 Cylinder averages of an invariant density

Let a uniformly expanding interval map have cylinders ${\Delta }_w$ whose maximum length at level $n$ is at most $C{\lambda }^{-n}$, with $\lambda >1$. Suppose an invariant probability density $h$ is Lipschitz continuous with constant $L$ and satisfies $h\ge h_{\ast }>0$. Its cylinder probabilities are $\nu ([w])={\int }_{{\Delta }_w}hdx$. Define the piecewise constant logarithmic averageThe average of $h$ over ${\Delta }_w$ differs from $h(x)$ by at most $L\mathrm{diam}({\Delta }_w)$. Since the logarithm has derivative at most $1/h_{\ast }$ on this range,Thus the cylinder data recover the logarithm of the density uniformly at a geometric rate.

Conversely, take consistent positive cylinder probabilities and suppose these $g_n$ converge uniformly to a finite function $g$. Put $h=e^g$. If $\Vert g_n-g{\Vert }_{\infty }\le {\delta }_n\to 0$, integrating over every level-$n$ cylinder compares its $h$-mass with its assigned probability within factors $e^{\pm {\delta }_n}$. Subdivide any fixed cylinder into these finer cylinders and sum: consistency gives the same multiplicative bounds for the fixed cylinder. Letting $n\to \infty$ proves ${\int }_{{\Delta }_w}hdx=\nu ([w])$.

Shrinking cylinders determine the Borel probability measure, so the geometric image of the symbolic law is exactly $h(x)dx$. Use a consistent convention at branch endpoints; bounded limiting cylinder densities imply zero mass at individual endpoints. If the symbolic law is shift invariant, the coding relation transfers that invariance to the interval map. This converse is much stronger than inspecting a density ratio along just one itinerary.