Hostname: page-component-7bb8b95d7b-dtkg6 Total loading time: 0 Render date: 2024-09-24T00:33:00.921Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamical multifractal zeta-functions and fine multifractal spectra of graph-directed self-conformal constructions

Published online by Cambridge University Press:  11 February 2015

V. MIJOVIĆ  and
L. OLSEN
V. MIJOVIĆ
Affiliation:
Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland email vm27@st-and.ac.uk, lo@st-and.ac.uk
L. OLSEN
Affiliation:
Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland email vm27@st-and.ac.uk, lo@st-and.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce multifractal pressure and dynamical multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the fine multifractal spectra of graph-directed self-conformal measures and the fine multifractal spectra of ergodic Birkhoff averages of continuous functions on graph-directed self-conformal sets.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 6 , September 2016 , pp. 1922 - 1971
Copyright
© Cambridge University Press, 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Artin, M. and Mazur, B.. On periodic points. Ann. of Math. (2) 21 (1965), 82–99.Google Scholar
Arbeiter, M. and Patzschke, N.. Random self-similar multifractals. Math. Nachr. 181 (1996), 5–42.CrossRefGoogle Scholar
Baker, S. P.. A multifractal zeta function for Gibbs measures supported on cookie-cutter sets. Nonlinearity 26 (2013), 1125–1142.Google Scholar
Baladi, V.. Dynamical zeta functions and generalized Fredholm determinants. Proc. XIth International Congress of Mathematical Physics (Paris, 1994). International Press, Cambridge, MA, 1995, pp. 249–260.Google Scholar
Baladi, V.. A brief introduction to dynamical zeta functions. Classical Nonintegrability, Quantum Chaos (DIMV-Seminar 27). Eds. Knauf, A. and Sinai, Ya. G.. Birkhäuser, Boston, 1997, pp. 3–20.Google Scholar
Barral, J. and Mensi, M.. Multifractal analysis of Birkhoff averages on ‘self-affine’ symbolic spaces. Nonlinearity 21 (2008), 2409–2425.Google Scholar
Barreira, L.. Thermodynamic Formalism and Applications to Dimension Theory. Birkhäuser, Boston, MA, 2011.Google Scholar
Bowen, R.. Hausdorff dimension of quasi-circles. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 11–26.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470) . Springer, Berlin, 1975.Google Scholar
Cole, J.. The geometry of graph directed self-conformal multifractals. PhD Thesis, University of St. Andrews, 1998.Google Scholar
Cole, J.. Relative multifractal analysis. Chaos Solitons Fractals 11 (2000), 2233–2250.Google Scholar
Conway, J.. A Course in Functional Analysis, 2nd edn. Springer, New York, 1990.Google Scholar
Dembo, A. and Zeitouni, O.. Large Deviations Techniques and Applications. Jones and Bartlett Publishers, Boston, MA, 1993.Google Scholar
Edgar, G. A. and Mauldin, R. D.. Multifractal decompositions of digraph recursive fractals. Proc. Lond. Math. Soc. 65 (1992), 604–628.CrossRefGoogle Scholar
Ellis, R.. Entropy, Large Deviations, and Statistical Mechanics (Grundlehren der Mathematischen Wissenschaften, 271) . Springer, New York, 1985.Google Scholar
Falconer, K. J.. Techniques in Fractal Geometry. Wiley, New York, 1997.Google Scholar
Fan, A.-H. and Feng, D.-J.. On the distribution of long-term time averages on symbolic space. J. Stat. Phys. 99 (2000), 813–856.Google Scholar
Fan, A.-H., Feng, D.-J. and Wu, J.. Recurrence, dimension and entropy. J. Lond. Math. Soc. (2) 64 (2001), 229–244.Google Scholar
Feng, D.-J., Lau, K.-S. and Wu, J.. Ergodic limits on the conformal repellers. Adv. Math. 169 (2002), 58–91.Google Scholar
Hutchinson, J.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
Jiang, D.-Q., Qian, M. and Qian, M.-P.. Mathematical Theory of Nonequilibrium Steady States. On the Frontier of Probability and Dynamical Systems (Lecture Notes in Mathematics, 1833) . Springer, Berlin, 2004.Google Scholar
Lapidus, M., Lévy-Véhel, J. and Rock, J.. Fractal strings and multifractal zeta functions. Lett. Math. Phys. 88 (2009), 101–129.Google Scholar
Lapidus, M. and Rock, J.. Towards zeta functions and complex dimensions of multifractals. Complex Var. Elliptic Equ. 54 (2009), 545–559.Google Scholar
Lapidus, M. L. and van Frankenhuysen, M.. Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions. Birkhäuser, Boston, MA, 2000.Google Scholar
Lapidus, M. L. and van Frankenhuysen, M.. Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (Springer Monographs in Mathematics) . Springer, New York, 2006.Google Scholar
Levy-Vehel, J. and Mendivil, F.. Multifractal and higher-dimensional zeta functions. Nonlinearity 24 (2011), 259–276.Google Scholar
Mauldin, R. D. and Urbanski, M.. Graph Directed Markov Systems. Geometry and Dynamics of Limit Sets (Cambridge Tracts in Mathematics, 148) . Cambridge University Press, Cambridge, 2003.CrossRefGoogle Scholar
Mijović, V. and Olsen, L.. Multifractal spectra and multifractal zeta-functions. Preprint, 2013.Google Scholar
Olsen, L.. Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. J. Math. Pures Appl. (9) 82 (2003), 1591–1649.Google Scholar
Olsen, L.. Multifractal tubes. Preprint, 2013.Google Scholar
Olsen, L.. Multifractal tubes: Multifractal zeta-functions, multifractal Steiner formulas and explicit formulas. Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I (Contemporary Mathematics, 600) . Eds. D. Carfi, M. Lapidus, E. Pearse et al. American Mathematical Society, Providence, RI, 2013.Google Scholar
Olsen, L.. Dynamical multifractal zeta-functions, multifractal pressure and fine multifractal spectra. J. Anal. Math., to appear.Google Scholar
Olivier, E.. Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for g-measures. Nonlinearity 24 (1999), 1571–1585.CrossRefGoogle Scholar
Olsen, L. and Winter, S.. Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: non-linearity, divergence points and Banach space valued spectra. Bull. Sci. Math. 131 (2007), 518–558.Google Scholar
Orey, S. and Pelikan, S.. Large deviation principles for stationary processes. Ann. Probab. 16 (1988), 1481–1495.Google Scholar
Orey, S. and Pelikan, S.. Deviations of trajectory averages and the defect in Pesin’s formula for Anosov diffeomorphisms. Trans. Amer. Math. Soc. 315 (1989), 741–753.Google Scholar
Patzschke, N.. Self-conformal multifractal measures. Adv. in Appl. Math. 19 (1997), 486–513.Google Scholar
Parry, W. and Pollicott, M.. An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann. of Math. (2) 118 (1983), 573–591.Google Scholar
Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990).Google Scholar
Pesin, Y.. Dimension Theory in Dynamical Systems (Contemporary Views and Applications) . The University of Chicago Press, Chicago, IL, 1997.Google Scholar
Rockafellar, R. T.. Convex Analysis. Princeton University Press, Princeton, NJ, 1970.Google Scholar
Ruelle, D.. Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics (Encyclopedia of Mathematics and its Applications, 5) . Addison-Wesley, Reading, MA, 1978.Google Scholar
Ruelle, D.. Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval (CRM Monograph Series, 4) . American Mathematical Society, Providence, RI, 1994.Google Scholar
Varadhan, S. R. S.. Asymptotic probabilities and differential equations. Comm. Pure Appl. Math. 19 (1966), 261–286.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) . Springer, New York, 1982.Google Scholar