Generic properties of invariant measures for simple piecewise monotonic transformations

1987 ◽  
Vol 59 (1) ◽  
pp. 64-80 ◽  
Author(s):  
Franz Hofbauer
Keyword(s):  
Invariant Measures ◽  
Generic Properties ◽  
Piecewise Monotonic

Local dimension for piecewise monotonic maps on the interval

1995 ◽  
Vol 15 (6) ◽  
pp. 1119-1142 ◽  
Author(s):  
Franz Hofbauer
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measures ◽  
Positive Characteristic ◽  
Characteristic Exponent ◽  
Local Dimension ◽  
Pressure Function ◽  
Piecewise Monotonic ◽  
Almost Everywhere ◽  
Conformal Measures

AbstractThe local dimension of invariant and conformal measures for piecewise monotonic transformations on the interval is considered. For ergodic invariant measures m with positive characteristic exponent χm we show that the local dimension exists almost everywhere and equals hm/χm For certain conformal measures we show a relation between a pressure function and the Hausdorff dimension of sets, on which the local dimension is constant.

CONDITIONALLY INVARIANT MEASURES FOR POSITION DEPENDENT RANDOM MAPS OF AN INTERVAL WITH HOLES

2006 ◽  
Vol 16 (02) ◽  
pp. 437-444
Author(s):  
WAEL BAHSOUN ◽  
PAWEŁ GÓRA
Keyword(s):  
Invariant Measures ◽  
Unit Interval ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Piecewise Monotonic ◽  
Laws Of Motion

We study position dependent random maps on the unit interval with holes where the possible laws of motion are piecewise monotonic transformations. The main result of this note is proving the existence of absolutely continuous conditionally invariant measures.

scholarly journals The structure of piecewise monotonic transformations

1981 ◽  
Vol 1 (2) ◽  
pp. 159-178 ◽  
Author(s):  
Franz Hofbauer
Keyword(s):  
Symbolic Dynamics ◽  
Invariant Measures ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Piecewise Monotonic ◽  
Piecewise Continuous ◽  
Absolutely Continuous Invariant ◽  
Nonwandering Set

AbstractTransformations on [0, 1] which are piecewise monotonic and piecewise continuous are considered. Using symbolic dynamics, the structure of their nonwandering set is determined. This is then used to prove results about maximal and absolutely continuous invariant measures.

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