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July  2021, 41(7): 3241-3271. doi: 10.3934/dcds.2020404

Another point of view on Kusuoka's measure

Ugo Bessi

Dipartimento di Matematica e Fisica, Largo S. Leonardo Murialdo 1, 00146, Roma

Received  December 2019 Revised  September 2020 Published  July 2021 Early access  December 2020

Fund Project: Work partially supported by the PRIN2009 grant "Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations"

Kusuoka's measure on fractals is a Gibbs measure of a very special kind, since its potential is discontinuous while the standard theory of Gibbs measures requires continuous (in its simplest version, Hölder) potentials. In this paper we shall see that for many fractals it is possible to build a class of matrix-valued Gibbs measures completely within the scope of the standard theory; there are naturally some minor modifications, but they are only due to the fact that we are dealing with matrix-valued functions and measures. We shall use these matrix-valued Gibbs measures to build self-similar bilinear forms on fractals. Moreover, we shall see that Kusuoka's measure and bilinear form can be recovered in a simple way from the matrix-valued Gibbs measure.

Keywords: Fractal sets, Kusuoka's measure, matrix-valued Gibbs measures.
Mathematics Subject Classification: Primary: 28A80; Secondary: 31C25.
Citation: Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3241-3271. doi: 10.3934/dcds.2020404
References:
[1]

M. T. Barlow and R. F. Bass, The construction of Brownian motion on the Sierpinski carpet, Ann. IHP, 25 (1989), 225-257. http://www.numdam.org/item?id=AIHPB_1989__25_3_225_0.  Google Scholar

[2]

M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpiski gasket, Probab. Th. Rel. Fields, 79 (1988), 543-623.  doi: 10.1007/BF00318785.  Google Scholar

[3]

R. Bell, C.-W. Ho and R. S. Strichartz, Energy measures of harmonic functions on the Sierpinski gasket, Indiana Univ. Math, 63 (2014), 831-868.  doi: 10.1512/iumj.2014.63.5256.  Google Scholar

[4]

G. Birkhoff, Lattice Theory, Third Edition, AMS Colloquium Publ., AMS, Providence, R. I., Vol. XXV, 1967.  Google Scholar

[5]

S. Chiari, J. Frisch, D. J. Kelleher and L. G. Rogers, Measurable Riemannian structure on higher dimensional harmonic Sierpinski gaskets, Preprint, (2017). Google Scholar

[6]

D.-J. Feng and A. Käenmäki, Equilibrium states for the pressure function for products of matrices, Discrete Continuous Dynam. Systems, 30 (2011), 699-708.  doi: 10.3934/dcds.2011.30.699.  Google Scholar

[7]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, Göttingen, 2011.  Google Scholar

[8]

S. Goldstein, Random walks and diffusions on fractals, in Percolation Theory and Ergodic Theory of Infinite Particle Systems, (ed. H. Kesten), Springer, New York, 121-129, 1987. doi: 10.1007/978-1-4613-8734-3_8.  Google Scholar

[9]

B. M. Hambly, V. Metz and A. Teplyaev, Self-similar energies on post-critically finite self-similar fractals, J. London. Math. Soc., 74 (2006), 93-112.  doi: 10.1112/S002461070602312X.  Google Scholar

[10]

F. Hirsch, Opérateurs carré du champ, in Sèminaire Bourbaki, 1978,167-182. doi: 10.1007/BFb0070761.  Google Scholar

[11]

A. Johansson, A. Öberg and M. Pollicott, Ergodic theory of Kusuoka's measures, J. Fractal Geom., 4 (2017), 185-214.  doi: 10.4171/JFG/49.  Google Scholar

[12]

N. Kajino, Analysis and geometry of the measurable Riemannian structure on the Sierpinski gasket, in Contemporary Math., Amer. Math. Soc., Providence, RI, 600 2013. doi: 10.1090/conm/600/11932.  Google Scholar

[13]

J. Kigami, Analysis on Fractals, Cambridge tracts in Math., Cambridge Univ. Press, Cambridge, 143 2001. doi: 10.1017/CBO9780511470943.  Google Scholar

[14]

P. Koskela and Y. Zhou, Geometry and analysis of Dirichlet forms, Adv. Math., 231 (2012), 2755-2801.  doi: 10.1016/j.aim.2012.08.004.  Google Scholar

[15]

S. Kusuoka, A diffusion process on a fractal, in Probabilistic methods in Mathematical Physics, (eds. K. Ito and N. Ikeda) Academic Press, Boston, MA, 1987,251-274.  Google Scholar

[16]

S. Kusuoka, Dirichlet forms on fractals and products of random matrices, Publ. Res. Inst. Math. Sci., 25 (1989), 659-680.  doi: 10.2977/prims/1195173187.  Google Scholar

[17]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[18]

I. D. Morris, Ergodic properties of matrix equilibrium state, Ergodic Theory and Dyn. Sys., 38 (2018), 2295-2320.  doi: 10.1017/etds.2016.117.  Google Scholar

[19]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Functional Analysis, 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093.  Google Scholar

[20]

U. Mosco, Variational fractals, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 683-712. http://www.numdam.org/item/?id=ASNSP_1997_4_25_3-4_683_0.  Google Scholar

[21]

R. Peirone, A p. c. f. self similar fractal with no self similar energy, J. Fractal Geom., 6 (2019), 393-404.  doi: 10.4171/JFG/82.  Google Scholar

[22]

R. Peirone, Existence of self-similar energies on finitely ramified fractals, J. Anal. Math., 123 (2014), 35-94.  doi: 10.1007/s11854-014-013-x.  Google Scholar

[23]

R. Peirone, Convergence of Dirichlet forms on fractals, in Topics on Concentration Phenomena and Problems with Multiple Scales, Lect. Notes Unione Mat. Ital. 2, Springer, Berlin, 2006,139-188. doi: 10.1007/978-3-540-36546-4_3.  Google Scholar

[24]

W. Perry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Asterisque, 187-188, 1990.  Google Scholar

[25]

M. Piraino, The weak Bernoulli property for matrix equilibrium states, Ergodic Theory Dynam. Systems, 40 (2020), 2219-2238. doi: 10.1017/etds.2018.129.  Google Scholar

[26]

W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974.  Google Scholar

[27]

A. Teplyaev, Harmonic coordinates on fractals with finitely ramified cell structure, Canad. J. Math., 60 (2008), 457-480.  doi: 10.4153/CJM-2008-022-3.  Google Scholar

[28]

M. Viana, Stochastic Analysis of Deterministic Systems, Mimeographed Notes. Google Scholar

show all references

References:
[1]

M. T. Barlow and R. F. Bass, The construction of Brownian motion on the Sierpinski carpet, Ann. IHP, 25 (1989), 225-257. http://www.numdam.org/item?id=AIHPB_1989__25_3_225_0.  Google Scholar

[2]

M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpiski gasket, Probab. Th. Rel. Fields, 79 (1988), 543-623.  doi: 10.1007/BF00318785.  Google Scholar

[3]

R. Bell, C.-W. Ho and R. S. Strichartz, Energy measures of harmonic functions on the Sierpinski gasket, Indiana Univ. Math, 63 (2014), 831-868.  doi: 10.1512/iumj.2014.63.5256.  Google Scholar

[4]

G. Birkhoff, Lattice Theory, Third Edition, AMS Colloquium Publ., AMS, Providence, R. I., Vol. XXV, 1967.  Google Scholar

[5]

S. Chiari, J. Frisch, D. J. Kelleher and L. G. Rogers, Measurable Riemannian structure on higher dimensional harmonic Sierpinski gaskets, Preprint, (2017). Google Scholar

[6]

D.-J. Feng and A. Käenmäki, Equilibrium states for the pressure function for products of matrices, Discrete Continuous Dynam. Systems, 30 (2011), 699-708.  doi: 10.3934/dcds.2011.30.699.  Google Scholar

[7]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, Göttingen, 2011.  Google Scholar

[8]

S. Goldstein, Random walks and diffusions on fractals, in Percolation Theory and Ergodic Theory of Infinite Particle Systems, (ed. H. Kesten), Springer, New York, 121-129, 1987. doi: 10.1007/978-1-4613-8734-3_8.  Google Scholar

[9]

B. M. Hambly, V. Metz and A. Teplyaev, Self-similar energies on post-critically finite self-similar fractals, J. London. Math. Soc., 74 (2006), 93-112.  doi: 10.1112/S002461070602312X.  Google Scholar

[10]

F. Hirsch, Opérateurs carré du champ, in Sèminaire Bourbaki, 1978,167-182. doi: 10.1007/BFb0070761.  Google Scholar

[11]

A. Johansson, A. Öberg and M. Pollicott, Ergodic theory of Kusuoka's measures, J. Fractal Geom., 4 (2017), 185-214.  doi: 10.4171/JFG/49.  Google Scholar

[12]

N. Kajino, Analysis and geometry of the measurable Riemannian structure on the Sierpinski gasket, in Contemporary Math., Amer. Math. Soc., Providence, RI, 600 2013. doi: 10.1090/conm/600/11932.  Google Scholar

[13]

J. Kigami, Analysis on Fractals, Cambridge tracts in Math., Cambridge Univ. Press, Cambridge, 143 2001. doi: 10.1017/CBO9780511470943.  Google Scholar

[14]

P. Koskela and Y. Zhou, Geometry and analysis of Dirichlet forms, Adv. Math., 231 (2012), 2755-2801.  doi: 10.1016/j.aim.2012.08.004.  Google Scholar

[15]

S. Kusuoka, A diffusion process on a fractal, in Probabilistic methods in Mathematical Physics, (eds. K. Ito and N. Ikeda) Academic Press, Boston, MA, 1987,251-274.  Google Scholar

[16]

S. Kusuoka, Dirichlet forms on fractals and products of random matrices, Publ. Res. Inst. Math. Sci., 25 (1989), 659-680.  doi: 10.2977/prims/1195173187.  Google Scholar

[17]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[18]

I. D. Morris, Ergodic properties of matrix equilibrium state, Ergodic Theory and Dyn. Sys., 38 (2018), 2295-2320.  doi: 10.1017/etds.2016.117.  Google Scholar

[19]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Functional Analysis, 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093.  Google Scholar

[20]

U. Mosco, Variational fractals, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 683-712. http://www.numdam.org/item/?id=ASNSP_1997_4_25_3-4_683_0.  Google Scholar

[21]

R. Peirone, A p. c. f. self similar fractal with no self similar energy, J. Fractal Geom., 6 (2019), 393-404.  doi: 10.4171/JFG/82.  Google Scholar

[22]

R. Peirone, Existence of self-similar energies on finitely ramified fractals, J. Anal. Math., 123 (2014), 35-94.  doi: 10.1007/s11854-014-013-x.  Google Scholar

[23]

R. Peirone, Convergence of Dirichlet forms on fractals, in Topics on Concentration Phenomena and Problems with Multiple Scales, Lect. Notes Unione Mat. Ital. 2, Springer, Berlin, 2006,139-188. doi: 10.1007/978-3-540-36546-4_3.  Google Scholar

[24]

W. Perry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Asterisque, 187-188, 1990.  Google Scholar

[25]

M. Piraino, The weak Bernoulli property for matrix equilibrium states, Ergodic Theory Dynam. Systems, 40 (2020), 2219-2238. doi: 10.1017/etds.2018.129.  Google Scholar

[26]

W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974.  Google Scholar

[27]

A. Teplyaev, Harmonic coordinates on fractals with finitely ramified cell structure, Canad. J. Math., 60 (2008), 457-480.  doi: 10.4153/CJM-2008-022-3.  Google Scholar

[28]

M. Viana, Stochastic Analysis of Deterministic Systems, Mimeographed Notes. Google Scholar

Figure 1.  The first stage of the harmonic gasket
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