Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers

Geometrical and Measure-Theoretic Structures of Maps With a Mostly Expanding Center

“…For diffeomorphisms in P HA 1 us (M ), based on the uncertain behavior on centers (E cu and E cs ), we introduce a new definition of the skeleton which is different from well-known definitions from [33,17,46] in recent years.…”

“…The Gibbs u-state is an important candidate for the physical measures. With the tool of Gibbs u-states, in recent decades much progress has been made on physical measures for systems with non-uniform hyperbolicity such as [8,17,2,4,3,43,6,33].…”

“…Mi and Yongluo. Cao also studied finiteness, the upper-semicontinuity of number, continuity, statistically stability of physical measures for the diffeomorphisms with mostly expanding and mostly contracting centers in [33]. Since characteristics (number,support, basin and variation) of the ergodic measures of maximal u-entropy and physical measures are similar for "weak"-mostly contracting center.…”

“…Our starting point was the observation that the structure of physical measures and ergodic measures of maximal partial entropy of type c-cu-states are closely related to invariant manifolds of hyperbolic periodic points by skeleton. Considering the absence of absolute continuity along stable manifolds and the equivalence between decomposition measures along unstable manifolds and Lebesgue measures along unstable manifolds about ergodic measures of maximal partial entropy of type c-custates, we propose a new definition of the skeleton which is different from those in both [17] and [33]. The definitions there will be upgraded to definition of strong-skeleton in this article.…”

“…Finally, by the qualitative construction and our results, new example we give shows that for such diffeomorphisms, they can have any given number ergodic measures of maximal u-entropy of type c-cu-states in the sense of support, there is uncountable ergodic measure of maximal u-entropy of type c-cu-states whose support is the same, and union of the basins of ergodic measure of maximal u-entropy of type c-cu-states is dense in whole space, number of c-cu-state is upper semicontinuous in the sense of support on the certain environment. Such example of C 1+ can also be applied to construction of more than one physical measures of diffeomorphisms with mostly expanding center and mostly contracting center in [33], and then used to study finiteness, the upper-semicontinuity of number, the statistically stability and continuity of physical measures. * 2020 Mathematics Subject Classification.…”