DCDS
The conditional variational principle for maps with the pseudo-orbit tracing property
Zheng Yin Ercai Chen
Discrete & Continuous Dynamical Systems - A 2019, 39(1): 463-481 doi: 10.3934/dcds.2019019
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
$\mathscr M_{\rm inv}(X,f)$
.
keywords: Topological entropy conditional variational principle

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