# American Institute of Mathematical Sciences

January  2019, 39(1): 463-481. doi: 10.3934/dcds.2019019

## The conditional variational principle for maps with the pseudo-orbit tracing property

 1 School of Mathematical Sciences, Anhui University, Hefei 230601, Anhui, China 2 School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, Jiangsu, China 3 Center of Nonlinear Science, Nanjing University, Nanjing 210093, Jiangsu, China

* Corresponding author.

Received  March 2018 Published  October 2018

Let
 $(X,d,f)$
be a topological dynamical system, where
 $(X,d)$
is a compact metric space and
 $f:X \to X$
is a continuous map. We define
 $n$
-ordered empirical measure of
 $x \in X$
by
 $\mathscr{E}_n(x) = \frac{1}{n}\sum\limits_{i = 0}^{n-1}δ_{f^ix},$
where
 $δ_y$
is the Dirac mass at
 $y$
. Denote by
 $V(x)$
the set of limit measures of the sequence of measures
 $\mathscr{E}_n(x)$
. In this paper, we obtain conditional variational principles for the topological entropy of
 $\Delta_{sub}(I): = \left\{ {x \in X:V(x)\subset I} \right\},$
and
 $\Delta_{cap}(I): = \left\{ {x \in X:V(x)\cap I≠\emptyset } \right\}.$
in a dynamical system with the pseudo-orbit tracing property, where
 $I$
is a certain subset of
 $\mathscr M_{\rm inv}(X,f)$
.
Citation: Zheng Yin, Ercai Chen. The conditional variational principle for maps with the pseudo-orbit tracing property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 463-481. doi: 10.3934/dcds.2019019
##### References:
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##### References:
 [1] N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North-Holland, Amsterdam, 1994. Google Scholar [2] J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696. doi: 10.1017/S0143385702001165. Google Scholar [3] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar [4] L. Chen and S. Li, Shadowing property for inverse limit spaces, Proc. Amer. Math. Soc., 115 (1992), 573-580. doi: 10.1090/S0002-9939-1992-1097338-X. Google Scholar [5] E. Coven, I. Kan and J. Yorke, Pseudo-orbit shadowing in the family of tents maps, Trans. Amer. Math. Soc., 308 (1988), 227-241. doi: 10.1090/S0002-9947-1988-0946440-2. Google Scholar [6] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, vol. 527, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/BFb0082364. Google Scholar [7] Y. Dong, P. Oprocha and X. Tian, On the irregular points for systems with the shadowing property, Ergodic Theory Dynam. Systems, 38 (2018), 2108-2131. doi: 10.1017/etds.2016.126. Google Scholar [8] T. Downarowicz, Survey of odometers and Toeplitz flows, in Algebraic and Topological Dynamics, Contemp. Math. 385, Amer. Math. Soc., Providence, RI, (2005), 7-37. doi: 10.1090/conm/385. Google Scholar [9] A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphism, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. doi: 10.1007/BF02684777. Google Scholar [10] D. Kwietniak, M. Lacka and P. Oprocha, A panorama of specification-like properties and their consequences, in Dynamics and Numbers, Contemp. Math. 669, Amer. Math. Soc., Providence, RI, (2016), 155-186. doi: 10.1090/conm/669. Google Scholar [11] J. Li and P. Oprocha, Properties of invariant measures in dynamical systems with shadowing property, Ergodic Theory Dynam. Systems, 38 (2018), 2257-2294. doi: 10.1017/etds.2016.125. Google Scholar [12] V. Mijović and L. Olsen, Dynamical multifractal zeta-functions and fine multifractal spectra of graph-directed self-conformal constructions, Ergodic Theory Dynam. Systems, 36 (2016), 1922-1971. doi: 10.1017/etds.2014.140. Google Scholar [13] L. Olsen, Divergence points of deformed empirical measures, Math. Res. Lett., 9 (2002), 701-713. doi: 10.4310/MRL.2002.v9.n6.a1. Google Scholar [14] L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591-1649. doi: 10.1016/j.matpur.2003.09.007. Google Scholar [15] L. Olsen and S. Winter, Normal and non-normal points of self-similar sets and divergence points of self-similar measures, J. Lond. Math. Soc., 67 (2003), 103-122. doi: 10.1112/S0024610702003630. Google Scholar [16] Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, University of Chicago Press, Chicago, 1997. doi: 10.7208/chicago/9780226662237.001.0001. Google Scholar [17] C. Pfister and W. Sullivan, Large deviations estimates for dynamical systems without the specification property. Applications to the $\beta$-shifts, Nonlinearity, 18 (2005), 237-261. doi: 10.1088/0951-7715/18/1/013. Google Scholar [18] C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929-956. doi: 10.1017/S0143385706000824. Google Scholar [19] F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergodic Theory Dynam. Systems, 23 (2003), 317-348. doi: 10.1017/S0143385702000913. Google Scholar [20] X. Tian and P. Varandas, Topological entropy of level sets of empirical measures for nonuniformly expanding maps, Discrete Continuous Dynam. Systems - A, 37 (2017), 5407-5431. doi: 10.3934/dcds.2017235. Google Scholar [21] X. Zhou and E. Chen, Multifractal analysis for the historic set in topological dynamical systems, Nonlinearity, 26 (2013), 1975-1997. doi: 10.1088/0951-7715/26/7/1975. Google Scholar
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