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Keywords:
Hausdorff dimension; packing dimension; Lorenz transformation; ergodic measure
Hausdorff dimension; packing dimension; Lorenz transformation; ergodic measure
Summary:
We extend the notions of Hausdorff and packing dimension introducing weights in their definition. These dimensions are computed for ergodic invariant probability measures of two-dimensional Lorenz transformations, which are transformations of the type occuring as first return maps to a certain cross section for the Lorenz differential equation. We give a formula of the dimensions of such measures in terms of entropy and Lyapunov exponents. This is done for two choices of the weights using the recurrence time of a set and equilibrium states respectively.
References:
[1] Afraimovich V.: Pesins dimension for Poincaré recurrences. Chaos 7 (1997), 12--20. DOI 10.1063/1.166237 | MR 1439803
[2] Afraimovich V., Chazottes J.-R., Saussol B.: Pointwise dimension for Poincaré recurrences associated with maps and special flows. Discrete Contin. Dyn. Syst. A 9 (2003), 263--280. MR 1952373
[3] Billingsley P.: Ergodic Theory and Information. Krieger New York (1978). MR 0524567
[4] Bowen R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, 470 Springer Berlin-Heidelberg-New York (1975). MR 2423393 | Zbl 0308.28010
[5] Cutler C.: The Hausdorff dimension distribution of finite measures in Euclidian space. Canad. J. Math. 38 (1986), 1459--1484. DOI 10.4153/CJM-1986-071-9 | MR 0873419
[6] Hofbauer F.: On intrinsic ergodicity of piecewise monotone transformations with positive entropy. Israel J. Math. 34 (1979), 213--237. DOI 10.1007/BF02760884 | MR 0570882
[7] Hofbauer F.: On intrinsic ergodicity of piecewise monotone transformations with positive entropy II. Israel J. Math. 38 (1981), 107--115. DOI 10.1007/BF02761854 | MR 0599481
[8] Hofbauer F.: Piecewise invertible dynamical systems. Probab. Theory Relat. Fields 72 (1986), 359--386. DOI 10.1007/BF00334191 | MR 0843500 | Zbl 0578.60069
[9] Hofbauer F.: An inequality for the Lyapunov exponent of an ergodic invariant measure for a piecewise monotonic map on the interval. Lyapunov Exponents (Oberwolfach, 1990), L. Arnold, H. Crauel, J.-P. Eckmann, Eds., Lecture Notes in Mathematics, 1486, Springer, Berlin, 1991, pp. 227--231. MR 1178961
[10] Hofbauer F.: Hausdorff dimension and pressure for piecewise monotonic maps of the interval. J. London Math. Soc. 47 (1993), 142--156. DOI 10.1112/jlms/s2-47.1.142 | MR 1200984 | Zbl 0725.54031
[11] Hofbauer F.: The recurrence dimension for piecewise monotonic maps of the interval. Ann. Scuola Norm. Super. Pisa Cl. Sci. 4 (2005), 439--449. MR 2185864 | Zbl 1170.37316
[12] Hofbauer F., Keller G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982), 119--140. DOI 10.1007/BF01215004 | MR 0656227 | Zbl 0485.28016
[13] Hofbauer F., Keller G.: Equilibrium states for piecewise monotonic transformations. Ergodic Theory Dynam. Systems 2 (1982), 23--43. MR 0684242 | Zbl 0499.28012
[14] Hofbauer F., Raith P.: The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval. Canad. Math. Bull. 35 (1992), 84--98. DOI 10.4153/CMB-1992-013-x | MR 1157469 | Zbl 0701.28005
[15] Hofbauer F., Raith P., Steinberger T.: Multifractal dimensions for invariant subsets of piecewise monotonic interval maps. Fund. Math. 176 (2003), 209--232. DOI 10.4064/fm176-3-2 | MR 1992820 | Zbl 1051.37011
[16] Hofbauer F., Urbański M.: Fractal properties of invariant subsets for piecewise monotonic maps of the interval. Trans. Amer. Math. Soc. 343 (1994), 659--673. DOI 10.1090/S0002-9947-1994-1232188-9 | MR 1232188
[17] Keller G.: Extended bounded variation and application to piecewise monotonic transformations. Probab. Theory Relat. Fields 69 (1985), 461--478. MR 0787608
[18] Ledrappier F.: Principe variationnel et systemes dynamiques symboliques. Probab. Theory Relat. Fields 30 (1974), 185--202. MR 0404584 | Zbl 0276.93004
[19] Mattila P.: Geometry of Sets and Measures in Euclidean space. Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995. MR 1333890
[20] Olsen L.: A multifractal formalism. Adv. Math. 116 (1995), 82--196. DOI 10.1006/aima.1995.1066 | MR 1361481 | Zbl 0841.28012
[21] Pesin Ya.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications. Chicago Lectures in Mathematics University of Chicago Press Chicago (1997). MR 1489237
[22] Rychlik M.: Bounded variation and invariant measures. Studia Math. 76 (1983), 69--80. MR 0728198 | Zbl 0575.28011
[23] Saussol B., Troubetzkoy S., Vaienti S.: Recurrence, dimensions and Lyapunov exponents. J. Statist. Phys. 106 (2002), 623--634. DOI 10.1023/A:1013710422755 | MR 1884547 | Zbl 1138.37300
[24] Saussol B., Troubetzkoy S., Vaienti S.: Recurrence and Lyapunov exponents. Moscow Math. J. 3 (2003), 189--203. MR 1996808 | Zbl 1083.37504
[25] Steinberger T.: Local dimension of ergodic measures for two-dimensional Lorenz transformations. Ergodic Theory Dynam. Systems 20 (2000), 911--923. MR 1764935 | Zbl 0965.37011
[26] Walters P.: Equilibrium states for $\beta$-transformations and related transformations. Math. Z. 159 (1978), 65--88. DOI 10.1007/BF01174569 | MR 0466492 | Zbl 0364.28016
[27] Walters P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics, 79, Springer, New York, 1982. DOI 10.1007/978-1-4612-5775-2 | MR 0648108 | Zbl 0958.28011
[28] Young L.-S.: Dimension, entropy and Lyapunov exponents. Ergodic Theory Dynam. Systems 2 (1982), 109--124. MR 0684248 | Zbl 0523.58024
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