# Numerical Approximation of Conditionally Invariant Measures via Maximum Entropy

@article{Bose2014NumericalAO, title={Numerical Approximation of Conditionally Invariant Measures via Maximum Entropy}, author={Christopher J. Bose and Rua Murray}, journal={arXiv: Dynamical Systems}, year={2014}, pages={81-104} }

It is well known that open dynamical systems can admit an uncountable number of (absolutely continuous) conditionally invariant measures (ACCIMs) for each prescribed escape rate. We propose and illustrate a convex optimisation-based selection scheme (essentially maximum entropy) for gaining numerical access to some of these measures. The work is similar to the maximum entropy (MAXENT) approach for calculating absolutely continuous invariant measures of nonsingular dynamical systems but contains…

## References

SHOWING 1-10 OF 24 REFERENCES

Duality and the Computation of Approximate Invariant Densities for Nonsingular Transformations

- Computer Science, MathematicsSIAM J. Optim.
- 2007

The method of the paper circumvents technical obstructions in the derivation of optimality conditions and yields an unexpected benefit: each finite moment approximation leads to rigorous bounds on the support of all invariant densities for $T$.

Markov extensions and conditionally invariant measures for certain logistic maps with small holes

- MathematicsErgodic Theory and Dynamical Systems
- 2005

We study the family of quadratic maps fa(x) = 1 − ax2 on the interval [−1, 1] with 0 [nle ] a [nle ] 2. When small holes are introduced into the system, we prove the existence of an absolutely…

Existence and convergence properties of physical measures for certain dynamical systems with holes

- Mathematics, PhysicsErgodic Theory and Dynamical Systems
- 2009

Abstract We study two classes of dynamical systems with holes: expanding maps of the interval and Collet–Eckmann maps with singularities. In both cases, we prove that there is a natural absolutely…

Spectral degeneracy and escape dynamics for intermittent maps with a hole

- Mathematics
- 2010

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small…

Minimum 'energy' approximations of invariant measures for nonsingular transformations

- Mathematics
- 2005

We study variational methods for
rigorous approximation of invariant
densities for a nonsingular map $T$ on a Borel measure space.
The general method takes the form of a convergent sequence of …

Efficient computation of topological entropy, pressure, conformal measures, and equilibrium states in one dimension.

- Mathematics, MedicinePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2007

This work describes a fast and accurate method to compute the pressure and equilibrium states for maps of the interval T:[0,1]-->[0, 1] with respect to potentials phi:-->R.Perron-Frobenius operator and recovers the topological entropy by setting phi identical with 0.

Escape rates and conditionally invariant measures

- Mathematics
- 2006

We consider dynamical systems on domains that are not invariant under the dynamics—for example, a system with a hole in the phase space—and raise issues regarding the meaning of escape rates and…

Approximating invariant densities of metastable systems

- MathematicsErgodic Theory and Dynamical Systems
- 2010

Abstract We consider a piecewise smooth expanding map on an interval which has two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability…

Metastability of certain intermittent maps

- Mathematics
- 2012

We study an intermittent map which has exactly two ergodic invariant densities. The densities are supported on two subintervals with a common boundary point. Due to certain perturbations, leakage of…

Lasota–Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set

- Mathematics
- 2003

Abstract Let T :I→I be a Lasota–Yorke map on the interval I, let Y be a nontrivial sub-interval of I and g 0 :I→ R + , be a strictly positive potential which belongs to BV and admits a conformal…