2 results
Expansion properties of double standard maps
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 8 / August 2023
- Published online by Cambridge University Press:
- 05 July 2022, pp. 2549-2588
- Print publication:
- August 2023
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For the family of double standard maps
$f_{a,b}=2x+a+({b}/{\pi }) \sin 2\pi x \pmod {1}$ we investigate the structure of the space of parameters a when 
$b=1$ and when 
$b\in [0,1)$. In the first case the maps have a critical point, but for a set of parameters 
$E_1$ of positive Lebesgue measure there is an invariant absolutely continuous measure for 
$f_{a,1}$. In the second case there is an open non-empty set 
$E_b$ of parameters for which the map 
$f_{a,b}$ is expanding. We show that as 
$b\nearrow 1$, the set 
$E_b$ accumulates on many points of 
$E_1$ in a regular way from the measure point of view.
Invariant densities for random systems of the interval
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 42 / Issue 1 / January 2022
- Published online by Cambridge University Press:
- 29 December 2020, pp. 141-179
- Print publication:
- January 2022
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For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. If the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random
$\beta $
-transformations and random Lüroth maps with a hole.
