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### Open Access Journals

DCDS

This paper investigates the exact rate of convergence in Ulam's method: a
well-known discretization scheme for approximating the invariant density of an absolutely
continuous invariant probability measure for piecewise expanding interval maps.
It is shown by example that the rate is no better than
$O(\frac{\log n}{n})$, where $n$ is the number
of cells in the discretization. The result is in agreement with upper estimates previously
established in a number of general settings, and shows that the conjectured rate of $O(\frac{1}{n})$ cannot be obtained, even for extremely regular maps.

DCDS

Let $T$ be a piecewise expanding interval map and $T_H$ be an abstract perturbation of $T$ into an interval map with a hole. Given a number

*ℓ*, 0 <*ℓ*< l, we compute an upper-bound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than -ln(1-*ℓ*). The two main ingredients of our approach are Ulam's method and an abstract perturbation result of Keller and Liverani.
DCDS

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
optimization problems on $L^p$, $1 \leq p < \infty$ with a convex objective
and finite moment constraints.
Provided $T$ admits an invariant density in the appropriate $L^p$ space,
weak convergence of the sequence of optimal solutions is
observed; norm
convergence can be obtained when the objective is a Kadec functional.
No regularity or expansiveness assumptions on $T$ need to be
made, and the method applies to maps on multidimensional
domains.
Objectives leading to norm convergence include Entropy,
'Energy' and 'Positively Constrained Energy'.

Explicit solutions for the finite moment problems in the case of the 'Energy' functional are derived using duality - the optimality condition is then a linear algebra problem. Strong duality is obtained even though the dual functional may not be coercive and the set of moment test functions is not assumed to be pseudo-Haar. Finally, some numerical studies are presented for the case of moment test functions derived from a finite partition of the dynamical phase space and the results are compared with Ulam's method.

Explicit solutions for the finite moment problems in the case of the 'Energy' functional are derived using duality - the optimality condition is then a linear algebra problem. Strong duality is obtained even though the dual functional may not be coercive and the set of moment test functions is not assumed to be pseudo-Haar. Finally, some numerical studies are presented for the case of moment test functions derived from a finite partition of the dynamical phase space and the results are compared with Ulam's method.

keywords:
Frobenius-Perron operator
,
entropy
,
normal convex integrand
,
duality.
,
Kadec
,
moment problem
,
Invariant measure

DCDS

We give a deterministic representation for position dependent random
maps and describe the structure of its set of invariant
measures. Our construction generalizes the skew product
representation of random maps with constant probabilities.
In particular, we establish one-to-one correspondence
between eigenfunctions corresponding
to eigenvalues of unit modulus for the Frobenius-Perron (transfer) operator
of the random map and for those of the skew. An immediate consequence
is one-to-one correspondence between
absolutely continuous invariant measures (acims) for the
position dependent random map and acims for its deterministic
representation.

## Year of publication

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