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:
[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. CovenI. 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. DongP. 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

show all references

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. CovenI. 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. DongP. 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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