The topological entropy of non-dense orbits and generalized Schmidt games

Abstract

We generalize the notion of Schmidt games to the setting of the general Caratheódory construction. The winning sets for such generalized Schmidt games usually have large corresponding Caratheódory dimensions (e.g., Hausdorff dimension and topological entropy). As an application, we show that for every $C^{1+\unicode[STIX]{x1D703}}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$ satisfying certain technical conditions, the topological entropy of the set of points with non-dense forward orbits is bounded below by the unstable metric entropy (in the sense of Ledrappier–Young) of certain invariant measures. This also gives a unified proof of the fact that the topological entropy of such a set is equal to the topological entropy of $f$, when $f$ is a toral automorphism or the time-one map of a certain non-quasiunipotent homogeneous flow.

Non-dense orbits of systems with the approximate product property

“…For example, see [1-3, 7, 11-13, 20, 22-24]. Similar results have also been established for more general hyperbolic or partially hyperbolic systems [10,15,21,37,39,[41][42][43][44]. Non-dense orbits are also closely related to irregular behaviours.…”

“…Such cases are not covered by our approach, while they are naturally covered by the approaches with the Schmidt games (e.g. [3,13,44]).…”

Unstable entropies and dimension theory of partially hyperbolic systems

“…The general concept of multifractal analysis is to decompose the phase space into subsets of points which have a similar dynamical behavior and to describe the size of these subsets from the geometrical or topological viewpoint. Sets with similar dynamical behavior include the basin set of an invariant measure or general saturated sets [8,30], recurrent and dense sets [9,41], non-dense sets [13,44,46,47], level sets and irregular sets of Birkhoff ergodic average [3-7, 19, 23, 26, 27, 30, 37, 38], level sets and irregular sets of Lyapunov exponents [2,15,28,40], which have been studied a lot by using various measurements such as Hausdorff dimension, topological entropy or pressure, Lebesgue measure and distributional chaos etc. Here the topological entropy used was introduced by Bowen [8] to characterize the dynamical complexity of arbitrary sets which are not necessarily compact nor invariant from the perspective of 'dimensional' nature.…”

Unstable entropies and Dimension Theory of Partially Hyperbolic Systems

“…The general concept of multifractal analysis is to decompose the phase space into subsets of points which have a similar dynamical behavior and to describe the size of these subsets from the geometrical or topological viewpoint. Sets with similar dynamical behavior include the basin set of an invariant measure or general saturated sets [8,30], recurrent and dense sets [42,9], non-dense sets [45,13,47,48], level sets and irregular sets of Birkhoff ergodic average [26,27,3,4,6,5,7,38,30,23,39,19,37], level sets and irregular sets of Lyapunov exponents [2,28,15,41], which have been studied a lot by using various measurements such as Hausdorff dimension, topological entropy or pressure, Lebesgue measure and distributional chaos etc. Here the topological entropy used was introduced by Bowen [8] to characterize the dynamical complexity of arbitrary sets which are not necessarily compact nor invariant from the perspective of "dimensional" nature.…”