In 1930 Hopf received a fellowship from the Rockefeller Foundation to study classical mechanics with Birkhoff at Harvard in the United States. He arrived Cambridge, Massachusetts in October of 1930 but his official affiliation was not the Harvard Mathematics Department but, instead, the Harvard College Observatory. While in the Harvard College Observatory he worked on many mathematical and astronomical subjects including topology and ergodic theory. In particular he studied the theory of measure and invariant integrals in ergodic theory and his paper On time average theorem in dynamics which appeared in the Proceedings of the National Academy of Sciences is considered by many as the first readable paper in modern ergodic theory. Another important contribution from this period was the Wiener-Hopf equations, which he developed in collaboration with Norbert Wiener from the Massachusetts Institute of Technology. By 1960, a discrete version of these equations was being extensively used in electrical engineering and geophysics, their use continuing until the present day. Other work which he undertook during this period was on stellar atmospheres and on elliptic partial differential equations.
On 14 December 1931, with the help of Norbert Wiener, Hopf joined the Department of Mathematics of the Massachusetts Institute of Technology accepting the position of Assistant Professor. Initially he had a three years contract but this was subsequently extended to four years (1931 to 1936). While at MIT, Hopf did much of his work on ergodic theory which he published in papers such as Complete Transitivity and the Ergodic Principle (1932), Proof of Gibbs Hypothesis on Statistical Equilibrium (1932) and On Causality, Statistics and Probability (1934). In this 1934 paper Hopf discussed the method of arbitrary functions as a foundation for probability and many related concepts. Using these concepts Hopf was able to give a unified presentation of many results in ergodic theory that he and others had found since 1931. He also published a book Mathematical problems of radiative equilibrium in 1934 which was reprinted in 1964. In addition of being an outstanding mathematician, Hopf had the ability to illuminate the most complex subjects for his colleagues and even for non specialists. Because of this talent many discoveries and demonstrations of other mathematicians became easier to understand when described by Hopf.
In 1936, at the end of the MIT contract, Hopf received an offer of full professorship in the University of Leipzig. As a result of this Hopf, with his wife Ilse, returned to Germany which, by this time, was already being ruled by the Nazi party. In Leipzig Hopf undertook research on quantic mechanics (1937), Geodesics on manifolds of negative curvature (1939), Statistik der geod (1939) and on the influence of curvature of a closed Riemannian manifold on its topology (1941).
One important event from this period was the publication of the book Ergodentheorie Ⓣ (1937), most of which was written when Hopf was still at the Massachusetts Institute of Technology. In that book containing only 81 pages, Hopf made a precise and elegant summary of ergodic theory. In 1940 Hopf was on the list of the invited lecturers to the International Congress of Mathematicians to be held in Cambridge, Massachusetts. Because of the start of World War II, however, this Congress was cancelled.
In 1942 Hopf was drafted to work in the German Aeronautical Institute. In 1944, one year before the end of World War II, Hopf was appointed to a professorship at the University of Munich. He held this post until 1947 by which time he had returned to the United States, where he presented the definitive solution of Hurewicz's problem. On 22 February 1949 Hopf became a US citizen. He joined Indiana University as a Professor in 1949, a position he held until he retired in 1972. In 1962 he was made Research Professor of Mathematics, staying in that position until his death.
An important publication from this period was An inequality for positive linear integral operators (1963) which appeared in the Journal for Mathematics and Mechanics. This paper is concerned with some extensions of Jentzsch's theorem on the existence of a positive eigenfunction for a positive integral operator.
In 1971 Hopf was the American Mathematical Society Gibbs Lecturer. Coming out of this lecture was a paper Ergodic theory and the geodesic flow on surfaces of constant negative curvature which he published in the Bulletin of the American Mathematical Society. Hopf wrote in the introduction to that paper:-
Famous investigations on the theory of surfaces of constant negative curvature have been carried out around the turn of the century by F Klein and H Poincaré in connection with complex function theory. The theory of the geodesics in the large on such surfaces was developed later in the famous memoirs by P Koebe. This theory is purely topological. The measure-theoretical point of view became dominant in the later thirties after the advent of ergodic theory, and the papers of G A Hedlund and E Hopf on the ergodic character of the geodesic flow came into being. The present paper is an elaboration of the author's Gibbs lecture of this year  and at the same time of the author's paper of 1939 on the subject, at least of its part concerning constant negative curvature.Hopf was never forgiven by many people for his moving to Germany in 1936, where the Nazi party was already in power. As a result most of his work to ergodic theory and topology was neglected or even attributed to others in the years following the end of World War II. An example of this was the dropping of Hopf's name from the discrete version of the so called Wiener-Hopf equations, which are currently referred to as "Wiener filter".
In  Icha summarises Hopf's mathematical achievements:-
His interests and principal achievements were in the fields of partial and ordinary differential equations, calculus of variations, ergodic theory, topological dynamics, integral equations, differential geometry, complex function theory and functional analysis. Hopf's work is also of the greatest importance to the hydrodynamics, theory of turbulence and radiative transfer theory.
Article by: J J O'Connor and E F Robertson based on a biography submitted by Osvaldo de Oliveira Duarte which in turn made substantial use of .