6533b82bfe1ef96bd128cd1a

RESEARCH PRODUCT

Robust existence of nonhyperbolic ergodic measures with positive entropy and full support

Christian BonattiLorenzo J. DíazDominik Kwietniak

subject

Transitive relationPure mathematicsHyperbolicityMathematics::Dynamical SystemsDense setContinuous function (set theory)[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS][MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]Scale (descriptive set theory)Dynamical Systems (math.DS)Measure (mathematics)Theoretical Computer SciencePositive entropyMathematics (miscellaneous)FOS: MathematicsErgodic theory37D25 37D35 37D30 28D99Mathematics - Dynamical SystemsMathematicsCriterion

description

We prove that for some manifolds $M$ the set of robustly transitive partially hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre direction contains a $C^1$-open and dense subset of diffeomorphisms with nonhyperbolic measures which are ergodic, fully supported and have positive entropy. To do so, we formulate abstract conditions sufficient for the construction of an ergodic, fully supported measure $\mu$ which has positive entropy and is such that for a continuous function $\phi\colon X\to\mathbb{R}$ the integral $\int\phi\,d\mu$ vanishes. The criterion is an extended version of the control at any scale with a long and sparse tail technique coming from the previous works.

yearjournalcountryeditionlanguage
2021-12-22
https://hal.science/hal-03860928