Adam Fieldsteel

Equipartition of interval partitions and an application to number theory

We give wider application and simpler proofs of results describing the rate at which the digits of one number theoretic expansion determine those of another The proofs are based on general measuretheoretic covering arguments and not on the dynamics of specic maps

Sets that force recurrence

We characterize those subsets S of the positive integers with the property that, whenever a point x in a dynamical system enters a compact set K along S, K contains a recurrent point. We do the same for uniform recurrence.

Factor orbit equivalence of compact group extensions

LetG be a compact metrizable group. We show that any two ergodic extensions of transformationsT1andT2 by rotations ofG are factor orbit equivalent relative toT1andT2, and the equivalence may be taken to have a certain natural form.

The relative isomorphism theorem for Bernoulli flows

AbstractIn this paper we extend the work of Thouvenot and others on Bernoulli splitting of ergodic transformations to ergodic flows of finite entropy. We prove that ifA is a factor of a flowS, whereS1 is ergodic andA has a Bernoulli complement inS1, thenA has a Bernoulli complement inS. Consequently, Bernoulli splitting for flows is stable under taking intermediate factors and certain

Stability of the weak Pinsker property for flows

\bar d

Dyadic equivalence to completely positive entropy

limits. In addition it follows that the property of isomorphism with a Bernoulli × zero entropy flow is similarly stable.

Stability of m -equivalence to the weak Pinsker property

For a fixed irrational α > 0 we say that probability measure-preserving transformations S and T are α-equivalent if they can be realized as cross-sections in a common flow such that the return time functions on the cross-sections both take values in {1, 1 +α} and have equal integrals. Similarly we call two flows F and G α-equivalent if F has a cross-section S and G has a cross-section T isomorphic to S and again both the return time functions take values in {1, 1 + α} and have equal integrals. The integer kα(S), equal to the least positive such such that exp2πikα-1 belongs to the point spectrum of S, is an invariant of α-equivalence.We obtain a characterization of a-equivalence as a particular type of restricted orbit equivalence and use this to prove that within the class of loosely Bernoulli maps ka(S) together with the entropy h(S) are complete invariants of α-equivalence. There is a corresponding a-equivalence theorem for flows which has as a consequence, for example, that up to an obvious entropy restriction, any weakly mixing cross-section of a loosely Bernoulli flow can also be realized as a cross-section with a {1,1 + α}-valued return time function.For the proof of the α-equivalence theorem we develop a relative Kakutani equivalence theorem for compact group extensions which is of interest in its own right. Finally, an example of Fieldsteel and Rudolph is used to show that in general kα(S) is not a complete invariant of α-equivalence within a given even Kakutani equivalence class.

Restricted orbit changes of ergodic ℤ d -actions to achieve mixing and completely positive entropy

An ergodic flow is said to have the weak Pinsker property if it admits a decreasing sequence of factors whose entropies tend to zero and each of which has a Bernoulli complement. We show that this property is preserved under taking factors and d -limits. In addition, we show that a flow has the weak Pinsker property whenever one ergodic transformation in the flow has this property.