April  2007, 17(2): 271-280. doi: 10.3934/dcds.2007.17.271

Sub-actions and maximizing measures for one-dimensional transformations with a critical point


Instituto de Matemática, UFRGS, Av. Bento Gon¸calves, 9500 – 91509-900, Porto Alegre, RS, Brazil

Received  December 2005 Revised  September 2006 Published  November 2006

We consider the set $\F$ of the $C^1$-maps $f:\S^1 \to \S^1$ which are of degree 2, uniformly expanding except for a small interval and such that the origin is a fixed critical point. Fixing a $f \in \F$, we show that, for a generic $\alpha$-Hölder potential $A:\S^1 \to \RR$, the $f$-invariant probability measure that maximizes the action given by the integral of $A$ is unique and uniquely ergodic on its support. Furthermore, restricting $f$ to a suitable subset of $\S^1$, we also show that this measure is supported on periodic orbits. Our main tool is the existence of a $\alpha$-Hölder sub-action function for each fixed $f \in \F$ and each fixed potential $A$. We also show that these results can be applied to the special potential $A=\log|f'|$.
Citation: Flávia M. Branco. Sub-actions and maximizing measures for one-dimensional transformations with a critical point. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 271-280. doi: 10.3934/dcds.2007.17.271

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