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 and Continuous Dynamical Systems, 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.

[2]

J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696.  doi: 10.1017/S0143385702001165.

[3]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

L. Olsen, Divergence points of deformed empirical measures, Math. Res. Lett., 9 (2002), 701-713.  doi: 10.4310/MRL.2002.v9.n6.a1.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North-Holland, Amsterdam, 1994.

[2]

J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696.  doi: 10.1017/S0143385702001165.

[3]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

L. Olsen, Divergence points of deformed empirical measures, Math. Res. Lett., 9 (2002), 701-713.  doi: 10.4310/MRL.2002.v9.n6.a1.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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