física De Líquidos Y Sistemas Biológicos (IFLYSIB)La Plata, ArgentinaView full profile → , FERNANDO VERICATvericat@iflysib.unlp.edu.arConicet-UNLP and Grupo De Aplicaciones Matemáticas Y Estadísticas De La Facultad De Ingeniería (Gamefi) UnlpInstituto De Física De Líquidos Y Sistemas Biológicos (IFLYSIB)La Plata, ArgentinaView full profile →

* Corresponding author · click or hover a name for details

pp. 267–295Vol. 17Issue 2July 2019DOI: 10.1080/1726037X.2019.1668150XML

Received:: 15 Nov 2018
Accepted:: 18 Mar 2019
Published Online:: 18 Nov 2019
Article type:: Article
Language:: EN
Article no.:: 1726037X.2019.1668150
Pages:: 267–295

Abstract

In a recent article [J. d' Analyse Math131, 207, 2017], Olsen intoduced a generalized notion of multifractal pressure, and also a multifractal dynamical zeta function, which essentilly consists in considering not all configurations, but those which are ”multifractally relevant”. In this way more precise information about the multifractal spectrum analyzed is encoded by the multifractal pressure and the multifratcal zeta function. He applied the theory for dynamical systems modelled by finite alphabet shifts, in particular for self conformal iterated systems. Here we continue with this line considering dynamical systems given by countable Markov shifts.

Keywords

Topological pressureMarkov shiftzeta function

Subject Classifications

37B1037C30

References

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L.Barreira, Y.Pesin and J.Schmeling, On a general goncept of multifractality: Multifractal spectra for dimension, entropies and Lyapunov exponents. Multifractal rigidity , Chaos , Vol. 7 , Number 1 , 27 - 38 1997 . doi: 10.1063/1.166232
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C.Ercai, T. K u per, S. Lin, Topological entropy for divergent points , Ergod. Th. and Dynam. Sys ., Vol. 25 , 1173 - 1208 ( 2005 ). doi: 10.1017/S0143385704000872
K. J.Falconer, Techniques in Fractal Geometry , John Wiley & sons , 1997 .
A.Fan, J.Schmeling and M.Wu, The multifractal spectra of statistics. Further developments in fractals and related fields. Mathematical foundations and connections. Outgrowth of the 2nd international conference on fractals and related fields, Porquerolles Island, France, June 2011
T.C.Halsey, M.Jensen, L.Kadanoff, I, Procaccia and B.Shraiman, Fractal measures and their singularities: The characterization of strange sets , Phys. Rev. A , Vol. 33 , 1141 - 1151 ( 1986 ). doi: 10.1103/PhysRevA.33.1141
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G.Iommi and T.Jordan, Multifractal analysis and Birkhoff averages for countable Markov maps , Ergod. Th. and Dynam. Sys ., Vol. 35 , Number 8 , 2559 - 2586 ( 2015 ). doi: 10.1017/etds.2015.44
G.Iommi, Multifractal analysis for countable Markov shifts , Ergod. Th. and Dynam. Sys ., Vol. 23 , 1881 - 1907 ( 2005 ). doi: 10.1017/S0143385705000350
T.Jordan and M.Pollicott, Multifractal analysis and the variance of Gibbs measures , Journal of the London Mathematical Society , Vol. 76 , 57 - 72 , ( 2007 ). doi: 10.1112/jlms/jdm023
M.Lapidus, J.Lévy-Vehel and J.Rock, Fractal strings and multifractal zeta functions , Lett. Math. Phys , Vol. 88 , 101 - 129 ( 2009 ). doi: 10.1007/s11005-009-0302-y
M.Lapidus and J.Rock, Towards zeta functions and complex dimensions of multifractals , Complex Var Elliptic Equ . Vol. 54 , 545 – 559 , ( 2009 ). doi: 10.1080/17476930802326758
J.Li, M.Wu and Y.Xiong, Hausdorff dimensions of the divergence points of self-similar measures with the open set condition , Nonlinearity , Vol. 25 93 - 105 ( 2012 ). doi: 10.1088/0951-7715/25/1/93
R. D.Mauldin and M.Urbanski, Gibbs states on the symbolic space over an infinite alphabet , Israel J. of Math ., Vol. 125 , 93 - 130 , ( 2001 ). doi: 10.1007/BF02773377
D.Mauldin and M.Urbanski, Dimension and measures in infinite iterated function systems ., Proceedings London Math. Soc ., Vol. 73 , 105 - 154 ( 1996 ). doi: 10.1112/plms/s3-73.1.105
A.Mesón and F.Vericat, On the topological entropy of the irregular part of statistics multifractal spectra , J. Dynam. Sys. and Geom. Theories , Vol. 11 , 1 - 12 , ( 2013 ). doi: 10.1080/1726037X.2013.810440
A.Mesón and F.Vericat, Multifratcal spectrum for barycentric averages , J. Dynam. and Control Sys ., Vol. 22 , 623 - 635 ( 2016 ). doi: 10.1007/s10883-015-9278-3
L.Olsen, Dynamical multifractal zeta functions, multifractal pressure and fine multifractal spectrum , J. d’Analyse Math ., Vol. 131 , 207 - 253 ., ( 2017 ) doi: 10.1007/s11854-017-0007-6
W.Parry and M.Pollicott, An analogue of the prime number theorem for closed orbits of Axiom-A flows , Ann. of Math ., Vol. 118 , 573 - 591 ( 1983 ). doi: 10.2307/2006982
Y.Pesin, Dimension theory in dynamical systems, contemporary views and applications , The University of Chicago Press , 1997 .
Y.Pesin and H.Weiss, A Multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric gonstructions , J. Statatistical Phys ., Vol. 86 , 233 - 275 ( 1997 ). doi: 10.1007/BF02180206
O.Sarig, Existence of Gibbs measures for countable Markov shifts , Proc. Amer. Math. Soc ., Vol. 131 , 1751 - 1758 ( 2003 ). doi: 10.1090/S0002-9939-03-06927-2
Sarig, O.: Thermodynamic formalism for countable Markov shifts , . , 81 – 117 .( 2015 ). doi: 10.1090/pspum/089/01485
H.Takahasi, Large deviations principles for countable Markov shifts, Preprint, (2018)
F.Takens and E.Verbitski, Multifractal analysis of local entropies for expansive homeomorphisms with specifications , Commun. Math ., Vol. 203 , 593 - 612 ( 1999 ). doi: 10.1007/s002200050627
F.Takens and E.Verbitski, On the variational principle for the topological entropy of certain non-compact sets , Fundamenta Mathematicae , Vol. 105 , 203 - 237 ( 2000 )
R.A.Wijsman, Convergence of sequences of convex sets, cones and functions , Trans. Amer. Mat. Soc ., Vol. 123 , 32 - 45 ( 1966 ). doi: 10.1090/S0002-9947-1966-0196599-8

Views: 68Downloads: 99Citations: 1

[1] M.Artin and B.Mazur, On periodic points , Ann. of Math , Vol. 81 , 82 - 99 ( 1965 ). doi: 10.2307/1970384

[2] L.Barreira, Y.Pesin and J.Schmeling, On a general goncept of multifractality: Multifractal spectra for dimension, entropies and Lyapunov exponents. Multifractal rigidity , Chaos , Vol. 7 , Number 1 , 27 - 38 1997 . doi: 10.1063/1.166232

[3] L.Barreira and B.Saussol, Multifractal Analysis of hyperbolic flows , Commun. Math. Phys , Vol. 214 , 339 - 371 , ( 2000 ), doi: 10.1007/s002200000268

[4] L.Barreira and B.Saussol, Variational principle and mixed multifractal spectra , Trans. Amer. Math. Soc ., 116 , 3919 - 3944 ( 2001 ). doi: 10.1090/S0002-9947-01-02844-6

[5] C.Ercai, T. K u per, S. Lin, Topological entropy for divergent points , Ergod. Th. and Dynam. Sys ., Vol. 25 , 1173 - 1208 ( 2005 ). doi: 10.1017/S0143385704000872

[6] K. J.Falconer, Techniques in Fractal Geometry , John Wiley & sons , 1997 .

[7] A.Fan, J.Schmeling and M.Wu, The multifractal spectra of statistics. Further developments in fractals and related fields. Mathematical foundations and connections. Outgrowth of the 2nd international conference on fractals and related fields, Porquerolles Island, France, June 2011

[8] T.C.Halsey, M.Jensen, L.Kadanoff, I, Procaccia and B.Shraiman, Fractal measures and their singularities: The characterization of strange sets , Phys. Rev. A , Vol. 33 , 1141 - 1151 ( 1986 ). doi: 10.1103/PhysRevA.33.1141

[9] H. G. E.Hentschel and I.Procaccia, The infinite number of generalized dimensions of fractals and strange attractors , Physica D , Vol. 8 , 435 - 444 ( 1983 ). doi: 10.1016/0167-2789(83)90235-X

[10] G.Iommi and T.Jordan, Multifractal analysis and Birkhoff averages for countable Markov maps , Ergod. Th. and Dynam. Sys ., Vol. 35 , Number 8 , 2559 - 2586 ( 2015 ). doi: 10.1017/etds.2015.44

[11] G.Iommi, Multifractal analysis for countable Markov shifts , Ergod. Th. and Dynam. Sys ., Vol. 23 , 1881 - 1907 ( 2005 ). doi: 10.1017/S0143385705000350

[12] T.Jordan and M.Pollicott, Multifractal analysis and the variance of Gibbs measures , Journal of the London Mathematical Society , Vol. 76 , 57 - 72 , ( 2007 ). doi: 10.1112/jlms/jdm023

[13] M.Lapidus, J.Lévy-Vehel and J.Rock, Fractal strings and multifractal zeta functions , Lett. Math. Phys , Vol. 88 , 101 - 129 ( 2009 ). doi: 10.1007/s11005-009-0302-y

[14] M.Lapidus and J.Rock, Towards zeta functions and complex dimensions of multifractals , Complex Var Elliptic Equ . Vol. 54 , 545 – 559 , ( 2009 ). doi: 10.1080/17476930802326758

[15] J.Li, M.Wu and Y.Xiong, Hausdorff dimensions of the divergence points of self-similar measures with the open set condition , Nonlinearity , Vol. 25 93 - 105 ( 2012 ). doi: 10.1088/0951-7715/25/1/93

[16] R. D.Mauldin and M.Urbanski, Gibbs states on the symbolic space over an infinite alphabet , Israel J. of Math ., Vol. 125 , 93 - 130 , ( 2001 ). doi: 10.1007/BF02773377

[17] D.Mauldin and M.Urbanski, Dimension and measures in infinite iterated function systems ., Proceedings London Math. Soc ., Vol. 73 , 105 - 154 ( 1996 ). doi: 10.1112/plms/s3-73.1.105

[18] A.Mesón and F.Vericat, On the topological entropy of the irregular part of statistics multifractal spectra , J. Dynam. Sys. and Geom. Theories , Vol. 11 , 1 - 12 , ( 2013 ). doi: 10.1080/1726037X.2013.810440

[19] A.Mesón and F.Vericat, Multifratcal spectrum for barycentric averages , J. Dynam. and Control Sys ., Vol. 22 , 623 - 635 ( 2016 ). doi: 10.1007/s10883-015-9278-3

[20] L.Olsen, Dynamical multifractal zeta functions, multifractal pressure and fine multifractal spectrum , J. d’Analyse Math ., Vol. 131 , 207 - 253 ., ( 2017 ) doi: 10.1007/s11854-017-0007-6

[21] W.Parry and M.Pollicott, An analogue of the prime number theorem for closed orbits of Axiom-A flows , Ann. of Math ., Vol. 118 , 573 - 591 ( 1983 ). doi: 10.2307/2006982

[22] Y.Pesin, Dimension theory in dynamical systems, contemporary views and applications , The University of Chicago Press , 1997 .

[23] Y.Pesin and H.Weiss, A Multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric gonstructions , J. Statatistical Phys ., Vol. 86 , 233 - 275 ( 1997 ). doi: 10.1007/BF02180206

[24] O.Sarig, Existence of Gibbs measures for countable Markov shifts , Proc. Amer. Math. Soc ., Vol. 131 , 1751 - 1758 ( 2003 ). doi: 10.1090/S0002-9939-03-06927-2

[25] Sarig, O.: Thermodynamic formalism for countable Markov shifts , . , 81 – 117 .( 2015 ). doi: 10.1090/pspum/089/01485

[26] H.Takahasi, Large deviations principles for countable Markov shifts, Preprint, (2018)

[27] F.Takens and E.Verbitski, Multifractal analysis of local entropies for expansive homeomorphisms with specifications , Commun. Math ., Vol. 203 , 593 - 612 ( 1999 ). doi: 10.1007/s002200050627

[28] F.Takens and E.Verbitski, On the variational principle for the topological entropy of certain non-compact sets , Fundamenta Mathematicae , Vol. 105 , 203 - 237 ( 2000 )

[29] R.A.Wijsman, Convergence of sequences of convex sets, cones and functions , Trans. Amer. Mat. Soc ., Vol. 123 , 32 - 45 ( 1966 ). doi: 10.1090/S0002-9947-1966-0196599-8

Journal of Dynamical Systems and Geometric Theories

WoS JIF 2026 : 0.5 (Q3)

On a Multifractal Pressure for Countable Markov Shifts

Abstract

Keywords

Subject Classifications

References