1988/01/01 by Ethan M. Coven, Ittai Kan, James A. Yorke · 6 citations
Mathematics · Physics and Astronomy · Computer Science · #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Cellular Automata and Applications
paper · doi:10.1090/s0002-9947-1988-0946440-2
We study the family of tent maps—continuous, unimodal, piecewise linear maps of the interval with slopes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="plus-or-minus s"> <mml:semantics> <mml:mrow> <mml:mo> ± </mml:mo> <mml:mi>s</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">± s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartRoot 2 EndRoot less-than-or-slanted-equals s less-than-or-slanted-equals 2"> <mml:semantics> <mml:mrow> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:mo> ⩽ </mml:mo> <mml:mi>s</mml:mi> <mml:mo> ⩽ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">√ 2 \leqslant s \leqslant 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that tent maps have the shadowing property (every pseudo-orbit can be approximated by an actual orbit) for almost all parameters <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , although they fail to have the shadowing property for an uncountable, dense set of parameters. We also show that for any tent map, every pseudo-orbit can be approximated by an actual orbit of a tent map with a perhaps slightly larger slope.