Observable Lyapunov irregular sets for planar piecewise expanding maps
Copyright policy )Observable Lyapunov irregular sets for planar piecewise expanding maps
For any integer $r$ with $1\leq r<\infty$, we present a one-parameter family $F_σ$ $(0<σ<1)$ of 2-dimensional piecewise $\mathcal C^r$ expanding maps such that each $F_σ$ has an observable (i.e. Lebesgue positive) Lyapunov irregular set. These maps are obtained by modifying the piecewise expanding map given in Tsujii (2000). In strong contrast to it, we also show that any Lyapunov irregular set of any 2-dimensional piecewise real analytic expanding map is not observable. This is based on the spectral analysis of piecewise expanding maps in Buzzi (2000).
16 pages, 1 figure
- Tokyo Metropolitan University Japan
- Kyushu University Japan
- Tokai University Japan
Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.), Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc., 37C05, 37C30, 37C40, Dynamical Systems (math.DS), Lasota-Yorke inequality, piecewise expanding maps, Lyapunov irregular sets, Dynamical systems involving smooth mappings and diffeomorphisms, FOS: Mathematics, Mathematics - Dynamical Systems, Birkhoff irregular sets, Smooth ergodic theory, invariant measures for smooth dynamical systems, Lyapunov exponent
Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.), Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc., 37C05, 37C30, 37C40, Dynamical Systems (math.DS), Lasota-Yorke inequality, piecewise expanding maps, Lyapunov irregular sets, Dynamical systems involving smooth mappings and diffeomorphisms, FOS: Mathematics, Mathematics - Dynamical Systems, Birkhoff irregular sets, Smooth ergodic theory, invariant measures for smooth dynamical systems, Lyapunov exponent
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